Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, B

Percentage Accurate: 96.6% → 96.6%

Time: 15.9s

Alternatives: 24

Speedup: 1.0×

• 41.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (* x (exp (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return x * exp(((y * (log(z) - t)) + (a * (log((1.0 - z)) - b))));
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = x * exp(((y * (log(z) - t)) + (a * (log((1.0d0 - z)) - b))))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return x * Math.exp(((y * (Math.log(z) - t)) + (a * (Math.log((1.0 - z)) - b))));
}

Python

def code(x, y, z, t, a, b):
	return x * math.exp(((y * (math.log(z) - t)) + (a * (math.log((1.0 - z)) - b))))

Julia

function code(x, y, z, t, a, b)
	return Float64(x * exp(Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = x * exp(((y * (log(z) - t)) + (a * (log((1.0 - z)) - b))));
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(x * N[Exp[N[(N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 96.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (* x (exp (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return x * exp(((y * (log(z) - t)) + (a * (log((1.0 - z)) - b))));
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = x * exp(((y * (log(z) - t)) + (a * (log((1.0d0 - z)) - b))))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return x * Math.exp(((y * (Math.log(z) - t)) + (a * (Math.log((1.0 - z)) - b))));
}

Python

def code(x, y, z, t, a, b):
	return x * math.exp(((y * (math.log(z) - t)) + (a * (math.log((1.0 - z)) - b))))

Julia

function code(x, y, z, t, a, b)
	return Float64(x * exp(Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = x * exp(((y * (log(z) - t)) + (a * (log((1.0 - z)) - b))));
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(x * N[Exp[N[(N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 96.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (* x (exp (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return x * exp(((y * (log(z) - t)) + (a * (log((1.0 - z)) - b))));
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = x * exp(((y * (log(z) - t)) + (a * (log((1.0d0 - z)) - b))))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return x * Math.exp(((y * (Math.log(z) - t)) + (a * (Math.log((1.0 - z)) - b))));
}

Python

def code(x, y, z, t, a, b):
	return x * math.exp(((y * (math.log(z) - t)) + (a * (math.log((1.0 - z)) - b))))

Julia

function code(x, y, z, t, a, b)
	return Float64(x * exp(Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = x * exp(((y * (log(z) - t)) + (a * (log((1.0 - z)) - b))));
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(x * N[Exp[N[(N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.2%

   \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 2: 64.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+141}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(t, \mathsf{fma}\left(t, \left(y \cdot y\right) \cdot \mathsf{fma}\left(t \cdot 0.16666666666666666, y, 0.5\right), y\right), 1\right)}\\ \mathbf{elif}\;t\_1 \leq -20000000000:\\ \;\;\;\;x \cdot \left(b \cdot \left(0.5 \cdot \left(b \cdot \left(a \cdot a\right)\right)\right)\right)\\ \mathbf{elif}\;t\_1 \leq 0.0002:\\ \;\;\;\;x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5 \cdot \left(a \cdot a\right), -a\right), 1\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+164}:\\ \;\;\;\;a \cdot \left(a \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, x \cdot 0.5, \frac{x}{-a}\right), \frac{x}{a \cdot a}\right)\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+213}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(t, \mathsf{fma}\left(t, \left(y \cdot y\right) \cdot \mathsf{fma}\left(t \cdot -0.16666666666666666, y, 0.5\right), -y\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, x \cdot \left(y \cdot \mathsf{fma}\left(t \cdot 0.5, y, -1\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b)))))
   (if (<= t_1 -4e+141)
     (/
      x
      (fma t (fma t (* (* y y) (fma (* t 0.16666666666666666) y 0.5)) y) 1.0))
     (if (<= t_1 -20000000000.0)
       (* x (* b (* 0.5 (* b (* a a)))))
       (if (<= t_1 0.0002)
         (* x (fma b (fma b (* 0.5 (* a a)) (- a)) 1.0))
         (if (<= t_1 5e+164)
           (* a (* a (fma b (fma b (* x 0.5) (/ x (- a))) (/ x (* a a)))))
           (if (<= t_1 1e+213)
             (*
              x
              (fma
               t
               (fma t (* (* y y) (fma (* t -0.16666666666666666) y 0.5)) (- y))
               1.0))
             (fma t (* x (* y (fma (* t 0.5) y -1.0))) x))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	double tmp;
	if (t_1 <= -4e+141) {
		tmp = x / fma(t, fma(t, ((y * y) * fma((t * 0.16666666666666666), y, 0.5)), y), 1.0);
	} else if (t_1 <= -20000000000.0) {
		tmp = x * (b * (0.5 * (b * (a * a))));
	} else if (t_1 <= 0.0002) {
		tmp = x * fma(b, fma(b, (0.5 * (a * a)), -a), 1.0);
	} else if (t_1 <= 5e+164) {
		tmp = a * (a * fma(b, fma(b, (x * 0.5), (x / -a)), (x / (a * a))));
	} else if (t_1 <= 1e+213) {
		tmp = x * fma(t, fma(t, ((y * y) * fma((t * -0.16666666666666666), y, 0.5)), -y), 1.0);
	} else {
		tmp = fma(t, (x * (y * fma((t * 0.5), y, -1.0))), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))
	tmp = 0.0
	if (t_1 <= -4e+141)
		tmp = Float64(x / fma(t, fma(t, Float64(Float64(y * y) * fma(Float64(t * 0.16666666666666666), y, 0.5)), y), 1.0));
	elseif (t_1 <= -20000000000.0)
		tmp = Float64(x * Float64(b * Float64(0.5 * Float64(b * Float64(a * a)))));
	elseif (t_1 <= 0.0002)
		tmp = Float64(x * fma(b, fma(b, Float64(0.5 * Float64(a * a)), Float64(-a)), 1.0));
	elseif (t_1 <= 5e+164)
		tmp = Float64(a * Float64(a * fma(b, fma(b, Float64(x * 0.5), Float64(x / Float64(-a))), Float64(x / Float64(a * a)))));
	elseif (t_1 <= 1e+213)
		tmp = Float64(x * fma(t, fma(t, Float64(Float64(y * y) * fma(Float64(t * -0.16666666666666666), y, 0.5)), Float64(-y)), 1.0));
	else
		tmp = fma(t, Float64(x * Float64(y * fma(Float64(t * 0.5), y, -1.0))), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+141], N[(x / N[(t * N[(t * N[(N[(y * y), $MachinePrecision] * N[(N[(t * 0.16666666666666666), $MachinePrecision] * y + 0.5), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -20000000000.0], N[(x * N[(b * N[(0.5 * N[(b * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0002], N[(x * N[(b * N[(b * N[(0.5 * N[(a * a), $MachinePrecision]), $MachinePrecision] + (-a)), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+164], N[(a * N[(a * N[(b * N[(b * N[(x * 0.5), $MachinePrecision] + N[(x / (-a)), $MachinePrecision]), $MachinePrecision] + N[(x / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+213], N[(x * N[(t * N[(t * N[(N[(y * y), $MachinePrecision] * N[(N[(t * -0.16666666666666666), $MachinePrecision] * y + 0.5), $MachinePrecision]), $MachinePrecision] + (-y)), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(t * N[(x * N[(y * N[(N[(t * 0.5), $MachinePrecision] * y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]]]]]

Derivation

1. Split input into 6 regimes

2. if (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -4.00000000000000007e141

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(t \cdot y\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(t \cdot y\right)}} \]

      2. *-commutative N/A

         \[\leadsto x \cdot e^{\mathsf{neg}\left(\color{blue}{y \cdot t}\right)} \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      5. neg-lowering-neg.f64 49.3

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(-t\right)}} \]

   5. Simplified 49.3%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}} \]
   6. Step-by-step derivation

      1. exp-prod N/A

         \[\leadsto x \cdot \color{blue}{{\left(e^{y}\right)}^{\left(\mathsf{neg}\left(t\right)\right)}} \]

      2. pow-neg N/A

         \[\leadsto x \cdot \color{blue}{\frac{1}{{\left(e^{y}\right)}^{t}}} \]

      3. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{x}{{\left(e^{y}\right)}^{t}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{{\left(e^{y}\right)}^{t}}} \]

      5. pow-exp N/A

         \[\leadsto \frac{x}{\color{blue}{e^{y \cdot t}}} \]

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{e^{y \cdot t}}} \]

      7. *-lowering-*.f64 49.3

         \[\leadsto \frac{x}{e^{\color{blue}{y \cdot t}}} \]

   7. Applied egg-rr 49.3%

      \[\leadsto \color{blue}{\frac{x}{e^{y \cdot t}}} \]

   8. Taylor expanded in t around 0

      \[\leadsto \frac{x}{\color{blue}{1 + t \cdot \left(y + t \cdot \left(\frac{1}{6} \cdot \left(t \cdot {y}^{3}\right) + \frac{1}{2} \cdot {y}^{2}\right)\right)}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{t \cdot \left(y + t \cdot \left(\frac{1}{6} \cdot \left(t \cdot {y}^{3}\right) + \frac{1}{2} \cdot {y}^{2}\right)\right) + 1}} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(t, y + t \cdot \left(\frac{1}{6} \cdot \left(t \cdot {y}^{3}\right) + \frac{1}{2} \cdot {y}^{2}\right), 1\right)}} \]

      3. +-commutative N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(t, \color{blue}{t \cdot \left(\frac{1}{6} \cdot \left(t \cdot {y}^{3}\right) + \frac{1}{2} \cdot {y}^{2}\right) + y}, 1\right)} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(t, \color{blue}{\mathsf{fma}\left(t, \frac{1}{6} \cdot \left(t \cdot {y}^{3}\right) + \frac{1}{2} \cdot {y}^{2}, y\right)}, 1\right)} \]

      5. associate-*r* N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(t, \mathsf{fma}\left(t, \color{blue}{\left(\frac{1}{6} \cdot t\right) \cdot {y}^{3}} + \frac{1}{2} \cdot {y}^{2}, y\right), 1\right)} \]

      6. cube-mult N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(t, \mathsf{fma}\left(t, \left(\frac{1}{6} \cdot t\right) \cdot \color{blue}{\left(y \cdot \left(y \cdot y\right)\right)} + \frac{1}{2} \cdot {y}^{2}, y\right), 1\right)} \]

      7. unpow2 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(t, \mathsf{fma}\left(t, \left(\frac{1}{6} \cdot t\right) \cdot \left(y \cdot \color{blue}{{y}^{2}}\right) + \frac{1}{2} \cdot {y}^{2}, y\right), 1\right)} \]

      8. associate-*r* N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(t, \mathsf{fma}\left(t, \color{blue}{\left(\left(\frac{1}{6} \cdot t\right) \cdot y\right) \cdot {y}^{2}} + \frac{1}{2} \cdot {y}^{2}, y\right), 1\right)} \]

      9. distribute-rgt-out N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(t, \mathsf{fma}\left(t, \color{blue}{{y}^{2} \cdot \left(\left(\frac{1}{6} \cdot t\right) \cdot y + \frac{1}{2}\right)}, y\right), 1\right)} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(t, \mathsf{fma}\left(t, \color{blue}{{y}^{2} \cdot \left(\left(\frac{1}{6} \cdot t\right) \cdot y + \frac{1}{2}\right)}, y\right), 1\right)} \]

      11. unpow2 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(t, \mathsf{fma}\left(t, \color{blue}{\left(y \cdot y\right)} \cdot \left(\left(\frac{1}{6} \cdot t\right) \cdot y + \frac{1}{2}\right), y\right), 1\right)} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(t, \mathsf{fma}\left(t, \color{blue}{\left(y \cdot y\right)} \cdot \left(\left(\frac{1}{6} \cdot t\right) \cdot y + \frac{1}{2}\right), y\right), 1\right)} \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(t, \mathsf{fma}\left(t, \left(y \cdot y\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{6} \cdot t, y, \frac{1}{2}\right)}, y\right), 1\right)} \]

      14. *-commutative N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(t, \mathsf{fma}\left(t, \left(y \cdot y\right) \cdot \mathsf{fma}\left(\color{blue}{t \cdot \frac{1}{6}}, y, \frac{1}{2}\right), y\right), 1\right)} \]

      15. *-lowering-*.f64 79.1

          \[\leadsto \frac{x}{\mathsf{fma}\left(t, \mathsf{fma}\left(t, \left(y \cdot y\right) \cdot \mathsf{fma}\left(\color{blue}{t \cdot 0.16666666666666666}, y, 0.5\right), y\right), 1\right)} \]

   10. Simplified 79.1%

       \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(t, \mathsf{fma}\left(t, \left(y \cdot y\right) \cdot \mathsf{fma}\left(t \cdot 0.16666666666666666, y, 0.5\right), y\right), 1\right)}} \]

   if -4.00000000000000007e141 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -2e10

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      3. *-lowering-*.f64 48.4

         \[\leadsto x \cdot e^{-\color{blue}{a \cdot b}} \]

   5. Simplified 48.4%

      \[\leadsto x \cdot e^{\color{blue}{-a \cdot b}} \]

   6. Taylor expanded in a around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right) + 1\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, -1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right)} \]

      3. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right) \]

      4. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) + \left(\mathsf{neg}\left(b\right)\right)}, 1\right) \]

      5. associate-*r* N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\frac{1}{2} \cdot a\right) \cdot {b}^{2}} + \left(\mathsf{neg}\left(b\right)\right), 1\right) \]

      6. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \left(\frac{1}{2} \cdot a\right) \cdot \color{blue}{\left(b \cdot b\right)} + \left(\mathsf{neg}\left(b\right)\right), 1\right) \]

      7. associate-*r* N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\left(\frac{1}{2} \cdot a\right) \cdot b\right) \cdot b} + \left(\mathsf{neg}\left(b\right)\right), 1\right) \]

      8. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \left(\left(\frac{1}{2} \cdot a\right) \cdot b\right) \cdot b + \color{blue}{-1 \cdot b}, 1\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{b \cdot \left(\left(\frac{1}{2} \cdot a\right) \cdot b + -1\right)}, 1\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{b \cdot \left(\left(\frac{1}{2} \cdot a\right) \cdot b + -1\right)}, 1\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, b \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot a, b, -1\right)}, 1\right) \]

      12. *-lowering-*.f64 2.7

          \[\leadsto x \cdot \mathsf{fma}\left(a, b \cdot \mathsf{fma}\left(\color{blue}{0.5 \cdot a}, b, -1\right), 1\right) \]

   8. Simplified 2.7%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, b \cdot \mathsf{fma}\left(0.5 \cdot a, b, -1\right), 1\right)} \]

   9. Taylor expanded in b around inf

      \[\leadsto x \cdot \color{blue}{\left({b}^{2} \cdot \left(-1 \cdot \frac{a}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto x \cdot \left(\color{blue}{\left(b \cdot b\right)} \cdot \left(-1 \cdot \frac{a}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right) \]

       2. associate-*l* N/A

          \[\leadsto x \cdot \color{blue}{\left(b \cdot \left(b \cdot \left(-1 \cdot \frac{a}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \color{blue}{\left(b \cdot \left(b \cdot \left(-1 \cdot \frac{a}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \left(b \cdot \color{blue}{\left(b \cdot \left(-1 \cdot \frac{a}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)}\right) \]

       5. associate-*r/ N/A

          \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(\color{blue}{\frac{-1 \cdot a}{b}} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(\frac{\color{blue}{a \cdot -1}}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       7. associate-/l* N/A

          \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(\color{blue}{a \cdot \frac{-1}{b}} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       8. metadata-eval N/A

          \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \frac{\color{blue}{\mathsf{neg}\left(1\right)}}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       9. distribute-neg-frac N/A

          \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{b}\right)\right)} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       10. *-commutative N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(\mathsf{neg}\left(\frac{1}{b}\right)\right) + \color{blue}{{a}^{2} \cdot \frac{1}{2}}\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(\mathsf{neg}\left(\frac{1}{b}\right)\right) + \color{blue}{\left(a \cdot a\right)} \cdot \frac{1}{2}\right)\right)\right) \]

       12. associate-*l* N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(\mathsf{neg}\left(\frac{1}{b}\right)\right) + \color{blue}{a \cdot \left(a \cdot \frac{1}{2}\right)}\right)\right)\right) \]

       13. *-commutative N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(\mathsf{neg}\left(\frac{1}{b}\right)\right) + a \cdot \color{blue}{\left(\frac{1}{2} \cdot a\right)}\right)\right)\right) \]

       14. distribute-lft-out N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \color{blue}{\left(a \cdot \left(\left(\mathsf{neg}\left(\frac{1}{b}\right)\right) + \frac{1}{2} \cdot a\right)\right)}\right)\right) \]

       15. +-commutative N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \color{blue}{\left(\frac{1}{2} \cdot a + \left(\mathsf{neg}\left(\frac{1}{b}\right)\right)\right)}\right)\right)\right) \]

       16. sub-neg N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \color{blue}{\left(\frac{1}{2} \cdot a - \frac{1}{b}\right)}\right)\right)\right) \]

       17. *-lowering-*.f64 N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \color{blue}{\left(a \cdot \left(\frac{1}{2} \cdot a - \frac{1}{b}\right)\right)}\right)\right) \]

       18. sub-neg N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \color{blue}{\left(\frac{1}{2} \cdot a + \left(\mathsf{neg}\left(\frac{1}{b}\right)\right)\right)}\right)\right)\right) \]

       19. *-commutative N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(\color{blue}{a \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(\frac{1}{b}\right)\right)\right)\right)\right)\right) \]

       20. accelerator-lowering-fma.f64 N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \color{blue}{\mathsf{fma}\left(a, \frac{1}{2}, \mathsf{neg}\left(\frac{1}{b}\right)\right)}\right)\right)\right) \]

       21. distribute-neg-frac N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \mathsf{fma}\left(a, \frac{1}{2}, \color{blue}{\frac{\mathsf{neg}\left(1\right)}{b}}\right)\right)\right)\right) \]

       22. metadata-eval N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \mathsf{fma}\left(a, \frac{1}{2}, \frac{\color{blue}{-1}}{b}\right)\right)\right)\right) \]

       23. /-lowering-/.f64 27.2

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \mathsf{fma}\left(a, 0.5, \color{blue}{\frac{-1}{b}}\right)\right)\right)\right) \]

   11. Simplified 27.2%

       \[\leadsto x \cdot \color{blue}{\left(b \cdot \left(b \cdot \left(a \cdot \mathsf{fma}\left(a, 0.5, \frac{-1}{b}\right)\right)\right)\right)} \]

   12. Taylor expanded in b around inf

       \[\leadsto x \cdot \left(b \cdot \color{blue}{\left(\frac{1}{2} \cdot \left({a}^{2} \cdot b\right)\right)}\right) \]
   13. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \left(b \cdot \color{blue}{\left(\frac{1}{2} \cdot \left({a}^{2} \cdot b\right)\right)}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \left(b \cdot \left(\frac{1}{2} \cdot \color{blue}{\left({a}^{2} \cdot b\right)}\right)\right) \]

       3. unpow2 N/A

          \[\leadsto x \cdot \left(b \cdot \left(\frac{1}{2} \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot b\right)\right)\right) \]

       4. *-lowering-*.f64 45.1

          \[\leadsto x \cdot \left(b \cdot \left(0.5 \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot b\right)\right)\right) \]

   14. Simplified 45.1%

       \[\leadsto x \cdot \left(b \cdot \color{blue}{\left(0.5 \cdot \left(\left(a \cdot a\right) \cdot b\right)\right)}\right) \]

   if -2e10 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 2.0000000000000001e-4

   1. Initial program 88.6%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      3. *-lowering-*.f64 86.0

         \[\leadsto x \cdot e^{-\color{blue}{a \cdot b}} \]

   5. Simplified 86.0%

      \[\leadsto x \cdot e^{\color{blue}{-a \cdot b}} \]

   6. Taylor expanded in b around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + b \cdot \left(-1 \cdot a + \frac{1}{2} \cdot \left({a}^{2} \cdot b\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(b \cdot \left(-1 \cdot a + \frac{1}{2} \cdot \left({a}^{2} \cdot b\right)\right) + 1\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(b, -1 \cdot a + \frac{1}{2} \cdot \left({a}^{2} \cdot b\right), 1\right)} \]

      3. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \color{blue}{\frac{1}{2} \cdot \left({a}^{2} \cdot b\right) + -1 \cdot a}, 1\right) \]

      4. *-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \color{blue}{\left({a}^{2} \cdot b\right) \cdot \frac{1}{2}} + -1 \cdot a, 1\right) \]

      5. *-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \color{blue}{\left(b \cdot {a}^{2}\right)} \cdot \frac{1}{2} + -1 \cdot a, 1\right) \]

      6. associate-*r* N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \color{blue}{b \cdot \left({a}^{2} \cdot \frac{1}{2}\right)} + -1 \cdot a, 1\right) \]

      7. *-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, b \cdot \color{blue}{\left(\frac{1}{2} \cdot {a}^{2}\right)} + -1 \cdot a, 1\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, \frac{1}{2} \cdot {a}^{2}, -1 \cdot a\right)}, 1\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\frac{1}{2} \cdot {a}^{2}}, -1 \cdot a\right), 1\right) \]

      10. unpow2 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{1}{2} \cdot \color{blue}{\left(a \cdot a\right)}, -1 \cdot a\right), 1\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{1}{2} \cdot \color{blue}{\left(a \cdot a\right)}, -1 \cdot a\right), 1\right) \]

      12. mul-1-neg N/A

          \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{1}{2} \cdot \left(a \cdot a\right), \color{blue}{\mathsf{neg}\left(a\right)}\right), 1\right) \]

      13. neg-lowering-neg.f64 89.4

          \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5 \cdot \left(a \cdot a\right), \color{blue}{-a}\right), 1\right) \]

   8. Simplified 89.4%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5 \cdot \left(a \cdot a\right), -a\right), 1\right)} \]

   if 2.0000000000000001e-4 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 4.9999999999999995e164

   1. Initial program 92.6%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      3. *-lowering-*.f64 58.9

         \[\leadsto x \cdot e^{-\color{blue}{a \cdot b}} \]

   5. Simplified 58.9%

      \[\leadsto x \cdot e^{\color{blue}{-a \cdot b}} \]

   6. Taylor expanded in a around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right) + 1\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, -1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right)} \]

      3. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right) \]

      4. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) + \left(\mathsf{neg}\left(b\right)\right)}, 1\right) \]

      5. associate-*r* N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\frac{1}{2} \cdot a\right) \cdot {b}^{2}} + \left(\mathsf{neg}\left(b\right)\right), 1\right) \]

      6. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \left(\frac{1}{2} \cdot a\right) \cdot \color{blue}{\left(b \cdot b\right)} + \left(\mathsf{neg}\left(b\right)\right), 1\right) \]

      7. associate-*r* N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\left(\frac{1}{2} \cdot a\right) \cdot b\right) \cdot b} + \left(\mathsf{neg}\left(b\right)\right), 1\right) \]

      8. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \left(\left(\frac{1}{2} \cdot a\right) \cdot b\right) \cdot b + \color{blue}{-1 \cdot b}, 1\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{b \cdot \left(\left(\frac{1}{2} \cdot a\right) \cdot b + -1\right)}, 1\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{b \cdot \left(\left(\frac{1}{2} \cdot a\right) \cdot b + -1\right)}, 1\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, b \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot a, b, -1\right)}, 1\right) \]

      12. *-lowering-*.f64 37.7

          \[\leadsto x \cdot \mathsf{fma}\left(a, b \cdot \mathsf{fma}\left(\color{blue}{0.5 \cdot a}, b, -1\right), 1\right) \]

   8. Simplified 37.7%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, b \cdot \mathsf{fma}\left(0.5 \cdot a, b, -1\right), 1\right)} \]

   9. Taylor expanded in a around inf

      \[\leadsto \color{blue}{{a}^{2} \cdot \left(-1 \cdot \frac{b \cdot x}{a} + \left(\frac{1}{2} \cdot \left({b}^{2} \cdot x\right) + \frac{x}{{a}^{2}}\right)\right)} \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \color{blue}{\left(a \cdot a\right)} \cdot \left(-1 \cdot \frac{b \cdot x}{a} + \left(\frac{1}{2} \cdot \left({b}^{2} \cdot x\right) + \frac{x}{{a}^{2}}\right)\right) \]

       2. associate-*l* N/A

          \[\leadsto \color{blue}{a \cdot \left(a \cdot \left(-1 \cdot \frac{b \cdot x}{a} + \left(\frac{1}{2} \cdot \left({b}^{2} \cdot x\right) + \frac{x}{{a}^{2}}\right)\right)\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{a \cdot \left(a \cdot \left(-1 \cdot \frac{b \cdot x}{a} + \left(\frac{1}{2} \cdot \left({b}^{2} \cdot x\right) + \frac{x}{{a}^{2}}\right)\right)\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto a \cdot \color{blue}{\left(a \cdot \left(-1 \cdot \frac{b \cdot x}{a} + \left(\frac{1}{2} \cdot \left({b}^{2} \cdot x\right) + \frac{x}{{a}^{2}}\right)\right)\right)} \]

       5. associate-+r+ N/A

          \[\leadsto a \cdot \left(a \cdot \color{blue}{\left(\left(-1 \cdot \frac{b \cdot x}{a} + \frac{1}{2} \cdot \left({b}^{2} \cdot x\right)\right) + \frac{x}{{a}^{2}}\right)}\right) \]

   11. Simplified 56.3%

       \[\leadsto \color{blue}{a \cdot \left(a \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, x \cdot 0.5, \frac{x}{-a}\right), \frac{x}{a \cdot a}\right)\right)} \]

   if 4.9999999999999995e164 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 9.99999999999999984e212

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(t \cdot y\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(t \cdot y\right)}} \]

      2. *-commutative N/A

         \[\leadsto x \cdot e^{\mathsf{neg}\left(\color{blue}{y \cdot t}\right)} \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      5. neg-lowering-neg.f64 60.8

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(-t\right)}} \]

   5. Simplified 60.8%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}} \]

   6. Taylor expanded in t around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + t \cdot \left(-1 \cdot y + t \cdot \left(\frac{-1}{6} \cdot \left(t \cdot {y}^{3}\right) + \frac{1}{2} \cdot {y}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(t \cdot \left(-1 \cdot y + t \cdot \left(\frac{-1}{6} \cdot \left(t \cdot {y}^{3}\right) + \frac{1}{2} \cdot {y}^{2}\right)\right) + 1\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(t, -1 \cdot y + t \cdot \left(\frac{-1}{6} \cdot \left(t \cdot {y}^{3}\right) + \frac{1}{2} \cdot {y}^{2}\right), 1\right)} \]

      3. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(t, \color{blue}{t \cdot \left(\frac{-1}{6} \cdot \left(t \cdot {y}^{3}\right) + \frac{1}{2} \cdot {y}^{2}\right) + -1 \cdot y}, 1\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(t, \color{blue}{\mathsf{fma}\left(t, \frac{-1}{6} \cdot \left(t \cdot {y}^{3}\right) + \frac{1}{2} \cdot {y}^{2}, -1 \cdot y\right)}, 1\right) \]

      5. associate-*r* N/A

         \[\leadsto x \cdot \mathsf{fma}\left(t, \mathsf{fma}\left(t, \color{blue}{\left(\frac{-1}{6} \cdot t\right) \cdot {y}^{3}} + \frac{1}{2} \cdot {y}^{2}, -1 \cdot y\right), 1\right) \]

      6. cube-mult N/A

         \[\leadsto x \cdot \mathsf{fma}\left(t, \mathsf{fma}\left(t, \left(\frac{-1}{6} \cdot t\right) \cdot \color{blue}{\left(y \cdot \left(y \cdot y\right)\right)} + \frac{1}{2} \cdot {y}^{2}, -1 \cdot y\right), 1\right) \]

      7. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(t, \mathsf{fma}\left(t, \left(\frac{-1}{6} \cdot t\right) \cdot \left(y \cdot \color{blue}{{y}^{2}}\right) + \frac{1}{2} \cdot {y}^{2}, -1 \cdot y\right), 1\right) \]

      8. associate-*r* N/A

         \[\leadsto x \cdot \mathsf{fma}\left(t, \mathsf{fma}\left(t, \color{blue}{\left(\left(\frac{-1}{6} \cdot t\right) \cdot y\right) \cdot {y}^{2}} + \frac{1}{2} \cdot {y}^{2}, -1 \cdot y\right), 1\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto x \cdot \mathsf{fma}\left(t, \mathsf{fma}\left(t, \color{blue}{{y}^{2} \cdot \left(\left(\frac{-1}{6} \cdot t\right) \cdot y + \frac{1}{2}\right)}, -1 \cdot y\right), 1\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(t, \mathsf{fma}\left(t, \color{blue}{{y}^{2} \cdot \left(\left(\frac{-1}{6} \cdot t\right) \cdot y + \frac{1}{2}\right)}, -1 \cdot y\right), 1\right) \]

      11. unpow2 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(t, \mathsf{fma}\left(t, \color{blue}{\left(y \cdot y\right)} \cdot \left(\left(\frac{-1}{6} \cdot t\right) \cdot y + \frac{1}{2}\right), -1 \cdot y\right), 1\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(t, \mathsf{fma}\left(t, \color{blue}{\left(y \cdot y\right)} \cdot \left(\left(\frac{-1}{6} \cdot t\right) \cdot y + \frac{1}{2}\right), -1 \cdot y\right), 1\right) \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(t, \mathsf{fma}\left(t, \left(y \cdot y\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{6} \cdot t, y, \frac{1}{2}\right)}, -1 \cdot y\right), 1\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(t, \mathsf{fma}\left(t, \left(y \cdot y\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{-1}{6} \cdot t}, y, \frac{1}{2}\right), -1 \cdot y\right), 1\right) \]

      15. mul-1-neg N/A

          \[\leadsto x \cdot \mathsf{fma}\left(t, \mathsf{fma}\left(t, \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{-1}{6} \cdot t, y, \frac{1}{2}\right), \color{blue}{\mathsf{neg}\left(y\right)}\right), 1\right) \]

      16. neg-lowering-neg.f64 80.2

          \[\leadsto x \cdot \mathsf{fma}\left(t, \mathsf{fma}\left(t, \left(y \cdot y\right) \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot t, y, 0.5\right), \color{blue}{-y}\right), 1\right) \]

   8. Simplified 80.2%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(t, \mathsf{fma}\left(t, \left(y \cdot y\right) \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot t, y, 0.5\right), -y\right), 1\right)} \]

   if 9.99999999999999984e212 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 98.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(t \cdot y\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(t \cdot y\right)}} \]

      2. *-commutative N/A

         \[\leadsto x \cdot e^{\mathsf{neg}\left(\color{blue}{y \cdot t}\right)} \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      5. neg-lowering-neg.f64 58.5

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(-t\right)}} \]

   5. Simplified 58.5%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{x + t \cdot \left(-1 \cdot \left(x \cdot y\right) + \frac{1}{2} \cdot \left(t \cdot \left(x \cdot {y}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(x \cdot y\right) + \frac{1}{2} \cdot \left(t \cdot \left(x \cdot {y}^{2}\right)\right)\right) + x} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, -1 \cdot \left(x \cdot y\right) + \frac{1}{2} \cdot \left(t \cdot \left(x \cdot {y}^{2}\right)\right), x\right)} \]

   8. Simplified 70.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, x \cdot \left(y \cdot \mathsf{fma}\left(0.5 \cdot t, y, -1\right)\right), x\right)} \]
3. Recombined 6 regimes into one program.

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -4 \cdot 10^{+141}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(t, \mathsf{fma}\left(t, \left(y \cdot y\right) \cdot \mathsf{fma}\left(t \cdot 0.16666666666666666, y, 0.5\right), y\right), 1\right)}\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -20000000000:\\ \;\;\;\;x \cdot \left(b \cdot \left(0.5 \cdot \left(b \cdot \left(a \cdot a\right)\right)\right)\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 0.0002:\\ \;\;\;\;x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5 \cdot \left(a \cdot a\right), -a\right), 1\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 5 \cdot 10^{+164}:\\ \;\;\;\;a \cdot \left(a \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, x \cdot 0.5, \frac{x}{-a}\right), \frac{x}{a \cdot a}\right)\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 10^{+213}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(t, \mathsf{fma}\left(t, \left(y \cdot y\right) \cdot \mathsf{fma}\left(t \cdot -0.16666666666666666, y, 0.5\right), -y\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, x \cdot \left(y \cdot \mathsf{fma}\left(t \cdot 0.5, y, -1\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 64.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+141}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(t, \mathsf{fma}\left(t, \left(y \cdot y\right) \cdot \mathsf{fma}\left(t \cdot 0.16666666666666666, y, 0.5\right), y\right), 1\right)}\\ \mathbf{elif}\;t\_1 \leq -20000000000:\\ \;\;\;\;x \cdot \left(b \cdot \left(0.5 \cdot \left(b \cdot \left(a \cdot a\right)\right)\right)\right)\\ \mathbf{elif}\;t\_1 \leq 2000000000000:\\ \;\;\;\;x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5 \cdot \left(a \cdot a\right), -a\right), 1\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+80}:\\ \;\;\;\;a \cdot \left(a \cdot \frac{x}{a \cdot a}\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+225}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(a, \left(\left(z + b\right) \cdot \left(z + b\right)\right) \cdot \left(a \cdot 0.5\right) - \left(z + b\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, x \cdot \left(y \cdot \mathsf{fma}\left(t \cdot 0.5, y, -1\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b)))))
   (if (<= t_1 -4e+141)
     (/
      x
      (fma t (fma t (* (* y y) (fma (* t 0.16666666666666666) y 0.5)) y) 1.0))
     (if (<= t_1 -20000000000.0)
       (* x (* b (* 0.5 (* b (* a a)))))
       (if (<= t_1 2000000000000.0)
         (* x (fma b (fma b (* 0.5 (* a a)) (- a)) 1.0))
         (if (<= t_1 1e+80)
           (* a (* a (/ x (* a a))))
           (if (<= t_1 2e+225)
             (* x (fma a (- (* (* (+ z b) (+ z b)) (* a 0.5)) (+ z b)) 1.0))
             (fma t (* x (* y (fma (* t 0.5) y -1.0))) x))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	double tmp;
	if (t_1 <= -4e+141) {
		tmp = x / fma(t, fma(t, ((y * y) * fma((t * 0.16666666666666666), y, 0.5)), y), 1.0);
	} else if (t_1 <= -20000000000.0) {
		tmp = x * (b * (0.5 * (b * (a * a))));
	} else if (t_1 <= 2000000000000.0) {
		tmp = x * fma(b, fma(b, (0.5 * (a * a)), -a), 1.0);
	} else if (t_1 <= 1e+80) {
		tmp = a * (a * (x / (a * a)));
	} else if (t_1 <= 2e+225) {
		tmp = x * fma(a, ((((z + b) * (z + b)) * (a * 0.5)) - (z + b)), 1.0);
	} else {
		tmp = fma(t, (x * (y * fma((t * 0.5), y, -1.0))), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))
	tmp = 0.0
	if (t_1 <= -4e+141)
		tmp = Float64(x / fma(t, fma(t, Float64(Float64(y * y) * fma(Float64(t * 0.16666666666666666), y, 0.5)), y), 1.0));
	elseif (t_1 <= -20000000000.0)
		tmp = Float64(x * Float64(b * Float64(0.5 * Float64(b * Float64(a * a)))));
	elseif (t_1 <= 2000000000000.0)
		tmp = Float64(x * fma(b, fma(b, Float64(0.5 * Float64(a * a)), Float64(-a)), 1.0));
	elseif (t_1 <= 1e+80)
		tmp = Float64(a * Float64(a * Float64(x / Float64(a * a))));
	elseif (t_1 <= 2e+225)
		tmp = Float64(x * fma(a, Float64(Float64(Float64(Float64(z + b) * Float64(z + b)) * Float64(a * 0.5)) - Float64(z + b)), 1.0));
	else
		tmp = fma(t, Float64(x * Float64(y * fma(Float64(t * 0.5), y, -1.0))), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+141], N[(x / N[(t * N[(t * N[(N[(y * y), $MachinePrecision] * N[(N[(t * 0.16666666666666666), $MachinePrecision] * y + 0.5), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -20000000000.0], N[(x * N[(b * N[(0.5 * N[(b * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2000000000000.0], N[(x * N[(b * N[(b * N[(0.5 * N[(a * a), $MachinePrecision]), $MachinePrecision] + (-a)), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+80], N[(a * N[(a * N[(x / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+225], N[(x * N[(a * N[(N[(N[(N[(z + b), $MachinePrecision] * N[(z + b), $MachinePrecision]), $MachinePrecision] * N[(a * 0.5), $MachinePrecision]), $MachinePrecision] - N[(z + b), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(t * N[(x * N[(y * N[(N[(t * 0.5), $MachinePrecision] * y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]]]]]

Derivation

1. Split input into 6 regimes

2. if (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -4.00000000000000007e141

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(t \cdot y\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(t \cdot y\right)}} \]

      2. *-commutative N/A

         \[\leadsto x \cdot e^{\mathsf{neg}\left(\color{blue}{y \cdot t}\right)} \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      5. neg-lowering-neg.f64 49.3

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(-t\right)}} \]

   5. Simplified 49.3%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}} \]
   6. Step-by-step derivation

      1. exp-prod N/A

         \[\leadsto x \cdot \color{blue}{{\left(e^{y}\right)}^{\left(\mathsf{neg}\left(t\right)\right)}} \]

      2. pow-neg N/A

         \[\leadsto x \cdot \color{blue}{\frac{1}{{\left(e^{y}\right)}^{t}}} \]

      3. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{x}{{\left(e^{y}\right)}^{t}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{{\left(e^{y}\right)}^{t}}} \]

      5. pow-exp N/A

         \[\leadsto \frac{x}{\color{blue}{e^{y \cdot t}}} \]

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{e^{y \cdot t}}} \]

      7. *-lowering-*.f64 49.3

         \[\leadsto \frac{x}{e^{\color{blue}{y \cdot t}}} \]

   7. Applied egg-rr 49.3%

      \[\leadsto \color{blue}{\frac{x}{e^{y \cdot t}}} \]

   8. Taylor expanded in t around 0

      \[\leadsto \frac{x}{\color{blue}{1 + t \cdot \left(y + t \cdot \left(\frac{1}{6} \cdot \left(t \cdot {y}^{3}\right) + \frac{1}{2} \cdot {y}^{2}\right)\right)}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{t \cdot \left(y + t \cdot \left(\frac{1}{6} \cdot \left(t \cdot {y}^{3}\right) + \frac{1}{2} \cdot {y}^{2}\right)\right) + 1}} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(t, y + t \cdot \left(\frac{1}{6} \cdot \left(t \cdot {y}^{3}\right) + \frac{1}{2} \cdot {y}^{2}\right), 1\right)}} \]

      3. +-commutative N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(t, \color{blue}{t \cdot \left(\frac{1}{6} \cdot \left(t \cdot {y}^{3}\right) + \frac{1}{2} \cdot {y}^{2}\right) + y}, 1\right)} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(t, \color{blue}{\mathsf{fma}\left(t, \frac{1}{6} \cdot \left(t \cdot {y}^{3}\right) + \frac{1}{2} \cdot {y}^{2}, y\right)}, 1\right)} \]

      5. associate-*r* N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(t, \mathsf{fma}\left(t, \color{blue}{\left(\frac{1}{6} \cdot t\right) \cdot {y}^{3}} + \frac{1}{2} \cdot {y}^{2}, y\right), 1\right)} \]

      6. cube-mult N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(t, \mathsf{fma}\left(t, \left(\frac{1}{6} \cdot t\right) \cdot \color{blue}{\left(y \cdot \left(y \cdot y\right)\right)} + \frac{1}{2} \cdot {y}^{2}, y\right), 1\right)} \]

      7. unpow2 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(t, \mathsf{fma}\left(t, \left(\frac{1}{6} \cdot t\right) \cdot \left(y \cdot \color{blue}{{y}^{2}}\right) + \frac{1}{2} \cdot {y}^{2}, y\right), 1\right)} \]

      8. associate-*r* N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(t, \mathsf{fma}\left(t, \color{blue}{\left(\left(\frac{1}{6} \cdot t\right) \cdot y\right) \cdot {y}^{2}} + \frac{1}{2} \cdot {y}^{2}, y\right), 1\right)} \]

      9. distribute-rgt-out N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(t, \mathsf{fma}\left(t, \color{blue}{{y}^{2} \cdot \left(\left(\frac{1}{6} \cdot t\right) \cdot y + \frac{1}{2}\right)}, y\right), 1\right)} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(t, \mathsf{fma}\left(t, \color{blue}{{y}^{2} \cdot \left(\left(\frac{1}{6} \cdot t\right) \cdot y + \frac{1}{2}\right)}, y\right), 1\right)} \]

      11. unpow2 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(t, \mathsf{fma}\left(t, \color{blue}{\left(y \cdot y\right)} \cdot \left(\left(\frac{1}{6} \cdot t\right) \cdot y + \frac{1}{2}\right), y\right), 1\right)} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(t, \mathsf{fma}\left(t, \color{blue}{\left(y \cdot y\right)} \cdot \left(\left(\frac{1}{6} \cdot t\right) \cdot y + \frac{1}{2}\right), y\right), 1\right)} \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(t, \mathsf{fma}\left(t, \left(y \cdot y\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{6} \cdot t, y, \frac{1}{2}\right)}, y\right), 1\right)} \]

      14. *-commutative N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(t, \mathsf{fma}\left(t, \left(y \cdot y\right) \cdot \mathsf{fma}\left(\color{blue}{t \cdot \frac{1}{6}}, y, \frac{1}{2}\right), y\right), 1\right)} \]

      15. *-lowering-*.f64 79.1

          \[\leadsto \frac{x}{\mathsf{fma}\left(t, \mathsf{fma}\left(t, \left(y \cdot y\right) \cdot \mathsf{fma}\left(\color{blue}{t \cdot 0.16666666666666666}, y, 0.5\right), y\right), 1\right)} \]

   10. Simplified 79.1%

       \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(t, \mathsf{fma}\left(t, \left(y \cdot y\right) \cdot \mathsf{fma}\left(t \cdot 0.16666666666666666, y, 0.5\right), y\right), 1\right)}} \]

   if -4.00000000000000007e141 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -2e10

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      3. *-lowering-*.f64 48.4

         \[\leadsto x \cdot e^{-\color{blue}{a \cdot b}} \]

   5. Simplified 48.4%

      \[\leadsto x \cdot e^{\color{blue}{-a \cdot b}} \]

   6. Taylor expanded in a around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right) + 1\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, -1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right)} \]

      3. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right) \]

      4. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) + \left(\mathsf{neg}\left(b\right)\right)}, 1\right) \]

      5. associate-*r* N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\frac{1}{2} \cdot a\right) \cdot {b}^{2}} + \left(\mathsf{neg}\left(b\right)\right), 1\right) \]

      6. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \left(\frac{1}{2} \cdot a\right) \cdot \color{blue}{\left(b \cdot b\right)} + \left(\mathsf{neg}\left(b\right)\right), 1\right) \]

      7. associate-*r* N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\left(\frac{1}{2} \cdot a\right) \cdot b\right) \cdot b} + \left(\mathsf{neg}\left(b\right)\right), 1\right) \]

      8. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \left(\left(\frac{1}{2} \cdot a\right) \cdot b\right) \cdot b + \color{blue}{-1 \cdot b}, 1\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{b \cdot \left(\left(\frac{1}{2} \cdot a\right) \cdot b + -1\right)}, 1\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{b \cdot \left(\left(\frac{1}{2} \cdot a\right) \cdot b + -1\right)}, 1\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, b \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot a, b, -1\right)}, 1\right) \]

      12. *-lowering-*.f64 2.7

          \[\leadsto x \cdot \mathsf{fma}\left(a, b \cdot \mathsf{fma}\left(\color{blue}{0.5 \cdot a}, b, -1\right), 1\right) \]

   8. Simplified 2.7%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, b \cdot \mathsf{fma}\left(0.5 \cdot a, b, -1\right), 1\right)} \]

   9. Taylor expanded in b around inf

      \[\leadsto x \cdot \color{blue}{\left({b}^{2} \cdot \left(-1 \cdot \frac{a}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto x \cdot \left(\color{blue}{\left(b \cdot b\right)} \cdot \left(-1 \cdot \frac{a}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right) \]

       2. associate-*l* N/A

          \[\leadsto x \cdot \color{blue}{\left(b \cdot \left(b \cdot \left(-1 \cdot \frac{a}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \color{blue}{\left(b \cdot \left(b \cdot \left(-1 \cdot \frac{a}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \left(b \cdot \color{blue}{\left(b \cdot \left(-1 \cdot \frac{a}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)}\right) \]

       5. associate-*r/ N/A

          \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(\color{blue}{\frac{-1 \cdot a}{b}} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(\frac{\color{blue}{a \cdot -1}}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       7. associate-/l* N/A

          \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(\color{blue}{a \cdot \frac{-1}{b}} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       8. metadata-eval N/A

          \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \frac{\color{blue}{\mathsf{neg}\left(1\right)}}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       9. distribute-neg-frac N/A

          \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{b}\right)\right)} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       10. *-commutative N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(\mathsf{neg}\left(\frac{1}{b}\right)\right) + \color{blue}{{a}^{2} \cdot \frac{1}{2}}\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(\mathsf{neg}\left(\frac{1}{b}\right)\right) + \color{blue}{\left(a \cdot a\right)} \cdot \frac{1}{2}\right)\right)\right) \]

       12. associate-*l* N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(\mathsf{neg}\left(\frac{1}{b}\right)\right) + \color{blue}{a \cdot \left(a \cdot \frac{1}{2}\right)}\right)\right)\right) \]

       13. *-commutative N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(\mathsf{neg}\left(\frac{1}{b}\right)\right) + a \cdot \color{blue}{\left(\frac{1}{2} \cdot a\right)}\right)\right)\right) \]

       14. distribute-lft-out N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \color{blue}{\left(a \cdot \left(\left(\mathsf{neg}\left(\frac{1}{b}\right)\right) + \frac{1}{2} \cdot a\right)\right)}\right)\right) \]

       15. +-commutative N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \color{blue}{\left(\frac{1}{2} \cdot a + \left(\mathsf{neg}\left(\frac{1}{b}\right)\right)\right)}\right)\right)\right) \]

       16. sub-neg N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \color{blue}{\left(\frac{1}{2} \cdot a - \frac{1}{b}\right)}\right)\right)\right) \]

       17. *-lowering-*.f64 N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \color{blue}{\left(a \cdot \left(\frac{1}{2} \cdot a - \frac{1}{b}\right)\right)}\right)\right) \]

       18. sub-neg N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \color{blue}{\left(\frac{1}{2} \cdot a + \left(\mathsf{neg}\left(\frac{1}{b}\right)\right)\right)}\right)\right)\right) \]

       19. *-commutative N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(\color{blue}{a \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(\frac{1}{b}\right)\right)\right)\right)\right)\right) \]

       20. accelerator-lowering-fma.f64 N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \color{blue}{\mathsf{fma}\left(a, \frac{1}{2}, \mathsf{neg}\left(\frac{1}{b}\right)\right)}\right)\right)\right) \]

       21. distribute-neg-frac N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \mathsf{fma}\left(a, \frac{1}{2}, \color{blue}{\frac{\mathsf{neg}\left(1\right)}{b}}\right)\right)\right)\right) \]

       22. metadata-eval N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \mathsf{fma}\left(a, \frac{1}{2}, \frac{\color{blue}{-1}}{b}\right)\right)\right)\right) \]

       23. /-lowering-/.f64 27.2

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \mathsf{fma}\left(a, 0.5, \color{blue}{\frac{-1}{b}}\right)\right)\right)\right) \]

   11. Simplified 27.2%

       \[\leadsto x \cdot \color{blue}{\left(b \cdot \left(b \cdot \left(a \cdot \mathsf{fma}\left(a, 0.5, \frac{-1}{b}\right)\right)\right)\right)} \]

   12. Taylor expanded in b around inf

       \[\leadsto x \cdot \left(b \cdot \color{blue}{\left(\frac{1}{2} \cdot \left({a}^{2} \cdot b\right)\right)}\right) \]
   13. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \left(b \cdot \color{blue}{\left(\frac{1}{2} \cdot \left({a}^{2} \cdot b\right)\right)}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \left(b \cdot \left(\frac{1}{2} \cdot \color{blue}{\left({a}^{2} \cdot b\right)}\right)\right) \]

       3. unpow2 N/A

          \[\leadsto x \cdot \left(b \cdot \left(\frac{1}{2} \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot b\right)\right)\right) \]

       4. *-lowering-*.f64 45.1

          \[\leadsto x \cdot \left(b \cdot \left(0.5 \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot b\right)\right)\right) \]

   14. Simplified 45.1%

       \[\leadsto x \cdot \left(b \cdot \color{blue}{\left(0.5 \cdot \left(\left(a \cdot a\right) \cdot b\right)\right)}\right) \]

   if -2e10 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 2e12

   1. Initial program 89.2%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      3. *-lowering-*.f64 86.7

         \[\leadsto x \cdot e^{-\color{blue}{a \cdot b}} \]

   5. Simplified 86.7%

      \[\leadsto x \cdot e^{\color{blue}{-a \cdot b}} \]

   6. Taylor expanded in b around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + b \cdot \left(-1 \cdot a + \frac{1}{2} \cdot \left({a}^{2} \cdot b\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(b \cdot \left(-1 \cdot a + \frac{1}{2} \cdot \left({a}^{2} \cdot b\right)\right) + 1\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(b, -1 \cdot a + \frac{1}{2} \cdot \left({a}^{2} \cdot b\right), 1\right)} \]

      3. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \color{blue}{\frac{1}{2} \cdot \left({a}^{2} \cdot b\right) + -1 \cdot a}, 1\right) \]

      4. *-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \color{blue}{\left({a}^{2} \cdot b\right) \cdot \frac{1}{2}} + -1 \cdot a, 1\right) \]

      5. *-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \color{blue}{\left(b \cdot {a}^{2}\right)} \cdot \frac{1}{2} + -1 \cdot a, 1\right) \]

      6. associate-*r* N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \color{blue}{b \cdot \left({a}^{2} \cdot \frac{1}{2}\right)} + -1 \cdot a, 1\right) \]

      7. *-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, b \cdot \color{blue}{\left(\frac{1}{2} \cdot {a}^{2}\right)} + -1 \cdot a, 1\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, \frac{1}{2} \cdot {a}^{2}, -1 \cdot a\right)}, 1\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\frac{1}{2} \cdot {a}^{2}}, -1 \cdot a\right), 1\right) \]

      10. unpow2 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{1}{2} \cdot \color{blue}{\left(a \cdot a\right)}, -1 \cdot a\right), 1\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{1}{2} \cdot \color{blue}{\left(a \cdot a\right)}, -1 \cdot a\right), 1\right) \]

      12. mul-1-neg N/A

          \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{1}{2} \cdot \left(a \cdot a\right), \color{blue}{\mathsf{neg}\left(a\right)}\right), 1\right) \]

      13. neg-lowering-neg.f64 86.5

          \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5 \cdot \left(a \cdot a\right), \color{blue}{-a}\right), 1\right) \]

   8. Simplified 86.5%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5 \cdot \left(a \cdot a\right), -a\right), 1\right)} \]

   if 2e12 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 1e80

   1. Initial program 85.9%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      3. *-lowering-*.f64 30.7

         \[\leadsto x \cdot e^{-\color{blue}{a \cdot b}} \]

   5. Simplified 30.7%

      \[\leadsto x \cdot e^{\color{blue}{-a \cdot b}} \]

   6. Taylor expanded in a around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right) + 1\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, -1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right)} \]

      3. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right) \]

      4. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) + \left(\mathsf{neg}\left(b\right)\right)}, 1\right) \]

      5. associate-*r* N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\frac{1}{2} \cdot a\right) \cdot {b}^{2}} + \left(\mathsf{neg}\left(b\right)\right), 1\right) \]

      6. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \left(\frac{1}{2} \cdot a\right) \cdot \color{blue}{\left(b \cdot b\right)} + \left(\mathsf{neg}\left(b\right)\right), 1\right) \]

      7. associate-*r* N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\left(\frac{1}{2} \cdot a\right) \cdot b\right) \cdot b} + \left(\mathsf{neg}\left(b\right)\right), 1\right) \]

      8. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \left(\left(\frac{1}{2} \cdot a\right) \cdot b\right) \cdot b + \color{blue}{-1 \cdot b}, 1\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{b \cdot \left(\left(\frac{1}{2} \cdot a\right) \cdot b + -1\right)}, 1\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{b \cdot \left(\left(\frac{1}{2} \cdot a\right) \cdot b + -1\right)}, 1\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, b \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot a, b, -1\right)}, 1\right) \]

      12. *-lowering-*.f64 10.4

          \[\leadsto x \cdot \mathsf{fma}\left(a, b \cdot \mathsf{fma}\left(\color{blue}{0.5 \cdot a}, b, -1\right), 1\right) \]

   8. Simplified 10.4%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, b \cdot \mathsf{fma}\left(0.5 \cdot a, b, -1\right), 1\right)} \]

   9. Taylor expanded in a around inf

      \[\leadsto \color{blue}{{a}^{2} \cdot \left(-1 \cdot \frac{b \cdot x}{a} + \left(\frac{1}{2} \cdot \left({b}^{2} \cdot x\right) + \frac{x}{{a}^{2}}\right)\right)} \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \color{blue}{\left(a \cdot a\right)} \cdot \left(-1 \cdot \frac{b \cdot x}{a} + \left(\frac{1}{2} \cdot \left({b}^{2} \cdot x\right) + \frac{x}{{a}^{2}}\right)\right) \]

       2. associate-*l* N/A

          \[\leadsto \color{blue}{a \cdot \left(a \cdot \left(-1 \cdot \frac{b \cdot x}{a} + \left(\frac{1}{2} \cdot \left({b}^{2} \cdot x\right) + \frac{x}{{a}^{2}}\right)\right)\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{a \cdot \left(a \cdot \left(-1 \cdot \frac{b \cdot x}{a} + \left(\frac{1}{2} \cdot \left({b}^{2} \cdot x\right) + \frac{x}{{a}^{2}}\right)\right)\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto a \cdot \color{blue}{\left(a \cdot \left(-1 \cdot \frac{b \cdot x}{a} + \left(\frac{1}{2} \cdot \left({b}^{2} \cdot x\right) + \frac{x}{{a}^{2}}\right)\right)\right)} \]

       5. associate-+r+ N/A

          \[\leadsto a \cdot \left(a \cdot \color{blue}{\left(\left(-1 \cdot \frac{b \cdot x}{a} + \frac{1}{2} \cdot \left({b}^{2} \cdot x\right)\right) + \frac{x}{{a}^{2}}\right)}\right) \]

   11. Simplified 51.1%

       \[\leadsto \color{blue}{a \cdot \left(a \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, x \cdot 0.5, \frac{x}{-a}\right), \frac{x}{a \cdot a}\right)\right)} \]

   12. Taylor expanded in b around 0

       \[\leadsto a \cdot \left(a \cdot \color{blue}{\frac{x}{{a}^{2}}}\right) \]
   13. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto a \cdot \left(a \cdot \color{blue}{\frac{x}{{a}^{2}}}\right) \]

       2. unpow2 N/A

          \[\leadsto a \cdot \left(a \cdot \frac{x}{\color{blue}{a \cdot a}}\right) \]

       3. *-lowering-*.f64 65.2

          \[\leadsto a \cdot \left(a \cdot \frac{x}{\color{blue}{a \cdot a}}\right) \]

   14. Simplified 65.2%

       \[\leadsto a \cdot \left(a \cdot \color{blue}{\frac{x}{a \cdot a}}\right) \]

   if 1e80 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 1.99999999999999986e225

   1. Initial program 97.7%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto x \cdot e^{\color{blue}{a \cdot \left(\log \left(1 - z\right) - b\right)}} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{a \cdot \left(\log \left(1 - z\right) - b\right)}} \]

      2. --lowering--.f64 N/A

         \[\leadsto x \cdot e^{a \cdot \color{blue}{\left(\log \left(1 - z\right) - b\right)}} \]

      3. sub-neg N/A

         \[\leadsto x \cdot e^{a \cdot \left(\log \color{blue}{\left(1 + \left(\mathsf{neg}\left(z\right)\right)\right)} - b\right)} \]

      4. accelerator-lowering-log1p.f64 N/A

         \[\leadsto x \cdot e^{a \cdot \left(\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(z\right)\right)} - b\right)} \]

      5. neg-lowering-neg.f64 66.2

         \[\leadsto x \cdot e^{a \cdot \left(\mathsf{log1p}\left(\color{blue}{-z}\right) - b\right)} \]

   5. Simplified 66.2%

      \[\leadsto x \cdot e^{\color{blue}{a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)}} \]

   6. Taylor expanded in z around 0

      \[\leadsto x \cdot e^{a \cdot \color{blue}{\left(-1 \cdot z - b\right)}} \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto x \cdot e^{a \cdot \color{blue}{\left(-1 \cdot z - b\right)}} \]

      2. mul-1-neg N/A

         \[\leadsto x \cdot e^{a \cdot \left(\color{blue}{\left(\mathsf{neg}\left(z\right)\right)} - b\right)} \]

      3. neg-lowering-neg.f64 66.2

         \[\leadsto x \cdot e^{a \cdot \left(\color{blue}{\left(-z\right)} - b\right)} \]

   8. Simplified 66.2%

      \[\leadsto x \cdot e^{a \cdot \color{blue}{\left(\left(-z\right) - b\right)}} \]

   9. Taylor expanded in a around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + a \cdot \left(-1 \cdot \left(b + z\right) + \frac{1}{2} \cdot \left(a \cdot {\left(b + z\right)}^{2}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto x \cdot \color{blue}{\left(a \cdot \left(-1 \cdot \left(b + z\right) + \frac{1}{2} \cdot \left(a \cdot {\left(b + z\right)}^{2}\right)\right) + 1\right)} \]

       2. accelerator-lowering-fma.f64 N/A

          \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, -1 \cdot \left(b + z\right) + \frac{1}{2} \cdot \left(a \cdot {\left(b + z\right)}^{2}\right), 1\right)} \]

       3. +-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\frac{1}{2} \cdot \left(a \cdot {\left(b + z\right)}^{2}\right) + -1 \cdot \left(b + z\right)}, 1\right) \]

       4. mul-1-neg N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \frac{1}{2} \cdot \left(a \cdot {\left(b + z\right)}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\left(b + z\right)\right)\right)}, 1\right) \]

       5. unsub-neg N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\frac{1}{2} \cdot \left(a \cdot {\left(b + z\right)}^{2}\right) - \left(b + z\right)}, 1\right) \]

       6. --lowering--.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\frac{1}{2} \cdot \left(a \cdot {\left(b + z\right)}^{2}\right) - \left(b + z\right)}, 1\right) \]

       7. *-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(a \cdot {\left(b + z\right)}^{2}\right) \cdot \frac{1}{2}} - \left(b + z\right), 1\right) \]

       8. *-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left({\left(b + z\right)}^{2} \cdot a\right)} \cdot \frac{1}{2} - \left(b + z\right), 1\right) \]

       9. associate-*l* N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{{\left(b + z\right)}^{2} \cdot \left(a \cdot \frac{1}{2}\right)} - \left(b + z\right), 1\right) \]

       10. *-commutative N/A

           \[\leadsto x \cdot \mathsf{fma}\left(a, {\left(b + z\right)}^{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot a\right)} - \left(b + z\right), 1\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{{\left(b + z\right)}^{2} \cdot \left(\frac{1}{2} \cdot a\right)} - \left(b + z\right), 1\right) \]

       12. unpow2 N/A

           \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\left(b + z\right) \cdot \left(b + z\right)\right)} \cdot \left(\frac{1}{2} \cdot a\right) - \left(b + z\right), 1\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\left(b + z\right) \cdot \left(b + z\right)\right)} \cdot \left(\frac{1}{2} \cdot a\right) - \left(b + z\right), 1\right) \]

       14. +-lowering-+.f64 N/A

           \[\leadsto x \cdot \mathsf{fma}\left(a, \left(\color{blue}{\left(b + z\right)} \cdot \left(b + z\right)\right) \cdot \left(\frac{1}{2} \cdot a\right) - \left(b + z\right), 1\right) \]

       15. +-lowering-+.f64 N/A

           \[\leadsto x \cdot \mathsf{fma}\left(a, \left(\left(b + z\right) \cdot \color{blue}{\left(b + z\right)}\right) \cdot \left(\frac{1}{2} \cdot a\right) - \left(b + z\right), 1\right) \]

       16. *-commutative N/A

           \[\leadsto x \cdot \mathsf{fma}\left(a, \left(\left(b + z\right) \cdot \left(b + z\right)\right) \cdot \color{blue}{\left(a \cdot \frac{1}{2}\right)} - \left(b + z\right), 1\right) \]

       17. *-lowering-*.f64 N/A

           \[\leadsto x \cdot \mathsf{fma}\left(a, \left(\left(b + z\right) \cdot \left(b + z\right)\right) \cdot \color{blue}{\left(a \cdot \frac{1}{2}\right)} - \left(b + z\right), 1\right) \]

       18. +-lowering-+.f64 59.7

           \[\leadsto x \cdot \mathsf{fma}\left(a, \left(\left(b + z\right) \cdot \left(b + z\right)\right) \cdot \left(a \cdot 0.5\right) - \color{blue}{\left(b + z\right)}, 1\right) \]

   11. Simplified 59.7%

       \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, \left(\left(b + z\right) \cdot \left(b + z\right)\right) \cdot \left(a \cdot 0.5\right) - \left(b + z\right), 1\right)} \]

   if 1.99999999999999986e225 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 97.7%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(t \cdot y\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(t \cdot y\right)}} \]

      2. *-commutative N/A

         \[\leadsto x \cdot e^{\mathsf{neg}\left(\color{blue}{y \cdot t}\right)} \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      5. neg-lowering-neg.f64 62.6

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(-t\right)}} \]

   5. Simplified 62.6%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{x + t \cdot \left(-1 \cdot \left(x \cdot y\right) + \frac{1}{2} \cdot \left(t \cdot \left(x \cdot {y}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(x \cdot y\right) + \frac{1}{2} \cdot \left(t \cdot \left(x \cdot {y}^{2}\right)\right)\right) + x} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, -1 \cdot \left(x \cdot y\right) + \frac{1}{2} \cdot \left(t \cdot \left(x \cdot {y}^{2}\right)\right), x\right)} \]

   8. Simplified 73.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, x \cdot \left(y \cdot \mathsf{fma}\left(0.5 \cdot t, y, -1\right)\right), x\right)} \]
3. Recombined 6 regimes into one program.

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -4 \cdot 10^{+141}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(t, \mathsf{fma}\left(t, \left(y \cdot y\right) \cdot \mathsf{fma}\left(t \cdot 0.16666666666666666, y, 0.5\right), y\right), 1\right)}\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -20000000000:\\ \;\;\;\;x \cdot \left(b \cdot \left(0.5 \cdot \left(b \cdot \left(a \cdot a\right)\right)\right)\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 2000000000000:\\ \;\;\;\;x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5 \cdot \left(a \cdot a\right), -a\right), 1\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 10^{+80}:\\ \;\;\;\;a \cdot \left(a \cdot \frac{x}{a \cdot a}\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 2 \cdot 10^{+225}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(a, \left(\left(z + b\right) \cdot \left(z + b\right)\right) \cdot \left(a \cdot 0.5\right) - \left(z + b\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, x \cdot \left(y \cdot \mathsf{fma}\left(t \cdot 0.5, y, -1\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 59.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+244}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(t \cdot t, y \cdot 0.5, t\right), 1\right)}\\ \mathbf{elif}\;t\_1 \leq -20000000000:\\ \;\;\;\;x \cdot \left(b \cdot \left(0.5 \cdot \left(b \cdot \left(a \cdot a\right)\right)\right)\right)\\ \mathbf{elif}\;t\_1 \leq 2000000000000:\\ \;\;\;\;x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5 \cdot \left(a \cdot a\right), -a\right), 1\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+80}:\\ \;\;\;\;a \cdot \left(a \cdot \frac{x}{a \cdot a}\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+225}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(a, \left(\left(z + b\right) \cdot \left(z + b\right)\right) \cdot \left(a \cdot 0.5\right) - \left(z + b\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, x \cdot \left(y \cdot \mathsf{fma}\left(t \cdot 0.5, y, -1\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b)))))
   (if (<= t_1 -4e+244)
     (/ x (fma y (fma (* t t) (* y 0.5) t) 1.0))
     (if (<= t_1 -20000000000.0)
       (* x (* b (* 0.5 (* b (* a a)))))
       (if (<= t_1 2000000000000.0)
         (* x (fma b (fma b (* 0.5 (* a a)) (- a)) 1.0))
         (if (<= t_1 1e+80)
           (* a (* a (/ x (* a a))))
           (if (<= t_1 2e+225)
             (* x (fma a (- (* (* (+ z b) (+ z b)) (* a 0.5)) (+ z b)) 1.0))
             (fma t (* x (* y (fma (* t 0.5) y -1.0))) x))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	double tmp;
	if (t_1 <= -4e+244) {
		tmp = x / fma(y, fma((t * t), (y * 0.5), t), 1.0);
	} else if (t_1 <= -20000000000.0) {
		tmp = x * (b * (0.5 * (b * (a * a))));
	} else if (t_1 <= 2000000000000.0) {
		tmp = x * fma(b, fma(b, (0.5 * (a * a)), -a), 1.0);
	} else if (t_1 <= 1e+80) {
		tmp = a * (a * (x / (a * a)));
	} else if (t_1 <= 2e+225) {
		tmp = x * fma(a, ((((z + b) * (z + b)) * (a * 0.5)) - (z + b)), 1.0);
	} else {
		tmp = fma(t, (x * (y * fma((t * 0.5), y, -1.0))), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))
	tmp = 0.0
	if (t_1 <= -4e+244)
		tmp = Float64(x / fma(y, fma(Float64(t * t), Float64(y * 0.5), t), 1.0));
	elseif (t_1 <= -20000000000.0)
		tmp = Float64(x * Float64(b * Float64(0.5 * Float64(b * Float64(a * a)))));
	elseif (t_1 <= 2000000000000.0)
		tmp = Float64(x * fma(b, fma(b, Float64(0.5 * Float64(a * a)), Float64(-a)), 1.0));
	elseif (t_1 <= 1e+80)
		tmp = Float64(a * Float64(a * Float64(x / Float64(a * a))));
	elseif (t_1 <= 2e+225)
		tmp = Float64(x * fma(a, Float64(Float64(Float64(Float64(z + b) * Float64(z + b)) * Float64(a * 0.5)) - Float64(z + b)), 1.0));
	else
		tmp = fma(t, Float64(x * Float64(y * fma(Float64(t * 0.5), y, -1.0))), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+244], N[(x / N[(y * N[(N[(t * t), $MachinePrecision] * N[(y * 0.5), $MachinePrecision] + t), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -20000000000.0], N[(x * N[(b * N[(0.5 * N[(b * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2000000000000.0], N[(x * N[(b * N[(b * N[(0.5 * N[(a * a), $MachinePrecision]), $MachinePrecision] + (-a)), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+80], N[(a * N[(a * N[(x / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+225], N[(x * N[(a * N[(N[(N[(N[(z + b), $MachinePrecision] * N[(z + b), $MachinePrecision]), $MachinePrecision] * N[(a * 0.5), $MachinePrecision]), $MachinePrecision] - N[(z + b), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(t * N[(x * N[(y * N[(N[(t * 0.5), $MachinePrecision] * y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]]]]]

Derivation

1. Split input into 6 regimes

2. if (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -4.0000000000000003e244

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(t \cdot y\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(t \cdot y\right)}} \]

      2. *-commutative N/A

         \[\leadsto x \cdot e^{\mathsf{neg}\left(\color{blue}{y \cdot t}\right)} \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      5. neg-lowering-neg.f64 52.7

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(-t\right)}} \]

   5. Simplified 52.7%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}} \]
   6. Step-by-step derivation

      1. exp-prod N/A

         \[\leadsto x \cdot \color{blue}{{\left(e^{y}\right)}^{\left(\mathsf{neg}\left(t\right)\right)}} \]

      2. pow-neg N/A

         \[\leadsto x \cdot \color{blue}{\frac{1}{{\left(e^{y}\right)}^{t}}} \]

      3. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{x}{{\left(e^{y}\right)}^{t}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{{\left(e^{y}\right)}^{t}}} \]

      5. pow-exp N/A

         \[\leadsto \frac{x}{\color{blue}{e^{y \cdot t}}} \]

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{e^{y \cdot t}}} \]

      7. *-lowering-*.f64 52.7

         \[\leadsto \frac{x}{e^{\color{blue}{y \cdot t}}} \]

   7. Applied egg-rr 52.7%

      \[\leadsto \color{blue}{\frac{x}{e^{y \cdot t}}} \]

   8. Taylor expanded in y around 0

      \[\leadsto \frac{x}{\color{blue}{1 + y \cdot \left(t + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right)\right)}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot \left(t + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right)\right) + 1}} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(y, t + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right), 1\right)}} \]

      3. +-commutative N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(y, \color{blue}{\frac{1}{2} \cdot \left({t}^{2} \cdot y\right) + t}, 1\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(y, \color{blue}{\left({t}^{2} \cdot y\right) \cdot \frac{1}{2}} + t, 1\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(y, \color{blue}{{t}^{2} \cdot \left(y \cdot \frac{1}{2}\right)} + t, 1\right)} \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left({t}^{2}, y \cdot \frac{1}{2}, t\right)}, 1\right)} \]

      7. unpow2 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(\color{blue}{t \cdot t}, y \cdot \frac{1}{2}, t\right), 1\right)} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(\color{blue}{t \cdot t}, y \cdot \frac{1}{2}, t\right), 1\right)} \]

      9. *-lowering-*.f64 65.5

         \[\leadsto \frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(t \cdot t, \color{blue}{y \cdot 0.5}, t\right), 1\right)} \]

   10. Simplified 65.5%

       \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(t \cdot t, y \cdot 0.5, t\right), 1\right)}} \]

   if -4.0000000000000003e244 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -2e10

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      3. *-lowering-*.f64 42.4

         \[\leadsto x \cdot e^{-\color{blue}{a \cdot b}} \]

   5. Simplified 42.4%

      \[\leadsto x \cdot e^{\color{blue}{-a \cdot b}} \]

   6. Taylor expanded in a around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right) + 1\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, -1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right)} \]

      3. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right) \]

      4. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) + \left(\mathsf{neg}\left(b\right)\right)}, 1\right) \]

      5. associate-*r* N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\frac{1}{2} \cdot a\right) \cdot {b}^{2}} + \left(\mathsf{neg}\left(b\right)\right), 1\right) \]

      6. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \left(\frac{1}{2} \cdot a\right) \cdot \color{blue}{\left(b \cdot b\right)} + \left(\mathsf{neg}\left(b\right)\right), 1\right) \]

      7. associate-*r* N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\left(\frac{1}{2} \cdot a\right) \cdot b\right) \cdot b} + \left(\mathsf{neg}\left(b\right)\right), 1\right) \]

      8. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \left(\left(\frac{1}{2} \cdot a\right) \cdot b\right) \cdot b + \color{blue}{-1 \cdot b}, 1\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{b \cdot \left(\left(\frac{1}{2} \cdot a\right) \cdot b + -1\right)}, 1\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{b \cdot \left(\left(\frac{1}{2} \cdot a\right) \cdot b + -1\right)}, 1\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, b \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot a, b, -1\right)}, 1\right) \]

      12. *-lowering-*.f64 3.0

          \[\leadsto x \cdot \mathsf{fma}\left(a, b \cdot \mathsf{fma}\left(\color{blue}{0.5 \cdot a}, b, -1\right), 1\right) \]

   8. Simplified 3.0%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, b \cdot \mathsf{fma}\left(0.5 \cdot a, b, -1\right), 1\right)} \]

   9. Taylor expanded in b around inf

      \[\leadsto x \cdot \color{blue}{\left({b}^{2} \cdot \left(-1 \cdot \frac{a}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto x \cdot \left(\color{blue}{\left(b \cdot b\right)} \cdot \left(-1 \cdot \frac{a}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right) \]

       2. associate-*l* N/A

          \[\leadsto x \cdot \color{blue}{\left(b \cdot \left(b \cdot \left(-1 \cdot \frac{a}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \color{blue}{\left(b \cdot \left(b \cdot \left(-1 \cdot \frac{a}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \left(b \cdot \color{blue}{\left(b \cdot \left(-1 \cdot \frac{a}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)}\right) \]

       5. associate-*r/ N/A

          \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(\color{blue}{\frac{-1 \cdot a}{b}} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(\frac{\color{blue}{a \cdot -1}}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       7. associate-/l* N/A

          \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(\color{blue}{a \cdot \frac{-1}{b}} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       8. metadata-eval N/A

          \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \frac{\color{blue}{\mathsf{neg}\left(1\right)}}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       9. distribute-neg-frac N/A

          \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{b}\right)\right)} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       10. *-commutative N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(\mathsf{neg}\left(\frac{1}{b}\right)\right) + \color{blue}{{a}^{2} \cdot \frac{1}{2}}\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(\mathsf{neg}\left(\frac{1}{b}\right)\right) + \color{blue}{\left(a \cdot a\right)} \cdot \frac{1}{2}\right)\right)\right) \]

       12. associate-*l* N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(\mathsf{neg}\left(\frac{1}{b}\right)\right) + \color{blue}{a \cdot \left(a \cdot \frac{1}{2}\right)}\right)\right)\right) \]

       13. *-commutative N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(\mathsf{neg}\left(\frac{1}{b}\right)\right) + a \cdot \color{blue}{\left(\frac{1}{2} \cdot a\right)}\right)\right)\right) \]

       14. distribute-lft-out N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \color{blue}{\left(a \cdot \left(\left(\mathsf{neg}\left(\frac{1}{b}\right)\right) + \frac{1}{2} \cdot a\right)\right)}\right)\right) \]

       15. +-commutative N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \color{blue}{\left(\frac{1}{2} \cdot a + \left(\mathsf{neg}\left(\frac{1}{b}\right)\right)\right)}\right)\right)\right) \]

       16. sub-neg N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \color{blue}{\left(\frac{1}{2} \cdot a - \frac{1}{b}\right)}\right)\right)\right) \]

       17. *-lowering-*.f64 N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \color{blue}{\left(a \cdot \left(\frac{1}{2} \cdot a - \frac{1}{b}\right)\right)}\right)\right) \]

       18. sub-neg N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \color{blue}{\left(\frac{1}{2} \cdot a + \left(\mathsf{neg}\left(\frac{1}{b}\right)\right)\right)}\right)\right)\right) \]

       19. *-commutative N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(\color{blue}{a \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(\frac{1}{b}\right)\right)\right)\right)\right)\right) \]

       20. accelerator-lowering-fma.f64 N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \color{blue}{\mathsf{fma}\left(a, \frac{1}{2}, \mathsf{neg}\left(\frac{1}{b}\right)\right)}\right)\right)\right) \]

       21. distribute-neg-frac N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \mathsf{fma}\left(a, \frac{1}{2}, \color{blue}{\frac{\mathsf{neg}\left(1\right)}{b}}\right)\right)\right)\right) \]

       22. metadata-eval N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \mathsf{fma}\left(a, \frac{1}{2}, \frac{\color{blue}{-1}}{b}\right)\right)\right)\right) \]

       23. /-lowering-/.f64 34.0

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \mathsf{fma}\left(a, 0.5, \color{blue}{\frac{-1}{b}}\right)\right)\right)\right) \]

   11. Simplified 34.0%

       \[\leadsto x \cdot \color{blue}{\left(b \cdot \left(b \cdot \left(a \cdot \mathsf{fma}\left(a, 0.5, \frac{-1}{b}\right)\right)\right)\right)} \]

   12. Taylor expanded in b around inf

       \[\leadsto x \cdot \left(b \cdot \color{blue}{\left(\frac{1}{2} \cdot \left({a}^{2} \cdot b\right)\right)}\right) \]
   13. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \left(b \cdot \color{blue}{\left(\frac{1}{2} \cdot \left({a}^{2} \cdot b\right)\right)}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \left(b \cdot \left(\frac{1}{2} \cdot \color{blue}{\left({a}^{2} \cdot b\right)}\right)\right) \]

       3. unpow2 N/A

          \[\leadsto x \cdot \left(b \cdot \left(\frac{1}{2} \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot b\right)\right)\right) \]

       4. *-lowering-*.f64 51.1

          \[\leadsto x \cdot \left(b \cdot \left(0.5 \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot b\right)\right)\right) \]

   14. Simplified 51.1%

       \[\leadsto x \cdot \left(b \cdot \color{blue}{\left(0.5 \cdot \left(\left(a \cdot a\right) \cdot b\right)\right)}\right) \]

   if -2e10 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 2e12

   1. Initial program 89.2%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      3. *-lowering-*.f64 86.7

         \[\leadsto x \cdot e^{-\color{blue}{a \cdot b}} \]

   5. Simplified 86.7%

      \[\leadsto x \cdot e^{\color{blue}{-a \cdot b}} \]

   6. Taylor expanded in b around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + b \cdot \left(-1 \cdot a + \frac{1}{2} \cdot \left({a}^{2} \cdot b\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(b \cdot \left(-1 \cdot a + \frac{1}{2} \cdot \left({a}^{2} \cdot b\right)\right) + 1\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(b, -1 \cdot a + \frac{1}{2} \cdot \left({a}^{2} \cdot b\right), 1\right)} \]

      3. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \color{blue}{\frac{1}{2} \cdot \left({a}^{2} \cdot b\right) + -1 \cdot a}, 1\right) \]

      4. *-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \color{blue}{\left({a}^{2} \cdot b\right) \cdot \frac{1}{2}} + -1 \cdot a, 1\right) \]

      5. *-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \color{blue}{\left(b \cdot {a}^{2}\right)} \cdot \frac{1}{2} + -1 \cdot a, 1\right) \]

      6. associate-*r* N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \color{blue}{b \cdot \left({a}^{2} \cdot \frac{1}{2}\right)} + -1 \cdot a, 1\right) \]

      7. *-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, b \cdot \color{blue}{\left(\frac{1}{2} \cdot {a}^{2}\right)} + -1 \cdot a, 1\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, \frac{1}{2} \cdot {a}^{2}, -1 \cdot a\right)}, 1\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\frac{1}{2} \cdot {a}^{2}}, -1 \cdot a\right), 1\right) \]

      10. unpow2 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{1}{2} \cdot \color{blue}{\left(a \cdot a\right)}, -1 \cdot a\right), 1\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{1}{2} \cdot \color{blue}{\left(a \cdot a\right)}, -1 \cdot a\right), 1\right) \]

      12. mul-1-neg N/A

          \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{1}{2} \cdot \left(a \cdot a\right), \color{blue}{\mathsf{neg}\left(a\right)}\right), 1\right) \]

      13. neg-lowering-neg.f64 86.5

          \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5 \cdot \left(a \cdot a\right), \color{blue}{-a}\right), 1\right) \]

   8. Simplified 86.5%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5 \cdot \left(a \cdot a\right), -a\right), 1\right)} \]

   if 2e12 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 1e80

   1. Initial program 85.9%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      3. *-lowering-*.f64 30.7

         \[\leadsto x \cdot e^{-\color{blue}{a \cdot b}} \]

   5. Simplified 30.7%

      \[\leadsto x \cdot e^{\color{blue}{-a \cdot b}} \]

   6. Taylor expanded in a around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right) + 1\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, -1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right)} \]

      3. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right) \]

      4. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) + \left(\mathsf{neg}\left(b\right)\right)}, 1\right) \]

      5. associate-*r* N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\frac{1}{2} \cdot a\right) \cdot {b}^{2}} + \left(\mathsf{neg}\left(b\right)\right), 1\right) \]

      6. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \left(\frac{1}{2} \cdot a\right) \cdot \color{blue}{\left(b \cdot b\right)} + \left(\mathsf{neg}\left(b\right)\right), 1\right) \]

      7. associate-*r* N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\left(\frac{1}{2} \cdot a\right) \cdot b\right) \cdot b} + \left(\mathsf{neg}\left(b\right)\right), 1\right) \]

      8. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \left(\left(\frac{1}{2} \cdot a\right) \cdot b\right) \cdot b + \color{blue}{-1 \cdot b}, 1\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{b \cdot \left(\left(\frac{1}{2} \cdot a\right) \cdot b + -1\right)}, 1\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{b \cdot \left(\left(\frac{1}{2} \cdot a\right) \cdot b + -1\right)}, 1\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, b \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot a, b, -1\right)}, 1\right) \]

      12. *-lowering-*.f64 10.4

          \[\leadsto x \cdot \mathsf{fma}\left(a, b \cdot \mathsf{fma}\left(\color{blue}{0.5 \cdot a}, b, -1\right), 1\right) \]

   8. Simplified 10.4%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, b \cdot \mathsf{fma}\left(0.5 \cdot a, b, -1\right), 1\right)} \]

   9. Taylor expanded in a around inf

      \[\leadsto \color{blue}{{a}^{2} \cdot \left(-1 \cdot \frac{b \cdot x}{a} + \left(\frac{1}{2} \cdot \left({b}^{2} \cdot x\right) + \frac{x}{{a}^{2}}\right)\right)} \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \color{blue}{\left(a \cdot a\right)} \cdot \left(-1 \cdot \frac{b \cdot x}{a} + \left(\frac{1}{2} \cdot \left({b}^{2} \cdot x\right) + \frac{x}{{a}^{2}}\right)\right) \]

       2. associate-*l* N/A

          \[\leadsto \color{blue}{a \cdot \left(a \cdot \left(-1 \cdot \frac{b \cdot x}{a} + \left(\frac{1}{2} \cdot \left({b}^{2} \cdot x\right) + \frac{x}{{a}^{2}}\right)\right)\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{a \cdot \left(a \cdot \left(-1 \cdot \frac{b \cdot x}{a} + \left(\frac{1}{2} \cdot \left({b}^{2} \cdot x\right) + \frac{x}{{a}^{2}}\right)\right)\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto a \cdot \color{blue}{\left(a \cdot \left(-1 \cdot \frac{b \cdot x}{a} + \left(\frac{1}{2} \cdot \left({b}^{2} \cdot x\right) + \frac{x}{{a}^{2}}\right)\right)\right)} \]

       5. associate-+r+ N/A

          \[\leadsto a \cdot \left(a \cdot \color{blue}{\left(\left(-1 \cdot \frac{b \cdot x}{a} + \frac{1}{2} \cdot \left({b}^{2} \cdot x\right)\right) + \frac{x}{{a}^{2}}\right)}\right) \]

   11. Simplified 51.1%

       \[\leadsto \color{blue}{a \cdot \left(a \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, x \cdot 0.5, \frac{x}{-a}\right), \frac{x}{a \cdot a}\right)\right)} \]

   12. Taylor expanded in b around 0

       \[\leadsto a \cdot \left(a \cdot \color{blue}{\frac{x}{{a}^{2}}}\right) \]
   13. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto a \cdot \left(a \cdot \color{blue}{\frac{x}{{a}^{2}}}\right) \]

       2. unpow2 N/A

          \[\leadsto a \cdot \left(a \cdot \frac{x}{\color{blue}{a \cdot a}}\right) \]

       3. *-lowering-*.f64 65.2

          \[\leadsto a \cdot \left(a \cdot \frac{x}{\color{blue}{a \cdot a}}\right) \]

   14. Simplified 65.2%

       \[\leadsto a \cdot \left(a \cdot \color{blue}{\frac{x}{a \cdot a}}\right) \]

   if 1e80 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 1.99999999999999986e225

   1. Initial program 97.7%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto x \cdot e^{\color{blue}{a \cdot \left(\log \left(1 - z\right) - b\right)}} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{a \cdot \left(\log \left(1 - z\right) - b\right)}} \]

      2. --lowering--.f64 N/A

         \[\leadsto x \cdot e^{a \cdot \color{blue}{\left(\log \left(1 - z\right) - b\right)}} \]

      3. sub-neg N/A

         \[\leadsto x \cdot e^{a \cdot \left(\log \color{blue}{\left(1 + \left(\mathsf{neg}\left(z\right)\right)\right)} - b\right)} \]

      4. accelerator-lowering-log1p.f64 N/A

         \[\leadsto x \cdot e^{a \cdot \left(\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(z\right)\right)} - b\right)} \]

      5. neg-lowering-neg.f64 66.2

         \[\leadsto x \cdot e^{a \cdot \left(\mathsf{log1p}\left(\color{blue}{-z}\right) - b\right)} \]

   5. Simplified 66.2%

      \[\leadsto x \cdot e^{\color{blue}{a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)}} \]

   6. Taylor expanded in z around 0

      \[\leadsto x \cdot e^{a \cdot \color{blue}{\left(-1 \cdot z - b\right)}} \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto x \cdot e^{a \cdot \color{blue}{\left(-1 \cdot z - b\right)}} \]

      2. mul-1-neg N/A

         \[\leadsto x \cdot e^{a \cdot \left(\color{blue}{\left(\mathsf{neg}\left(z\right)\right)} - b\right)} \]

      3. neg-lowering-neg.f64 66.2

         \[\leadsto x \cdot e^{a \cdot \left(\color{blue}{\left(-z\right)} - b\right)} \]

   8. Simplified 66.2%

      \[\leadsto x \cdot e^{a \cdot \color{blue}{\left(\left(-z\right) - b\right)}} \]

   9. Taylor expanded in a around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + a \cdot \left(-1 \cdot \left(b + z\right) + \frac{1}{2} \cdot \left(a \cdot {\left(b + z\right)}^{2}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto x \cdot \color{blue}{\left(a \cdot \left(-1 \cdot \left(b + z\right) + \frac{1}{2} \cdot \left(a \cdot {\left(b + z\right)}^{2}\right)\right) + 1\right)} \]

       2. accelerator-lowering-fma.f64 N/A

          \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, -1 \cdot \left(b + z\right) + \frac{1}{2} \cdot \left(a \cdot {\left(b + z\right)}^{2}\right), 1\right)} \]

       3. +-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\frac{1}{2} \cdot \left(a \cdot {\left(b + z\right)}^{2}\right) + -1 \cdot \left(b + z\right)}, 1\right) \]

       4. mul-1-neg N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \frac{1}{2} \cdot \left(a \cdot {\left(b + z\right)}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\left(b + z\right)\right)\right)}, 1\right) \]

       5. unsub-neg N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\frac{1}{2} \cdot \left(a \cdot {\left(b + z\right)}^{2}\right) - \left(b + z\right)}, 1\right) \]

       6. --lowering--.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\frac{1}{2} \cdot \left(a \cdot {\left(b + z\right)}^{2}\right) - \left(b + z\right)}, 1\right) \]

       7. *-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(a \cdot {\left(b + z\right)}^{2}\right) \cdot \frac{1}{2}} - \left(b + z\right), 1\right) \]

       8. *-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left({\left(b + z\right)}^{2} \cdot a\right)} \cdot \frac{1}{2} - \left(b + z\right), 1\right) \]

       9. associate-*l* N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{{\left(b + z\right)}^{2} \cdot \left(a \cdot \frac{1}{2}\right)} - \left(b + z\right), 1\right) \]

       10. *-commutative N/A

           \[\leadsto x \cdot \mathsf{fma}\left(a, {\left(b + z\right)}^{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot a\right)} - \left(b + z\right), 1\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{{\left(b + z\right)}^{2} \cdot \left(\frac{1}{2} \cdot a\right)} - \left(b + z\right), 1\right) \]

       12. unpow2 N/A

           \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\left(b + z\right) \cdot \left(b + z\right)\right)} \cdot \left(\frac{1}{2} \cdot a\right) - \left(b + z\right), 1\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\left(b + z\right) \cdot \left(b + z\right)\right)} \cdot \left(\frac{1}{2} \cdot a\right) - \left(b + z\right), 1\right) \]

       14. +-lowering-+.f64 N/A

           \[\leadsto x \cdot \mathsf{fma}\left(a, \left(\color{blue}{\left(b + z\right)} \cdot \left(b + z\right)\right) \cdot \left(\frac{1}{2} \cdot a\right) - \left(b + z\right), 1\right) \]

       15. +-lowering-+.f64 N/A

           \[\leadsto x \cdot \mathsf{fma}\left(a, \left(\left(b + z\right) \cdot \color{blue}{\left(b + z\right)}\right) \cdot \left(\frac{1}{2} \cdot a\right) - \left(b + z\right), 1\right) \]

       16. *-commutative N/A

           \[\leadsto x \cdot \mathsf{fma}\left(a, \left(\left(b + z\right) \cdot \left(b + z\right)\right) \cdot \color{blue}{\left(a \cdot \frac{1}{2}\right)} - \left(b + z\right), 1\right) \]

       17. *-lowering-*.f64 N/A

           \[\leadsto x \cdot \mathsf{fma}\left(a, \left(\left(b + z\right) \cdot \left(b + z\right)\right) \cdot \color{blue}{\left(a \cdot \frac{1}{2}\right)} - \left(b + z\right), 1\right) \]

       18. +-lowering-+.f64 59.7

           \[\leadsto x \cdot \mathsf{fma}\left(a, \left(\left(b + z\right) \cdot \left(b + z\right)\right) \cdot \left(a \cdot 0.5\right) - \color{blue}{\left(b + z\right)}, 1\right) \]

   11. Simplified 59.7%

       \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, \left(\left(b + z\right) \cdot \left(b + z\right)\right) \cdot \left(a \cdot 0.5\right) - \left(b + z\right), 1\right)} \]

   if 1.99999999999999986e225 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 97.7%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(t \cdot y\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(t \cdot y\right)}} \]

      2. *-commutative N/A

         \[\leadsto x \cdot e^{\mathsf{neg}\left(\color{blue}{y \cdot t}\right)} \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      5. neg-lowering-neg.f64 62.6

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(-t\right)}} \]

   5. Simplified 62.6%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{x + t \cdot \left(-1 \cdot \left(x \cdot y\right) + \frac{1}{2} \cdot \left(t \cdot \left(x \cdot {y}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(x \cdot y\right) + \frac{1}{2} \cdot \left(t \cdot \left(x \cdot {y}^{2}\right)\right)\right) + x} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, -1 \cdot \left(x \cdot y\right) + \frac{1}{2} \cdot \left(t \cdot \left(x \cdot {y}^{2}\right)\right), x\right)} \]

   8. Simplified 73.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, x \cdot \left(y \cdot \mathsf{fma}\left(0.5 \cdot t, y, -1\right)\right), x\right)} \]
3. Recombined 6 regimes into one program.

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -4 \cdot 10^{+244}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(t \cdot t, y \cdot 0.5, t\right), 1\right)}\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -20000000000:\\ \;\;\;\;x \cdot \left(b \cdot \left(0.5 \cdot \left(b \cdot \left(a \cdot a\right)\right)\right)\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 2000000000000:\\ \;\;\;\;x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5 \cdot \left(a \cdot a\right), -a\right), 1\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 10^{+80}:\\ \;\;\;\;a \cdot \left(a \cdot \frac{x}{a \cdot a}\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 2 \cdot 10^{+225}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(a, \left(\left(z + b\right) \cdot \left(z + b\right)\right) \cdot \left(a \cdot 0.5\right) - \left(z + b\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, x \cdot \left(y \cdot \mathsf{fma}\left(t \cdot 0.5, y, -1\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 56.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \left(b \cdot \left(0.5 \cdot \left(b \cdot \left(a \cdot a\right)\right)\right)\right)\\ t_2 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ t_3 := a \cdot \left(a \cdot \frac{x}{a \cdot a}\right)\\ \mathbf{if}\;t\_2 \leq -1.5 \cdot 10^{+286}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(t, y, 1\right)}\\ \mathbf{elif}\;t\_2 \leq -4 \cdot 10^{+244}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq -20000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2000000000000:\\ \;\;\;\;\mathsf{fma}\left(\left(-z\right) - b, x \cdot a, x\right)\\ \mathbf{elif}\;t\_2 \leq 10^{+104}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (* b (* 0.5 (* b (* a a))))))
        (t_2 (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b))))
        (t_3 (* a (* a (/ x (* a a))))))
   (if (<= t_2 -1.5e+286)
     (/ x (fma t y 1.0))
     (if (<= t_2 -4e+244)
       t_3
       (if (<= t_2 -20000000000.0)
         t_1
         (if (<= t_2 2000000000000.0)
           (fma (- (- z) b) (* x a) x)
           (if (<= t_2 1e+104) t_3 t_1)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * (b * (0.5 * (b * (a * a))));
	double t_2 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	double t_3 = a * (a * (x / (a * a)));
	double tmp;
	if (t_2 <= -1.5e+286) {
		tmp = x / fma(t, y, 1.0);
	} else if (t_2 <= -4e+244) {
		tmp = t_3;
	} else if (t_2 <= -20000000000.0) {
		tmp = t_1;
	} else if (t_2 <= 2000000000000.0) {
		tmp = fma((-z - b), (x * a), x);
	} else if (t_2 <= 1e+104) {
		tmp = t_3;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x * Float64(b * Float64(0.5 * Float64(b * Float64(a * a)))))
	t_2 = Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))
	t_3 = Float64(a * Float64(a * Float64(x / Float64(a * a))))
	tmp = 0.0
	if (t_2 <= -1.5e+286)
		tmp = Float64(x / fma(t, y, 1.0));
	elseif (t_2 <= -4e+244)
		tmp = t_3;
	elseif (t_2 <= -20000000000.0)
		tmp = t_1;
	elseif (t_2 <= 2000000000000.0)
		tmp = fma(Float64(Float64(-z) - b), Float64(x * a), x);
	elseif (t_2 <= 1e+104)
		tmp = t_3;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x * N[(b * N[(0.5 * N[(b * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(a * N[(a * N[(x / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1.5e+286], N[(x / N[(t * y + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -4e+244], t$95$3, If[LessEqual[t$95$2, -20000000000.0], t$95$1, If[LessEqual[t$95$2, 2000000000000.0], N[(N[((-z) - b), $MachinePrecision] * N[(x * a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$2, 1e+104], t$95$3, t$95$1]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -1.4999999999999999e286

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(t \cdot y\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(t \cdot y\right)}} \]

      2. *-commutative N/A

         \[\leadsto x \cdot e^{\mathsf{neg}\left(\color{blue}{y \cdot t}\right)} \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      5. neg-lowering-neg.f64 66.3

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(-t\right)}} \]

   5. Simplified 66.3%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}} \]
   6. Step-by-step derivation

      1. exp-prod N/A

         \[\leadsto x \cdot \color{blue}{{\left(e^{y}\right)}^{\left(\mathsf{neg}\left(t\right)\right)}} \]

      2. pow-neg N/A

         \[\leadsto x \cdot \color{blue}{\frac{1}{{\left(e^{y}\right)}^{t}}} \]

      3. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{x}{{\left(e^{y}\right)}^{t}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{{\left(e^{y}\right)}^{t}}} \]

      5. pow-exp N/A

         \[\leadsto \frac{x}{\color{blue}{e^{y \cdot t}}} \]

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{e^{y \cdot t}}} \]

      7. *-lowering-*.f64 66.3

         \[\leadsto \frac{x}{e^{\color{blue}{y \cdot t}}} \]

   7. Applied egg-rr 66.3%

      \[\leadsto \color{blue}{\frac{x}{e^{y \cdot t}}} \]

   8. Taylor expanded in y around 0

      \[\leadsto \frac{x}{\color{blue}{1 + t \cdot y}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{t \cdot y + 1}} \]

      2. accelerator-lowering-fma.f64 59.4

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(t, y, 1\right)}} \]

   10. Simplified 59.4%

       \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(t, y, 1\right)}} \]

   if -1.4999999999999999e286 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -4.0000000000000003e244 or 2e12 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 1e104

   1. Initial program 93.8%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      3. *-lowering-*.f64 48.4

         \[\leadsto x \cdot e^{-\color{blue}{a \cdot b}} \]

   5. Simplified 48.4%

      \[\leadsto x \cdot e^{\color{blue}{-a \cdot b}} \]

   6. Taylor expanded in a around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right) + 1\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, -1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right)} \]

      3. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right) \]

      4. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) + \left(\mathsf{neg}\left(b\right)\right)}, 1\right) \]

      5. associate-*r* N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\frac{1}{2} \cdot a\right) \cdot {b}^{2}} + \left(\mathsf{neg}\left(b\right)\right), 1\right) \]

      6. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \left(\frac{1}{2} \cdot a\right) \cdot \color{blue}{\left(b \cdot b\right)} + \left(\mathsf{neg}\left(b\right)\right), 1\right) \]

      7. associate-*r* N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\left(\frac{1}{2} \cdot a\right) \cdot b\right) \cdot b} + \left(\mathsf{neg}\left(b\right)\right), 1\right) \]

      8. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \left(\left(\frac{1}{2} \cdot a\right) \cdot b\right) \cdot b + \color{blue}{-1 \cdot b}, 1\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{b \cdot \left(\left(\frac{1}{2} \cdot a\right) \cdot b + -1\right)}, 1\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{b \cdot \left(\left(\frac{1}{2} \cdot a\right) \cdot b + -1\right)}, 1\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, b \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot a, b, -1\right)}, 1\right) \]

      12. *-lowering-*.f64 12.0

          \[\leadsto x \cdot \mathsf{fma}\left(a, b \cdot \mathsf{fma}\left(\color{blue}{0.5 \cdot a}, b, -1\right), 1\right) \]

   8. Simplified 12.0%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, b \cdot \mathsf{fma}\left(0.5 \cdot a, b, -1\right), 1\right)} \]

   9. Taylor expanded in a around inf

      \[\leadsto \color{blue}{{a}^{2} \cdot \left(-1 \cdot \frac{b \cdot x}{a} + \left(\frac{1}{2} \cdot \left({b}^{2} \cdot x\right) + \frac{x}{{a}^{2}}\right)\right)} \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \color{blue}{\left(a \cdot a\right)} \cdot \left(-1 \cdot \frac{b \cdot x}{a} + \left(\frac{1}{2} \cdot \left({b}^{2} \cdot x\right) + \frac{x}{{a}^{2}}\right)\right) \]

       2. associate-*l* N/A

          \[\leadsto \color{blue}{a \cdot \left(a \cdot \left(-1 \cdot \frac{b \cdot x}{a} + \left(\frac{1}{2} \cdot \left({b}^{2} \cdot x\right) + \frac{x}{{a}^{2}}\right)\right)\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{a \cdot \left(a \cdot \left(-1 \cdot \frac{b \cdot x}{a} + \left(\frac{1}{2} \cdot \left({b}^{2} \cdot x\right) + \frac{x}{{a}^{2}}\right)\right)\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto a \cdot \color{blue}{\left(a \cdot \left(-1 \cdot \frac{b \cdot x}{a} + \left(\frac{1}{2} \cdot \left({b}^{2} \cdot x\right) + \frac{x}{{a}^{2}}\right)\right)\right)} \]

       5. associate-+r+ N/A

          \[\leadsto a \cdot \left(a \cdot \color{blue}{\left(\left(-1 \cdot \frac{b \cdot x}{a} + \frac{1}{2} \cdot \left({b}^{2} \cdot x\right)\right) + \frac{x}{{a}^{2}}\right)}\right) \]

   11. Simplified 32.6%

       \[\leadsto \color{blue}{a \cdot \left(a \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, x \cdot 0.5, \frac{x}{-a}\right), \frac{x}{a \cdot a}\right)\right)} \]

   12. Taylor expanded in b around 0

       \[\leadsto a \cdot \left(a \cdot \color{blue}{\frac{x}{{a}^{2}}}\right) \]
   13. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto a \cdot \left(a \cdot \color{blue}{\frac{x}{{a}^{2}}}\right) \]

       2. unpow2 N/A

          \[\leadsto a \cdot \left(a \cdot \frac{x}{\color{blue}{a \cdot a}}\right) \]

       3. *-lowering-*.f64 57.5

          \[\leadsto a \cdot \left(a \cdot \frac{x}{\color{blue}{a \cdot a}}\right) \]

   14. Simplified 57.5%

       \[\leadsto a \cdot \left(a \cdot \color{blue}{\frac{x}{a \cdot a}}\right) \]

   if -4.0000000000000003e244 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -2e10 or 1e104 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 98.6%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      3. *-lowering-*.f64 47.0

         \[\leadsto x \cdot e^{-\color{blue}{a \cdot b}} \]

   5. Simplified 47.0%

      \[\leadsto x \cdot e^{\color{blue}{-a \cdot b}} \]

   6. Taylor expanded in a around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right) + 1\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, -1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right)} \]

      3. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right) \]

      4. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) + \left(\mathsf{neg}\left(b\right)\right)}, 1\right) \]

      5. associate-*r* N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\frac{1}{2} \cdot a\right) \cdot {b}^{2}} + \left(\mathsf{neg}\left(b\right)\right), 1\right) \]

      6. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \left(\frac{1}{2} \cdot a\right) \cdot \color{blue}{\left(b \cdot b\right)} + \left(\mathsf{neg}\left(b\right)\right), 1\right) \]

      7. associate-*r* N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\left(\frac{1}{2} \cdot a\right) \cdot b\right) \cdot b} + \left(\mathsf{neg}\left(b\right)\right), 1\right) \]

      8. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \left(\left(\frac{1}{2} \cdot a\right) \cdot b\right) \cdot b + \color{blue}{-1 \cdot b}, 1\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{b \cdot \left(\left(\frac{1}{2} \cdot a\right) \cdot b + -1\right)}, 1\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{b \cdot \left(\left(\frac{1}{2} \cdot a\right) \cdot b + -1\right)}, 1\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, b \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot a, b, -1\right)}, 1\right) \]

      12. *-lowering-*.f64 29.5

          \[\leadsto x \cdot \mathsf{fma}\left(a, b \cdot \mathsf{fma}\left(\color{blue}{0.5 \cdot a}, b, -1\right), 1\right) \]

   8. Simplified 29.5%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, b \cdot \mathsf{fma}\left(0.5 \cdot a, b, -1\right), 1\right)} \]

   9. Taylor expanded in b around inf

      \[\leadsto x \cdot \color{blue}{\left({b}^{2} \cdot \left(-1 \cdot \frac{a}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto x \cdot \left(\color{blue}{\left(b \cdot b\right)} \cdot \left(-1 \cdot \frac{a}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right) \]

       2. associate-*l* N/A

          \[\leadsto x \cdot \color{blue}{\left(b \cdot \left(b \cdot \left(-1 \cdot \frac{a}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \color{blue}{\left(b \cdot \left(b \cdot \left(-1 \cdot \frac{a}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \left(b \cdot \color{blue}{\left(b \cdot \left(-1 \cdot \frac{a}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)}\right) \]

       5. associate-*r/ N/A

          \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(\color{blue}{\frac{-1 \cdot a}{b}} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(\frac{\color{blue}{a \cdot -1}}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       7. associate-/l* N/A

          \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(\color{blue}{a \cdot \frac{-1}{b}} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       8. metadata-eval N/A

          \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \frac{\color{blue}{\mathsf{neg}\left(1\right)}}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       9. distribute-neg-frac N/A

          \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{b}\right)\right)} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       10. *-commutative N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(\mathsf{neg}\left(\frac{1}{b}\right)\right) + \color{blue}{{a}^{2} \cdot \frac{1}{2}}\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(\mathsf{neg}\left(\frac{1}{b}\right)\right) + \color{blue}{\left(a \cdot a\right)} \cdot \frac{1}{2}\right)\right)\right) \]

       12. associate-*l* N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(\mathsf{neg}\left(\frac{1}{b}\right)\right) + \color{blue}{a \cdot \left(a \cdot \frac{1}{2}\right)}\right)\right)\right) \]

       13. *-commutative N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(\mathsf{neg}\left(\frac{1}{b}\right)\right) + a \cdot \color{blue}{\left(\frac{1}{2} \cdot a\right)}\right)\right)\right) \]

       14. distribute-lft-out N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \color{blue}{\left(a \cdot \left(\left(\mathsf{neg}\left(\frac{1}{b}\right)\right) + \frac{1}{2} \cdot a\right)\right)}\right)\right) \]

       15. +-commutative N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \color{blue}{\left(\frac{1}{2} \cdot a + \left(\mathsf{neg}\left(\frac{1}{b}\right)\right)\right)}\right)\right)\right) \]

       16. sub-neg N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \color{blue}{\left(\frac{1}{2} \cdot a - \frac{1}{b}\right)}\right)\right)\right) \]

       17. *-lowering-*.f64 N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \color{blue}{\left(a \cdot \left(\frac{1}{2} \cdot a - \frac{1}{b}\right)\right)}\right)\right) \]

       18. sub-neg N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \color{blue}{\left(\frac{1}{2} \cdot a + \left(\mathsf{neg}\left(\frac{1}{b}\right)\right)\right)}\right)\right)\right) \]

       19. *-commutative N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(\color{blue}{a \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(\frac{1}{b}\right)\right)\right)\right)\right)\right) \]

       20. accelerator-lowering-fma.f64 N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \color{blue}{\mathsf{fma}\left(a, \frac{1}{2}, \mathsf{neg}\left(\frac{1}{b}\right)\right)}\right)\right)\right) \]

       21. distribute-neg-frac N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \mathsf{fma}\left(a, \frac{1}{2}, \color{blue}{\frac{\mathsf{neg}\left(1\right)}{b}}\right)\right)\right)\right) \]

       22. metadata-eval N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \mathsf{fma}\left(a, \frac{1}{2}, \frac{\color{blue}{-1}}{b}\right)\right)\right)\right) \]

       23. /-lowering-/.f64 46.5

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \mathsf{fma}\left(a, 0.5, \color{blue}{\frac{-1}{b}}\right)\right)\right)\right) \]

   11. Simplified 46.5%

       \[\leadsto x \cdot \color{blue}{\left(b \cdot \left(b \cdot \left(a \cdot \mathsf{fma}\left(a, 0.5, \frac{-1}{b}\right)\right)\right)\right)} \]

   12. Taylor expanded in b around inf

       \[\leadsto x \cdot \left(b \cdot \color{blue}{\left(\frac{1}{2} \cdot \left({a}^{2} \cdot b\right)\right)}\right) \]
   13. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \left(b \cdot \color{blue}{\left(\frac{1}{2} \cdot \left({a}^{2} \cdot b\right)\right)}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \left(b \cdot \left(\frac{1}{2} \cdot \color{blue}{\left({a}^{2} \cdot b\right)}\right)\right) \]

       3. unpow2 N/A

          \[\leadsto x \cdot \left(b \cdot \left(\frac{1}{2} \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot b\right)\right)\right) \]

       4. *-lowering-*.f64 51.8

          \[\leadsto x \cdot \left(b \cdot \left(0.5 \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot b\right)\right)\right) \]

   14. Simplified 51.8%

       \[\leadsto x \cdot \left(b \cdot \color{blue}{\left(0.5 \cdot \left(\left(a \cdot a\right) \cdot b\right)\right)}\right) \]

   if -2e10 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 2e12

   1. Initial program 89.2%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto x \cdot e^{\color{blue}{a \cdot \left(\log \left(1 - z\right) - b\right)}} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{a \cdot \left(\log \left(1 - z\right) - b\right)}} \]

      2. --lowering--.f64 N/A

         \[\leadsto x \cdot e^{a \cdot \color{blue}{\left(\log \left(1 - z\right) - b\right)}} \]

      3. sub-neg N/A

         \[\leadsto x \cdot e^{a \cdot \left(\log \color{blue}{\left(1 + \left(\mathsf{neg}\left(z\right)\right)\right)} - b\right)} \]

      4. accelerator-lowering-log1p.f64 N/A

         \[\leadsto x \cdot e^{a \cdot \left(\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(z\right)\right)} - b\right)} \]

      5. neg-lowering-neg.f64 97.2

         \[\leadsto x \cdot e^{a \cdot \left(\mathsf{log1p}\left(\color{blue}{-z}\right) - b\right)} \]

   5. Simplified 97.2%

      \[\leadsto x \cdot e^{\color{blue}{a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)}} \]

   6. Taylor expanded in z around 0

      \[\leadsto x \cdot e^{a \cdot \color{blue}{\left(-1 \cdot z - b\right)}} \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto x \cdot e^{a \cdot \color{blue}{\left(-1 \cdot z - b\right)}} \]

      2. mul-1-neg N/A

         \[\leadsto x \cdot e^{a \cdot \left(\color{blue}{\left(\mathsf{neg}\left(z\right)\right)} - b\right)} \]

      3. neg-lowering-neg.f64 97.2

         \[\leadsto x \cdot e^{a \cdot \left(\color{blue}{\left(-z\right)} - b\right)} \]

   8. Simplified 97.2%

      \[\leadsto x \cdot e^{a \cdot \color{blue}{\left(\left(-z\right) - b\right)}} \]

   9. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x + -1 \cdot \left(a \cdot \left(x \cdot \left(b + z\right)\right)\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{-1 \cdot \left(a \cdot \left(x \cdot \left(b + z\right)\right)\right) + x} \]

       2. mul-1-neg N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(a \cdot \left(x \cdot \left(b + z\right)\right)\right)\right)} + x \]

       3. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(x \cdot \left(b + z\right)\right) \cdot a}\right)\right) + x \]

       4. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\left(b + z\right) \cdot x\right)} \cdot a\right)\right) + x \]

       5. associate-*l* N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(b + z\right) \cdot \left(x \cdot a\right)}\right)\right) + x \]

       6. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(\left(b + z\right) \cdot \color{blue}{\left(a \cdot x\right)}\right)\right) + x \]

       7. distribute-lft-neg-in N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(b + z\right)\right)\right) \cdot \left(a \cdot x\right)} + x \]

       8. mul-1-neg N/A

          \[\leadsto \color{blue}{\left(-1 \cdot \left(b + z\right)\right)} \cdot \left(a \cdot x\right) + x \]

       9. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot \left(b + z\right), a \cdot x, x\right)} \]

       10. distribute-lft-in N/A

           \[\leadsto \mathsf{fma}\left(\color{blue}{-1 \cdot b + -1 \cdot z}, a \cdot x, x\right) \]

       11. +-commutative N/A

           \[\leadsto \mathsf{fma}\left(\color{blue}{-1 \cdot z + -1 \cdot b}, a \cdot x, x\right) \]

       12. mul-1-neg N/A

           \[\leadsto \mathsf{fma}\left(-1 \cdot z + \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}, a \cdot x, x\right) \]

       13. sub-neg N/A

           \[\leadsto \mathsf{fma}\left(\color{blue}{-1 \cdot z - b}, a \cdot x, x\right) \]

       14. --lowering--.f64 N/A

           \[\leadsto \mathsf{fma}\left(\color{blue}{-1 \cdot z - b}, a \cdot x, x\right) \]

       15. mul-1-neg N/A

           \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(z\right)\right)} - b, a \cdot x, x\right) \]

       16. neg-lowering-neg.f64 N/A

           \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(z\right)\right)} - b, a \cdot x, x\right) \]

       17. *-commutative N/A

           \[\leadsto \mathsf{fma}\left(\left(\mathsf{neg}\left(z\right)\right) - b, \color{blue}{x \cdot a}, x\right) \]

       18. *-lowering-*.f64 82.5

           \[\leadsto \mathsf{fma}\left(\left(-z\right) - b, \color{blue}{x \cdot a}, x\right) \]

   11. Simplified 82.5%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(-z\right) - b, x \cdot a, x\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -1.5 \cdot 10^{+286}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(t, y, 1\right)}\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -4 \cdot 10^{+244}:\\ \;\;\;\;a \cdot \left(a \cdot \frac{x}{a \cdot a}\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -20000000000:\\ \;\;\;\;x \cdot \left(b \cdot \left(0.5 \cdot \left(b \cdot \left(a \cdot a\right)\right)\right)\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 2000000000000:\\ \;\;\;\;\mathsf{fma}\left(\left(-z\right) - b, x \cdot a, x\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 10^{+104}:\\ \;\;\;\;a \cdot \left(a \cdot \frac{x}{a \cdot a}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(b \cdot \left(0.5 \cdot \left(b \cdot \left(a \cdot a\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 54.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t\_1 \leq -1.5 \cdot 10^{+286}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(t, y, 1\right)}\\ \mathbf{elif}\;t\_1 \leq -4 \cdot 10^{+244}:\\ \;\;\;\;a \cdot \left(a \cdot \frac{x}{a \cdot a}\right)\\ \mathbf{elif}\;t\_1 \leq -20000000000:\\ \;\;\;\;x \cdot \left(b \cdot \left(0.5 \cdot \left(b \cdot \left(a \cdot a\right)\right)\right)\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+225}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5 \cdot \left(a \cdot a\right), -a\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, x \cdot \left(y \cdot \mathsf{fma}\left(t \cdot 0.5, y, -1\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b)))))
   (if (<= t_1 -1.5e+286)
     (/ x (fma t y 1.0))
     (if (<= t_1 -4e+244)
       (* a (* a (/ x (* a a))))
       (if (<= t_1 -20000000000.0)
         (* x (* b (* 0.5 (* b (* a a)))))
         (if (<= t_1 2e+225)
           (* x (fma b (fma b (* 0.5 (* a a)) (- a)) 1.0))
           (fma t (* x (* y (fma (* t 0.5) y -1.0))) x)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	double tmp;
	if (t_1 <= -1.5e+286) {
		tmp = x / fma(t, y, 1.0);
	} else if (t_1 <= -4e+244) {
		tmp = a * (a * (x / (a * a)));
	} else if (t_1 <= -20000000000.0) {
		tmp = x * (b * (0.5 * (b * (a * a))));
	} else if (t_1 <= 2e+225) {
		tmp = x * fma(b, fma(b, (0.5 * (a * a)), -a), 1.0);
	} else {
		tmp = fma(t, (x * (y * fma((t * 0.5), y, -1.0))), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))
	tmp = 0.0
	if (t_1 <= -1.5e+286)
		tmp = Float64(x / fma(t, y, 1.0));
	elseif (t_1 <= -4e+244)
		tmp = Float64(a * Float64(a * Float64(x / Float64(a * a))));
	elseif (t_1 <= -20000000000.0)
		tmp = Float64(x * Float64(b * Float64(0.5 * Float64(b * Float64(a * a)))));
	elseif (t_1 <= 2e+225)
		tmp = Float64(x * fma(b, fma(b, Float64(0.5 * Float64(a * a)), Float64(-a)), 1.0));
	else
		tmp = fma(t, Float64(x * Float64(y * fma(Float64(t * 0.5), y, -1.0))), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1.5e+286], N[(x / N[(t * y + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -4e+244], N[(a * N[(a * N[(x / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -20000000000.0], N[(x * N[(b * N[(0.5 * N[(b * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+225], N[(x * N[(b * N[(b * N[(0.5 * N[(a * a), $MachinePrecision]), $MachinePrecision] + (-a)), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(t * N[(x * N[(y * N[(N[(t * 0.5), $MachinePrecision] * y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -1.4999999999999999e286

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(t \cdot y\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(t \cdot y\right)}} \]

      2. *-commutative N/A

         \[\leadsto x \cdot e^{\mathsf{neg}\left(\color{blue}{y \cdot t}\right)} \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      5. neg-lowering-neg.f64 66.3

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(-t\right)}} \]

   5. Simplified 66.3%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}} \]
   6. Step-by-step derivation

      1. exp-prod N/A

         \[\leadsto x \cdot \color{blue}{{\left(e^{y}\right)}^{\left(\mathsf{neg}\left(t\right)\right)}} \]

      2. pow-neg N/A

         \[\leadsto x \cdot \color{blue}{\frac{1}{{\left(e^{y}\right)}^{t}}} \]

      3. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{x}{{\left(e^{y}\right)}^{t}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{{\left(e^{y}\right)}^{t}}} \]

      5. pow-exp N/A

         \[\leadsto \frac{x}{\color{blue}{e^{y \cdot t}}} \]

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{e^{y \cdot t}}} \]

      7. *-lowering-*.f64 66.3

         \[\leadsto \frac{x}{e^{\color{blue}{y \cdot t}}} \]

   7. Applied egg-rr 66.3%

      \[\leadsto \color{blue}{\frac{x}{e^{y \cdot t}}} \]

   8. Taylor expanded in y around 0

      \[\leadsto \frac{x}{\color{blue}{1 + t \cdot y}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{t \cdot y + 1}} \]

      2. accelerator-lowering-fma.f64 59.4

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(t, y, 1\right)}} \]

   10. Simplified 59.4%

       \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(t, y, 1\right)}} \]

   if -1.4999999999999999e286 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -4.0000000000000003e244

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      3. *-lowering-*.f64 55.1

         \[\leadsto x \cdot e^{-\color{blue}{a \cdot b}} \]

   5. Simplified 55.1%

      \[\leadsto x \cdot e^{\color{blue}{-a \cdot b}} \]

   6. Taylor expanded in a around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right) + 1\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, -1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right)} \]

      3. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right) \]

      4. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) + \left(\mathsf{neg}\left(b\right)\right)}, 1\right) \]

      5. associate-*r* N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\frac{1}{2} \cdot a\right) \cdot {b}^{2}} + \left(\mathsf{neg}\left(b\right)\right), 1\right) \]

      6. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \left(\frac{1}{2} \cdot a\right) \cdot \color{blue}{\left(b \cdot b\right)} + \left(\mathsf{neg}\left(b\right)\right), 1\right) \]

      7. associate-*r* N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\left(\frac{1}{2} \cdot a\right) \cdot b\right) \cdot b} + \left(\mathsf{neg}\left(b\right)\right), 1\right) \]

      8. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \left(\left(\frac{1}{2} \cdot a\right) \cdot b\right) \cdot b + \color{blue}{-1 \cdot b}, 1\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{b \cdot \left(\left(\frac{1}{2} \cdot a\right) \cdot b + -1\right)}, 1\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{b \cdot \left(\left(\frac{1}{2} \cdot a\right) \cdot b + -1\right)}, 1\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, b \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot a, b, -1\right)}, 1\right) \]

      12. *-lowering-*.f64 2.3

          \[\leadsto x \cdot \mathsf{fma}\left(a, b \cdot \mathsf{fma}\left(\color{blue}{0.5 \cdot a}, b, -1\right), 1\right) \]

   8. Simplified 2.3%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, b \cdot \mathsf{fma}\left(0.5 \cdot a, b, -1\right), 1\right)} \]

   9. Taylor expanded in a around inf

      \[\leadsto \color{blue}{{a}^{2} \cdot \left(-1 \cdot \frac{b \cdot x}{a} + \left(\frac{1}{2} \cdot \left({b}^{2} \cdot x\right) + \frac{x}{{a}^{2}}\right)\right)} \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \color{blue}{\left(a \cdot a\right)} \cdot \left(-1 \cdot \frac{b \cdot x}{a} + \left(\frac{1}{2} \cdot \left({b}^{2} \cdot x\right) + \frac{x}{{a}^{2}}\right)\right) \]

       2. associate-*l* N/A

          \[\leadsto \color{blue}{a \cdot \left(a \cdot \left(-1 \cdot \frac{b \cdot x}{a} + \left(\frac{1}{2} \cdot \left({b}^{2} \cdot x\right) + \frac{x}{{a}^{2}}\right)\right)\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{a \cdot \left(a \cdot \left(-1 \cdot \frac{b \cdot x}{a} + \left(\frac{1}{2} \cdot \left({b}^{2} \cdot x\right) + \frac{x}{{a}^{2}}\right)\right)\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto a \cdot \color{blue}{\left(a \cdot \left(-1 \cdot \frac{b \cdot x}{a} + \left(\frac{1}{2} \cdot \left({b}^{2} \cdot x\right) + \frac{x}{{a}^{2}}\right)\right)\right)} \]

       5. associate-+r+ N/A

          \[\leadsto a \cdot \left(a \cdot \color{blue}{\left(\left(-1 \cdot \frac{b \cdot x}{a} + \frac{1}{2} \cdot \left({b}^{2} \cdot x\right)\right) + \frac{x}{{a}^{2}}\right)}\right) \]

   11. Simplified 9.4%

       \[\leadsto \color{blue}{a \cdot \left(a \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, x \cdot 0.5, \frac{x}{-a}\right), \frac{x}{a \cdot a}\right)\right)} \]

   12. Taylor expanded in b around 0

       \[\leadsto a \cdot \left(a \cdot \color{blue}{\frac{x}{{a}^{2}}}\right) \]
   13. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto a \cdot \left(a \cdot \color{blue}{\frac{x}{{a}^{2}}}\right) \]

       2. unpow2 N/A

          \[\leadsto a \cdot \left(a \cdot \frac{x}{\color{blue}{a \cdot a}}\right) \]

       3. *-lowering-*.f64 55.3

          \[\leadsto a \cdot \left(a \cdot \frac{x}{\color{blue}{a \cdot a}}\right) \]

   14. Simplified 55.3%

       \[\leadsto a \cdot \left(a \cdot \color{blue}{\frac{x}{a \cdot a}}\right) \]

   if -4.0000000000000003e244 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -2e10

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      3. *-lowering-*.f64 42.4

         \[\leadsto x \cdot e^{-\color{blue}{a \cdot b}} \]

   5. Simplified 42.4%

      \[\leadsto x \cdot e^{\color{blue}{-a \cdot b}} \]

   6. Taylor expanded in a around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right) + 1\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, -1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right)} \]

      3. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right) \]

      4. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) + \left(\mathsf{neg}\left(b\right)\right)}, 1\right) \]

      5. associate-*r* N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\frac{1}{2} \cdot a\right) \cdot {b}^{2}} + \left(\mathsf{neg}\left(b\right)\right), 1\right) \]

      6. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \left(\frac{1}{2} \cdot a\right) \cdot \color{blue}{\left(b \cdot b\right)} + \left(\mathsf{neg}\left(b\right)\right), 1\right) \]

      7. associate-*r* N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\left(\frac{1}{2} \cdot a\right) \cdot b\right) \cdot b} + \left(\mathsf{neg}\left(b\right)\right), 1\right) \]

      8. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \left(\left(\frac{1}{2} \cdot a\right) \cdot b\right) \cdot b + \color{blue}{-1 \cdot b}, 1\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{b \cdot \left(\left(\frac{1}{2} \cdot a\right) \cdot b + -1\right)}, 1\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{b \cdot \left(\left(\frac{1}{2} \cdot a\right) \cdot b + -1\right)}, 1\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, b \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot a, b, -1\right)}, 1\right) \]

      12. *-lowering-*.f64 3.0

          \[\leadsto x \cdot \mathsf{fma}\left(a, b \cdot \mathsf{fma}\left(\color{blue}{0.5 \cdot a}, b, -1\right), 1\right) \]

   8. Simplified 3.0%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, b \cdot \mathsf{fma}\left(0.5 \cdot a, b, -1\right), 1\right)} \]

   9. Taylor expanded in b around inf

      \[\leadsto x \cdot \color{blue}{\left({b}^{2} \cdot \left(-1 \cdot \frac{a}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto x \cdot \left(\color{blue}{\left(b \cdot b\right)} \cdot \left(-1 \cdot \frac{a}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right) \]

       2. associate-*l* N/A

          \[\leadsto x \cdot \color{blue}{\left(b \cdot \left(b \cdot \left(-1 \cdot \frac{a}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \color{blue}{\left(b \cdot \left(b \cdot \left(-1 \cdot \frac{a}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \left(b \cdot \color{blue}{\left(b \cdot \left(-1 \cdot \frac{a}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)}\right) \]

       5. associate-*r/ N/A

          \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(\color{blue}{\frac{-1 \cdot a}{b}} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(\frac{\color{blue}{a \cdot -1}}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       7. associate-/l* N/A

          \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(\color{blue}{a \cdot \frac{-1}{b}} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       8. metadata-eval N/A

          \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \frac{\color{blue}{\mathsf{neg}\left(1\right)}}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       9. distribute-neg-frac N/A

          \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{b}\right)\right)} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       10. *-commutative N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(\mathsf{neg}\left(\frac{1}{b}\right)\right) + \color{blue}{{a}^{2} \cdot \frac{1}{2}}\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(\mathsf{neg}\left(\frac{1}{b}\right)\right) + \color{blue}{\left(a \cdot a\right)} \cdot \frac{1}{2}\right)\right)\right) \]

       12. associate-*l* N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(\mathsf{neg}\left(\frac{1}{b}\right)\right) + \color{blue}{a \cdot \left(a \cdot \frac{1}{2}\right)}\right)\right)\right) \]

       13. *-commutative N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(\mathsf{neg}\left(\frac{1}{b}\right)\right) + a \cdot \color{blue}{\left(\frac{1}{2} \cdot a\right)}\right)\right)\right) \]

       14. distribute-lft-out N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \color{blue}{\left(a \cdot \left(\left(\mathsf{neg}\left(\frac{1}{b}\right)\right) + \frac{1}{2} \cdot a\right)\right)}\right)\right) \]

       15. +-commutative N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \color{blue}{\left(\frac{1}{2} \cdot a + \left(\mathsf{neg}\left(\frac{1}{b}\right)\right)\right)}\right)\right)\right) \]

       16. sub-neg N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \color{blue}{\left(\frac{1}{2} \cdot a - \frac{1}{b}\right)}\right)\right)\right) \]

       17. *-lowering-*.f64 N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \color{blue}{\left(a \cdot \left(\frac{1}{2} \cdot a - \frac{1}{b}\right)\right)}\right)\right) \]

       18. sub-neg N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \color{blue}{\left(\frac{1}{2} \cdot a + \left(\mathsf{neg}\left(\frac{1}{b}\right)\right)\right)}\right)\right)\right) \]

       19. *-commutative N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(\color{blue}{a \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(\frac{1}{b}\right)\right)\right)\right)\right)\right) \]

       20. accelerator-lowering-fma.f64 N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \color{blue}{\mathsf{fma}\left(a, \frac{1}{2}, \mathsf{neg}\left(\frac{1}{b}\right)\right)}\right)\right)\right) \]

       21. distribute-neg-frac N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \mathsf{fma}\left(a, \frac{1}{2}, \color{blue}{\frac{\mathsf{neg}\left(1\right)}{b}}\right)\right)\right)\right) \]

       22. metadata-eval N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \mathsf{fma}\left(a, \frac{1}{2}, \frac{\color{blue}{-1}}{b}\right)\right)\right)\right) \]

       23. /-lowering-/.f64 34.0

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \mathsf{fma}\left(a, 0.5, \color{blue}{\frac{-1}{b}}\right)\right)\right)\right) \]

   11. Simplified 34.0%

       \[\leadsto x \cdot \color{blue}{\left(b \cdot \left(b \cdot \left(a \cdot \mathsf{fma}\left(a, 0.5, \frac{-1}{b}\right)\right)\right)\right)} \]

   12. Taylor expanded in b around inf

       \[\leadsto x \cdot \left(b \cdot \color{blue}{\left(\frac{1}{2} \cdot \left({a}^{2} \cdot b\right)\right)}\right) \]
   13. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \left(b \cdot \color{blue}{\left(\frac{1}{2} \cdot \left({a}^{2} \cdot b\right)\right)}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \left(b \cdot \left(\frac{1}{2} \cdot \color{blue}{\left({a}^{2} \cdot b\right)}\right)\right) \]

       3. unpow2 N/A

          \[\leadsto x \cdot \left(b \cdot \left(\frac{1}{2} \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot b\right)\right)\right) \]

       4. *-lowering-*.f64 51.1

          \[\leadsto x \cdot \left(b \cdot \left(0.5 \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot b\right)\right)\right) \]

   14. Simplified 51.1%

       \[\leadsto x \cdot \left(b \cdot \color{blue}{\left(0.5 \cdot \left(\left(a \cdot a\right) \cdot b\right)\right)}\right) \]

   if -2e10 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 1.99999999999999986e225

   1. Initial program 92.1%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      3. *-lowering-*.f64 69.2

         \[\leadsto x \cdot e^{-\color{blue}{a \cdot b}} \]

   5. Simplified 69.2%

      \[\leadsto x \cdot e^{\color{blue}{-a \cdot b}} \]

   6. Taylor expanded in b around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + b \cdot \left(-1 \cdot a + \frac{1}{2} \cdot \left({a}^{2} \cdot b\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(b \cdot \left(-1 \cdot a + \frac{1}{2} \cdot \left({a}^{2} \cdot b\right)\right) + 1\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(b, -1 \cdot a + \frac{1}{2} \cdot \left({a}^{2} \cdot b\right), 1\right)} \]

      3. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \color{blue}{\frac{1}{2} \cdot \left({a}^{2} \cdot b\right) + -1 \cdot a}, 1\right) \]

      4. *-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \color{blue}{\left({a}^{2} \cdot b\right) \cdot \frac{1}{2}} + -1 \cdot a, 1\right) \]

      5. *-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \color{blue}{\left(b \cdot {a}^{2}\right)} \cdot \frac{1}{2} + -1 \cdot a, 1\right) \]

      6. associate-*r* N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \color{blue}{b \cdot \left({a}^{2} \cdot \frac{1}{2}\right)} + -1 \cdot a, 1\right) \]

      7. *-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, b \cdot \color{blue}{\left(\frac{1}{2} \cdot {a}^{2}\right)} + -1 \cdot a, 1\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, \frac{1}{2} \cdot {a}^{2}, -1 \cdot a\right)}, 1\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\frac{1}{2} \cdot {a}^{2}}, -1 \cdot a\right), 1\right) \]

      10. unpow2 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{1}{2} \cdot \color{blue}{\left(a \cdot a\right)}, -1 \cdot a\right), 1\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{1}{2} \cdot \color{blue}{\left(a \cdot a\right)}, -1 \cdot a\right), 1\right) \]

      12. mul-1-neg N/A

          \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{1}{2} \cdot \left(a \cdot a\right), \color{blue}{\mathsf{neg}\left(a\right)}\right), 1\right) \]

      13. neg-lowering-neg.f64 65.7

          \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5 \cdot \left(a \cdot a\right), \color{blue}{-a}\right), 1\right) \]

   8. Simplified 65.7%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5 \cdot \left(a \cdot a\right), -a\right), 1\right)} \]

   if 1.99999999999999986e225 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 97.7%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(t \cdot y\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(t \cdot y\right)}} \]

      2. *-commutative N/A

         \[\leadsto x \cdot e^{\mathsf{neg}\left(\color{blue}{y \cdot t}\right)} \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      5. neg-lowering-neg.f64 62.6

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(-t\right)}} \]

   5. Simplified 62.6%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{x + t \cdot \left(-1 \cdot \left(x \cdot y\right) + \frac{1}{2} \cdot \left(t \cdot \left(x \cdot {y}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(x \cdot y\right) + \frac{1}{2} \cdot \left(t \cdot \left(x \cdot {y}^{2}\right)\right)\right) + x} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, -1 \cdot \left(x \cdot y\right) + \frac{1}{2} \cdot \left(t \cdot \left(x \cdot {y}^{2}\right)\right), x\right)} \]

   8. Simplified 73.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, x \cdot \left(y \cdot \mathsf{fma}\left(0.5 \cdot t, y, -1\right)\right), x\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -1.5 \cdot 10^{+286}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(t, y, 1\right)}\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -4 \cdot 10^{+244}:\\ \;\;\;\;a \cdot \left(a \cdot \frac{x}{a \cdot a}\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -20000000000:\\ \;\;\;\;x \cdot \left(b \cdot \left(0.5 \cdot \left(b \cdot \left(a \cdot a\right)\right)\right)\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 2 \cdot 10^{+225}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5 \cdot \left(a \cdot a\right), -a\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, x \cdot \left(y \cdot \mathsf{fma}\left(t \cdot 0.5, y, -1\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 54.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t\_1 \leq -1.5 \cdot 10^{+286}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(t, y, 1\right)}\\ \mathbf{elif}\;t\_1 \leq -4 \cdot 10^{+244}:\\ \;\;\;\;a \cdot \left(a \cdot \frac{x}{a \cdot a}\right)\\ \mathbf{elif}\;t\_1 \leq -20000000000:\\ \;\;\;\;x \cdot \left(b \cdot \left(0.5 \cdot \left(b \cdot \left(a \cdot a\right)\right)\right)\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+225}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(b, b \cdot \left(0.5 \cdot \left(a \cdot a\right)\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, x \cdot \left(y \cdot \mathsf{fma}\left(t \cdot 0.5, y, -1\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b)))))
   (if (<= t_1 -1.5e+286)
     (/ x (fma t y 1.0))
     (if (<= t_1 -4e+244)
       (* a (* a (/ x (* a a))))
       (if (<= t_1 -20000000000.0)
         (* x (* b (* 0.5 (* b (* a a)))))
         (if (<= t_1 2e+225)
           (* x (fma b (* b (* 0.5 (* a a))) 1.0))
           (fma t (* x (* y (fma (* t 0.5) y -1.0))) x)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	double tmp;
	if (t_1 <= -1.5e+286) {
		tmp = x / fma(t, y, 1.0);
	} else if (t_1 <= -4e+244) {
		tmp = a * (a * (x / (a * a)));
	} else if (t_1 <= -20000000000.0) {
		tmp = x * (b * (0.5 * (b * (a * a))));
	} else if (t_1 <= 2e+225) {
		tmp = x * fma(b, (b * (0.5 * (a * a))), 1.0);
	} else {
		tmp = fma(t, (x * (y * fma((t * 0.5), y, -1.0))), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))
	tmp = 0.0
	if (t_1 <= -1.5e+286)
		tmp = Float64(x / fma(t, y, 1.0));
	elseif (t_1 <= -4e+244)
		tmp = Float64(a * Float64(a * Float64(x / Float64(a * a))));
	elseif (t_1 <= -20000000000.0)
		tmp = Float64(x * Float64(b * Float64(0.5 * Float64(b * Float64(a * a)))));
	elseif (t_1 <= 2e+225)
		tmp = Float64(x * fma(b, Float64(b * Float64(0.5 * Float64(a * a))), 1.0));
	else
		tmp = fma(t, Float64(x * Float64(y * fma(Float64(t * 0.5), y, -1.0))), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1.5e+286], N[(x / N[(t * y + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -4e+244], N[(a * N[(a * N[(x / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -20000000000.0], N[(x * N[(b * N[(0.5 * N[(b * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+225], N[(x * N[(b * N[(b * N[(0.5 * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(t * N[(x * N[(y * N[(N[(t * 0.5), $MachinePrecision] * y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -1.4999999999999999e286

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(t \cdot y\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(t \cdot y\right)}} \]

      2. *-commutative N/A

         \[\leadsto x \cdot e^{\mathsf{neg}\left(\color{blue}{y \cdot t}\right)} \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      5. neg-lowering-neg.f64 66.3

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(-t\right)}} \]

   5. Simplified 66.3%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}} \]
   6. Step-by-step derivation

      1. exp-prod N/A

         \[\leadsto x \cdot \color{blue}{{\left(e^{y}\right)}^{\left(\mathsf{neg}\left(t\right)\right)}} \]

      2. pow-neg N/A

         \[\leadsto x \cdot \color{blue}{\frac{1}{{\left(e^{y}\right)}^{t}}} \]

      3. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{x}{{\left(e^{y}\right)}^{t}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{{\left(e^{y}\right)}^{t}}} \]

      5. pow-exp N/A

         \[\leadsto \frac{x}{\color{blue}{e^{y \cdot t}}} \]

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{e^{y \cdot t}}} \]

      7. *-lowering-*.f64 66.3

         \[\leadsto \frac{x}{e^{\color{blue}{y \cdot t}}} \]

   7. Applied egg-rr 66.3%

      \[\leadsto \color{blue}{\frac{x}{e^{y \cdot t}}} \]

   8. Taylor expanded in y around 0

      \[\leadsto \frac{x}{\color{blue}{1 + t \cdot y}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{t \cdot y + 1}} \]

      2. accelerator-lowering-fma.f64 59.4

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(t, y, 1\right)}} \]

   10. Simplified 59.4%

       \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(t, y, 1\right)}} \]

   if -1.4999999999999999e286 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -4.0000000000000003e244

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      3. *-lowering-*.f64 55.1

         \[\leadsto x \cdot e^{-\color{blue}{a \cdot b}} \]

   5. Simplified 55.1%

      \[\leadsto x \cdot e^{\color{blue}{-a \cdot b}} \]

   6. Taylor expanded in a around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right) + 1\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, -1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right)} \]

      3. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right) \]

      4. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) + \left(\mathsf{neg}\left(b\right)\right)}, 1\right) \]

      5. associate-*r* N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\frac{1}{2} \cdot a\right) \cdot {b}^{2}} + \left(\mathsf{neg}\left(b\right)\right), 1\right) \]

      6. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \left(\frac{1}{2} \cdot a\right) \cdot \color{blue}{\left(b \cdot b\right)} + \left(\mathsf{neg}\left(b\right)\right), 1\right) \]

      7. associate-*r* N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\left(\frac{1}{2} \cdot a\right) \cdot b\right) \cdot b} + \left(\mathsf{neg}\left(b\right)\right), 1\right) \]

      8. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \left(\left(\frac{1}{2} \cdot a\right) \cdot b\right) \cdot b + \color{blue}{-1 \cdot b}, 1\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{b \cdot \left(\left(\frac{1}{2} \cdot a\right) \cdot b + -1\right)}, 1\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{b \cdot \left(\left(\frac{1}{2} \cdot a\right) \cdot b + -1\right)}, 1\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, b \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot a, b, -1\right)}, 1\right) \]

      12. *-lowering-*.f64 2.3

          \[\leadsto x \cdot \mathsf{fma}\left(a, b \cdot \mathsf{fma}\left(\color{blue}{0.5 \cdot a}, b, -1\right), 1\right) \]

   8. Simplified 2.3%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, b \cdot \mathsf{fma}\left(0.5 \cdot a, b, -1\right), 1\right)} \]

   9. Taylor expanded in a around inf

      \[\leadsto \color{blue}{{a}^{2} \cdot \left(-1 \cdot \frac{b \cdot x}{a} + \left(\frac{1}{2} \cdot \left({b}^{2} \cdot x\right) + \frac{x}{{a}^{2}}\right)\right)} \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \color{blue}{\left(a \cdot a\right)} \cdot \left(-1 \cdot \frac{b \cdot x}{a} + \left(\frac{1}{2} \cdot \left({b}^{2} \cdot x\right) + \frac{x}{{a}^{2}}\right)\right) \]

       2. associate-*l* N/A

          \[\leadsto \color{blue}{a \cdot \left(a \cdot \left(-1 \cdot \frac{b \cdot x}{a} + \left(\frac{1}{2} \cdot \left({b}^{2} \cdot x\right) + \frac{x}{{a}^{2}}\right)\right)\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{a \cdot \left(a \cdot \left(-1 \cdot \frac{b \cdot x}{a} + \left(\frac{1}{2} \cdot \left({b}^{2} \cdot x\right) + \frac{x}{{a}^{2}}\right)\right)\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto a \cdot \color{blue}{\left(a \cdot \left(-1 \cdot \frac{b \cdot x}{a} + \left(\frac{1}{2} \cdot \left({b}^{2} \cdot x\right) + \frac{x}{{a}^{2}}\right)\right)\right)} \]

       5. associate-+r+ N/A

          \[\leadsto a \cdot \left(a \cdot \color{blue}{\left(\left(-1 \cdot \frac{b \cdot x}{a} + \frac{1}{2} \cdot \left({b}^{2} \cdot x\right)\right) + \frac{x}{{a}^{2}}\right)}\right) \]

   11. Simplified 9.4%

       \[\leadsto \color{blue}{a \cdot \left(a \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, x \cdot 0.5, \frac{x}{-a}\right), \frac{x}{a \cdot a}\right)\right)} \]

   12. Taylor expanded in b around 0

       \[\leadsto a \cdot \left(a \cdot \color{blue}{\frac{x}{{a}^{2}}}\right) \]
   13. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto a \cdot \left(a \cdot \color{blue}{\frac{x}{{a}^{2}}}\right) \]

       2. unpow2 N/A

          \[\leadsto a \cdot \left(a \cdot \frac{x}{\color{blue}{a \cdot a}}\right) \]

       3. *-lowering-*.f64 55.3

          \[\leadsto a \cdot \left(a \cdot \frac{x}{\color{blue}{a \cdot a}}\right) \]

   14. Simplified 55.3%

       \[\leadsto a \cdot \left(a \cdot \color{blue}{\frac{x}{a \cdot a}}\right) \]

   if -4.0000000000000003e244 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -2e10

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      3. *-lowering-*.f64 42.4

         \[\leadsto x \cdot e^{-\color{blue}{a \cdot b}} \]

   5. Simplified 42.4%

      \[\leadsto x \cdot e^{\color{blue}{-a \cdot b}} \]

   6. Taylor expanded in a around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right) + 1\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, -1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right)} \]

      3. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right) \]

      4. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) + \left(\mathsf{neg}\left(b\right)\right)}, 1\right) \]

      5. associate-*r* N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\frac{1}{2} \cdot a\right) \cdot {b}^{2}} + \left(\mathsf{neg}\left(b\right)\right), 1\right) \]

      6. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \left(\frac{1}{2} \cdot a\right) \cdot \color{blue}{\left(b \cdot b\right)} + \left(\mathsf{neg}\left(b\right)\right), 1\right) \]

      7. associate-*r* N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\left(\frac{1}{2} \cdot a\right) \cdot b\right) \cdot b} + \left(\mathsf{neg}\left(b\right)\right), 1\right) \]

      8. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \left(\left(\frac{1}{2} \cdot a\right) \cdot b\right) \cdot b + \color{blue}{-1 \cdot b}, 1\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{b \cdot \left(\left(\frac{1}{2} \cdot a\right) \cdot b + -1\right)}, 1\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{b \cdot \left(\left(\frac{1}{2} \cdot a\right) \cdot b + -1\right)}, 1\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, b \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot a, b, -1\right)}, 1\right) \]

      12. *-lowering-*.f64 3.0

          \[\leadsto x \cdot \mathsf{fma}\left(a, b \cdot \mathsf{fma}\left(\color{blue}{0.5 \cdot a}, b, -1\right), 1\right) \]

   8. Simplified 3.0%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, b \cdot \mathsf{fma}\left(0.5 \cdot a, b, -1\right), 1\right)} \]

   9. Taylor expanded in b around inf

      \[\leadsto x \cdot \color{blue}{\left({b}^{2} \cdot \left(-1 \cdot \frac{a}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto x \cdot \left(\color{blue}{\left(b \cdot b\right)} \cdot \left(-1 \cdot \frac{a}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right) \]

       2. associate-*l* N/A

          \[\leadsto x \cdot \color{blue}{\left(b \cdot \left(b \cdot \left(-1 \cdot \frac{a}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \color{blue}{\left(b \cdot \left(b \cdot \left(-1 \cdot \frac{a}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \left(b \cdot \color{blue}{\left(b \cdot \left(-1 \cdot \frac{a}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)}\right) \]

       5. associate-*r/ N/A

          \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(\color{blue}{\frac{-1 \cdot a}{b}} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(\frac{\color{blue}{a \cdot -1}}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       7. associate-/l* N/A

          \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(\color{blue}{a \cdot \frac{-1}{b}} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       8. metadata-eval N/A

          \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \frac{\color{blue}{\mathsf{neg}\left(1\right)}}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       9. distribute-neg-frac N/A

          \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{b}\right)\right)} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       10. *-commutative N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(\mathsf{neg}\left(\frac{1}{b}\right)\right) + \color{blue}{{a}^{2} \cdot \frac{1}{2}}\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(\mathsf{neg}\left(\frac{1}{b}\right)\right) + \color{blue}{\left(a \cdot a\right)} \cdot \frac{1}{2}\right)\right)\right) \]

       12. associate-*l* N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(\mathsf{neg}\left(\frac{1}{b}\right)\right) + \color{blue}{a \cdot \left(a \cdot \frac{1}{2}\right)}\right)\right)\right) \]

       13. *-commutative N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(\mathsf{neg}\left(\frac{1}{b}\right)\right) + a \cdot \color{blue}{\left(\frac{1}{2} \cdot a\right)}\right)\right)\right) \]

       14. distribute-lft-out N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \color{blue}{\left(a \cdot \left(\left(\mathsf{neg}\left(\frac{1}{b}\right)\right) + \frac{1}{2} \cdot a\right)\right)}\right)\right) \]

       15. +-commutative N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \color{blue}{\left(\frac{1}{2} \cdot a + \left(\mathsf{neg}\left(\frac{1}{b}\right)\right)\right)}\right)\right)\right) \]

       16. sub-neg N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \color{blue}{\left(\frac{1}{2} \cdot a - \frac{1}{b}\right)}\right)\right)\right) \]

       17. *-lowering-*.f64 N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \color{blue}{\left(a \cdot \left(\frac{1}{2} \cdot a - \frac{1}{b}\right)\right)}\right)\right) \]

       18. sub-neg N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \color{blue}{\left(\frac{1}{2} \cdot a + \left(\mathsf{neg}\left(\frac{1}{b}\right)\right)\right)}\right)\right)\right) \]

       19. *-commutative N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(\color{blue}{a \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(\frac{1}{b}\right)\right)\right)\right)\right)\right) \]

       20. accelerator-lowering-fma.f64 N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \color{blue}{\mathsf{fma}\left(a, \frac{1}{2}, \mathsf{neg}\left(\frac{1}{b}\right)\right)}\right)\right)\right) \]

       21. distribute-neg-frac N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \mathsf{fma}\left(a, \frac{1}{2}, \color{blue}{\frac{\mathsf{neg}\left(1\right)}{b}}\right)\right)\right)\right) \]

       22. metadata-eval N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \mathsf{fma}\left(a, \frac{1}{2}, \frac{\color{blue}{-1}}{b}\right)\right)\right)\right) \]

       23. /-lowering-/.f64 34.0

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \mathsf{fma}\left(a, 0.5, \color{blue}{\frac{-1}{b}}\right)\right)\right)\right) \]

   11. Simplified 34.0%

       \[\leadsto x \cdot \color{blue}{\left(b \cdot \left(b \cdot \left(a \cdot \mathsf{fma}\left(a, 0.5, \frac{-1}{b}\right)\right)\right)\right)} \]

   12. Taylor expanded in b around inf

       \[\leadsto x \cdot \left(b \cdot \color{blue}{\left(\frac{1}{2} \cdot \left({a}^{2} \cdot b\right)\right)}\right) \]
   13. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \left(b \cdot \color{blue}{\left(\frac{1}{2} \cdot \left({a}^{2} \cdot b\right)\right)}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \left(b \cdot \left(\frac{1}{2} \cdot \color{blue}{\left({a}^{2} \cdot b\right)}\right)\right) \]

       3. unpow2 N/A

          \[\leadsto x \cdot \left(b \cdot \left(\frac{1}{2} \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot b\right)\right)\right) \]

       4. *-lowering-*.f64 51.1

          \[\leadsto x \cdot \left(b \cdot \left(0.5 \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot b\right)\right)\right) \]

   14. Simplified 51.1%

       \[\leadsto x \cdot \left(b \cdot \color{blue}{\left(0.5 \cdot \left(\left(a \cdot a\right) \cdot b\right)\right)}\right) \]

   if -2e10 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 1.99999999999999986e225

   1. Initial program 92.1%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      3. *-lowering-*.f64 69.2

         \[\leadsto x \cdot e^{-\color{blue}{a \cdot b}} \]

   5. Simplified 69.2%

      \[\leadsto x \cdot e^{\color{blue}{-a \cdot b}} \]

   6. Taylor expanded in b around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + b \cdot \left(-1 \cdot a + \frac{1}{2} \cdot \left({a}^{2} \cdot b\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(b \cdot \left(-1 \cdot a + \frac{1}{2} \cdot \left({a}^{2} \cdot b\right)\right) + 1\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(b, -1 \cdot a + \frac{1}{2} \cdot \left({a}^{2} \cdot b\right), 1\right)} \]

      3. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \color{blue}{\frac{1}{2} \cdot \left({a}^{2} \cdot b\right) + -1 \cdot a}, 1\right) \]

      4. *-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \color{blue}{\left({a}^{2} \cdot b\right) \cdot \frac{1}{2}} + -1 \cdot a, 1\right) \]

      5. *-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \color{blue}{\left(b \cdot {a}^{2}\right)} \cdot \frac{1}{2} + -1 \cdot a, 1\right) \]

      6. associate-*r* N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \color{blue}{b \cdot \left({a}^{2} \cdot \frac{1}{2}\right)} + -1 \cdot a, 1\right) \]

      7. *-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, b \cdot \color{blue}{\left(\frac{1}{2} \cdot {a}^{2}\right)} + -1 \cdot a, 1\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, \frac{1}{2} \cdot {a}^{2}, -1 \cdot a\right)}, 1\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\frac{1}{2} \cdot {a}^{2}}, -1 \cdot a\right), 1\right) \]

      10. unpow2 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{1}{2} \cdot \color{blue}{\left(a \cdot a\right)}, -1 \cdot a\right), 1\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{1}{2} \cdot \color{blue}{\left(a \cdot a\right)}, -1 \cdot a\right), 1\right) \]

      12. mul-1-neg N/A

          \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{1}{2} \cdot \left(a \cdot a\right), \color{blue}{\mathsf{neg}\left(a\right)}\right), 1\right) \]

      13. neg-lowering-neg.f64 65.7

          \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5 \cdot \left(a \cdot a\right), \color{blue}{-a}\right), 1\right) \]

   8. Simplified 65.7%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5 \cdot \left(a \cdot a\right), -a\right), 1\right)} \]

   9. Taylor expanded in b around inf

      \[\leadsto x \cdot \mathsf{fma}\left(b, \color{blue}{\frac{1}{2} \cdot \left({a}^{2} \cdot b\right)}, 1\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(b, \color{blue}{\left({a}^{2} \cdot b\right) \cdot \frac{1}{2}}, 1\right) \]

       2. *-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(b, \color{blue}{\left(b \cdot {a}^{2}\right)} \cdot \frac{1}{2}, 1\right) \]

       3. associate-*r* N/A

          \[\leadsto x \cdot \mathsf{fma}\left(b, \color{blue}{b \cdot \left({a}^{2} \cdot \frac{1}{2}\right)}, 1\right) \]

       4. *-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(b, b \cdot \color{blue}{\left(\frac{1}{2} \cdot {a}^{2}\right)}, 1\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(b, \color{blue}{b \cdot \left(\frac{1}{2} \cdot {a}^{2}\right)}, 1\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(b, b \cdot \color{blue}{\left(\frac{1}{2} \cdot {a}^{2}\right)}, 1\right) \]

       7. unpow2 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(b, b \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(a \cdot a\right)}\right), 1\right) \]

       8. *-lowering-*.f64 65.3

          \[\leadsto x \cdot \mathsf{fma}\left(b, b \cdot \left(0.5 \cdot \color{blue}{\left(a \cdot a\right)}\right), 1\right) \]

   11. Simplified 65.3%

       \[\leadsto x \cdot \mathsf{fma}\left(b, \color{blue}{b \cdot \left(0.5 \cdot \left(a \cdot a\right)\right)}, 1\right) \]

   if 1.99999999999999986e225 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 97.7%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(t \cdot y\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(t \cdot y\right)}} \]

      2. *-commutative N/A

         \[\leadsto x \cdot e^{\mathsf{neg}\left(\color{blue}{y \cdot t}\right)} \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      5. neg-lowering-neg.f64 62.6

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(-t\right)}} \]

   5. Simplified 62.6%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{x + t \cdot \left(-1 \cdot \left(x \cdot y\right) + \frac{1}{2} \cdot \left(t \cdot \left(x \cdot {y}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(x \cdot y\right) + \frac{1}{2} \cdot \left(t \cdot \left(x \cdot {y}^{2}\right)\right)\right) + x} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, -1 \cdot \left(x \cdot y\right) + \frac{1}{2} \cdot \left(t \cdot \left(x \cdot {y}^{2}\right)\right), x\right)} \]

   8. Simplified 73.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, x \cdot \left(y \cdot \mathsf{fma}\left(0.5 \cdot t, y, -1\right)\right), x\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -1.5 \cdot 10^{+286}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(t, y, 1\right)}\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -4 \cdot 10^{+244}:\\ \;\;\;\;a \cdot \left(a \cdot \frac{x}{a \cdot a}\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -20000000000:\\ \;\;\;\;x \cdot \left(b \cdot \left(0.5 \cdot \left(b \cdot \left(a \cdot a\right)\right)\right)\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 2 \cdot 10^{+225}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(b, b \cdot \left(0.5 \cdot \left(a \cdot a\right)\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, x \cdot \left(y \cdot \mathsf{fma}\left(t \cdot 0.5, y, -1\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 54.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t\_1 \leq -1.5 \cdot 10^{+286}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(t, y, 1\right)}\\ \mathbf{elif}\;t\_1 \leq -4 \cdot 10^{+244}:\\ \;\;\;\;a \cdot \left(a \cdot \frac{x}{a \cdot a}\right)\\ \mathbf{elif}\;t\_1 \leq -20000000000:\\ \;\;\;\;x \cdot \left(b \cdot \left(0.5 \cdot \left(b \cdot \left(a \cdot a\right)\right)\right)\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+225}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(b, b \cdot \left(0.5 \cdot \left(a \cdot a\right)\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(t, y \cdot \mathsf{fma}\left(t \cdot 0.5, y, -1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b)))))
   (if (<= t_1 -1.5e+286)
     (/ x (fma t y 1.0))
     (if (<= t_1 -4e+244)
       (* a (* a (/ x (* a a))))
       (if (<= t_1 -20000000000.0)
         (* x (* b (* 0.5 (* b (* a a)))))
         (if (<= t_1 2e+225)
           (* x (fma b (* b (* 0.5 (* a a))) 1.0))
           (* x (fma t (* y (fma (* t 0.5) y -1.0)) 1.0))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	double tmp;
	if (t_1 <= -1.5e+286) {
		tmp = x / fma(t, y, 1.0);
	} else if (t_1 <= -4e+244) {
		tmp = a * (a * (x / (a * a)));
	} else if (t_1 <= -20000000000.0) {
		tmp = x * (b * (0.5 * (b * (a * a))));
	} else if (t_1 <= 2e+225) {
		tmp = x * fma(b, (b * (0.5 * (a * a))), 1.0);
	} else {
		tmp = x * fma(t, (y * fma((t * 0.5), y, -1.0)), 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))
	tmp = 0.0
	if (t_1 <= -1.5e+286)
		tmp = Float64(x / fma(t, y, 1.0));
	elseif (t_1 <= -4e+244)
		tmp = Float64(a * Float64(a * Float64(x / Float64(a * a))));
	elseif (t_1 <= -20000000000.0)
		tmp = Float64(x * Float64(b * Float64(0.5 * Float64(b * Float64(a * a)))));
	elseif (t_1 <= 2e+225)
		tmp = Float64(x * fma(b, Float64(b * Float64(0.5 * Float64(a * a))), 1.0));
	else
		tmp = Float64(x * fma(t, Float64(y * fma(Float64(t * 0.5), y, -1.0)), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1.5e+286], N[(x / N[(t * y + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -4e+244], N[(a * N[(a * N[(x / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -20000000000.0], N[(x * N[(b * N[(0.5 * N[(b * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+225], N[(x * N[(b * N[(b * N[(0.5 * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(t * N[(y * N[(N[(t * 0.5), $MachinePrecision] * y + -1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -1.4999999999999999e286

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(t \cdot y\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(t \cdot y\right)}} \]

      2. *-commutative N/A

         \[\leadsto x \cdot e^{\mathsf{neg}\left(\color{blue}{y \cdot t}\right)} \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      5. neg-lowering-neg.f64 66.3

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(-t\right)}} \]

   5. Simplified 66.3%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}} \]
   6. Step-by-step derivation

      1. exp-prod N/A

         \[\leadsto x \cdot \color{blue}{{\left(e^{y}\right)}^{\left(\mathsf{neg}\left(t\right)\right)}} \]

      2. pow-neg N/A

         \[\leadsto x \cdot \color{blue}{\frac{1}{{\left(e^{y}\right)}^{t}}} \]

      3. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{x}{{\left(e^{y}\right)}^{t}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{{\left(e^{y}\right)}^{t}}} \]

      5. pow-exp N/A

         \[\leadsto \frac{x}{\color{blue}{e^{y \cdot t}}} \]

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{e^{y \cdot t}}} \]

      7. *-lowering-*.f64 66.3

         \[\leadsto \frac{x}{e^{\color{blue}{y \cdot t}}} \]

   7. Applied egg-rr 66.3%

      \[\leadsto \color{blue}{\frac{x}{e^{y \cdot t}}} \]

   8. Taylor expanded in y around 0

      \[\leadsto \frac{x}{\color{blue}{1 + t \cdot y}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{t \cdot y + 1}} \]

      2. accelerator-lowering-fma.f64 59.4

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(t, y, 1\right)}} \]

   10. Simplified 59.4%

       \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(t, y, 1\right)}} \]

   if -1.4999999999999999e286 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -4.0000000000000003e244

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      3. *-lowering-*.f64 55.1

         \[\leadsto x \cdot e^{-\color{blue}{a \cdot b}} \]

   5. Simplified 55.1%

      \[\leadsto x \cdot e^{\color{blue}{-a \cdot b}} \]

   6. Taylor expanded in a around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right) + 1\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, -1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right)} \]

      3. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right) \]

      4. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) + \left(\mathsf{neg}\left(b\right)\right)}, 1\right) \]

      5. associate-*r* N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\frac{1}{2} \cdot a\right) \cdot {b}^{2}} + \left(\mathsf{neg}\left(b\right)\right), 1\right) \]

      6. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \left(\frac{1}{2} \cdot a\right) \cdot \color{blue}{\left(b \cdot b\right)} + \left(\mathsf{neg}\left(b\right)\right), 1\right) \]

      7. associate-*r* N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\left(\frac{1}{2} \cdot a\right) \cdot b\right) \cdot b} + \left(\mathsf{neg}\left(b\right)\right), 1\right) \]

      8. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \left(\left(\frac{1}{2} \cdot a\right) \cdot b\right) \cdot b + \color{blue}{-1 \cdot b}, 1\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{b \cdot \left(\left(\frac{1}{2} \cdot a\right) \cdot b + -1\right)}, 1\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{b \cdot \left(\left(\frac{1}{2} \cdot a\right) \cdot b + -1\right)}, 1\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, b \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot a, b, -1\right)}, 1\right) \]

      12. *-lowering-*.f64 2.3

          \[\leadsto x \cdot \mathsf{fma}\left(a, b \cdot \mathsf{fma}\left(\color{blue}{0.5 \cdot a}, b, -1\right), 1\right) \]

   8. Simplified 2.3%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, b \cdot \mathsf{fma}\left(0.5 \cdot a, b, -1\right), 1\right)} \]

   9. Taylor expanded in a around inf

      \[\leadsto \color{blue}{{a}^{2} \cdot \left(-1 \cdot \frac{b \cdot x}{a} + \left(\frac{1}{2} \cdot \left({b}^{2} \cdot x\right) + \frac{x}{{a}^{2}}\right)\right)} \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \color{blue}{\left(a \cdot a\right)} \cdot \left(-1 \cdot \frac{b \cdot x}{a} + \left(\frac{1}{2} \cdot \left({b}^{2} \cdot x\right) + \frac{x}{{a}^{2}}\right)\right) \]

       2. associate-*l* N/A

          \[\leadsto \color{blue}{a \cdot \left(a \cdot \left(-1 \cdot \frac{b \cdot x}{a} + \left(\frac{1}{2} \cdot \left({b}^{2} \cdot x\right) + \frac{x}{{a}^{2}}\right)\right)\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{a \cdot \left(a \cdot \left(-1 \cdot \frac{b \cdot x}{a} + \left(\frac{1}{2} \cdot \left({b}^{2} \cdot x\right) + \frac{x}{{a}^{2}}\right)\right)\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto a \cdot \color{blue}{\left(a \cdot \left(-1 \cdot \frac{b \cdot x}{a} + \left(\frac{1}{2} \cdot \left({b}^{2} \cdot x\right) + \frac{x}{{a}^{2}}\right)\right)\right)} \]

       5. associate-+r+ N/A

          \[\leadsto a \cdot \left(a \cdot \color{blue}{\left(\left(-1 \cdot \frac{b \cdot x}{a} + \frac{1}{2} \cdot \left({b}^{2} \cdot x\right)\right) + \frac{x}{{a}^{2}}\right)}\right) \]

   11. Simplified 9.4%

       \[\leadsto \color{blue}{a \cdot \left(a \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, x \cdot 0.5, \frac{x}{-a}\right), \frac{x}{a \cdot a}\right)\right)} \]

   12. Taylor expanded in b around 0

       \[\leadsto a \cdot \left(a \cdot \color{blue}{\frac{x}{{a}^{2}}}\right) \]
   13. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto a \cdot \left(a \cdot \color{blue}{\frac{x}{{a}^{2}}}\right) \]

       2. unpow2 N/A

          \[\leadsto a \cdot \left(a \cdot \frac{x}{\color{blue}{a \cdot a}}\right) \]

       3. *-lowering-*.f64 55.3

          \[\leadsto a \cdot \left(a \cdot \frac{x}{\color{blue}{a \cdot a}}\right) \]

   14. Simplified 55.3%

       \[\leadsto a \cdot \left(a \cdot \color{blue}{\frac{x}{a \cdot a}}\right) \]

   if -4.0000000000000003e244 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -2e10

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      3. *-lowering-*.f64 42.4

         \[\leadsto x \cdot e^{-\color{blue}{a \cdot b}} \]

   5. Simplified 42.4%

      \[\leadsto x \cdot e^{\color{blue}{-a \cdot b}} \]

   6. Taylor expanded in a around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right) + 1\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, -1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right)} \]

      3. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right) \]

      4. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) + \left(\mathsf{neg}\left(b\right)\right)}, 1\right) \]

      5. associate-*r* N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\frac{1}{2} \cdot a\right) \cdot {b}^{2}} + \left(\mathsf{neg}\left(b\right)\right), 1\right) \]

      6. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \left(\frac{1}{2} \cdot a\right) \cdot \color{blue}{\left(b \cdot b\right)} + \left(\mathsf{neg}\left(b\right)\right), 1\right) \]

      7. associate-*r* N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\left(\frac{1}{2} \cdot a\right) \cdot b\right) \cdot b} + \left(\mathsf{neg}\left(b\right)\right), 1\right) \]

      8. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \left(\left(\frac{1}{2} \cdot a\right) \cdot b\right) \cdot b + \color{blue}{-1 \cdot b}, 1\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{b \cdot \left(\left(\frac{1}{2} \cdot a\right) \cdot b + -1\right)}, 1\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{b \cdot \left(\left(\frac{1}{2} \cdot a\right) \cdot b + -1\right)}, 1\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, b \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot a, b, -1\right)}, 1\right) \]

      12. *-lowering-*.f64 3.0

          \[\leadsto x \cdot \mathsf{fma}\left(a, b \cdot \mathsf{fma}\left(\color{blue}{0.5 \cdot a}, b, -1\right), 1\right) \]

   8. Simplified 3.0%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, b \cdot \mathsf{fma}\left(0.5 \cdot a, b, -1\right), 1\right)} \]

   9. Taylor expanded in b around inf

      \[\leadsto x \cdot \color{blue}{\left({b}^{2} \cdot \left(-1 \cdot \frac{a}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto x \cdot \left(\color{blue}{\left(b \cdot b\right)} \cdot \left(-1 \cdot \frac{a}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right) \]

       2. associate-*l* N/A

          \[\leadsto x \cdot \color{blue}{\left(b \cdot \left(b \cdot \left(-1 \cdot \frac{a}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \color{blue}{\left(b \cdot \left(b \cdot \left(-1 \cdot \frac{a}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \left(b \cdot \color{blue}{\left(b \cdot \left(-1 \cdot \frac{a}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)}\right) \]

       5. associate-*r/ N/A

          \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(\color{blue}{\frac{-1 \cdot a}{b}} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(\frac{\color{blue}{a \cdot -1}}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       7. associate-/l* N/A

          \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(\color{blue}{a \cdot \frac{-1}{b}} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       8. metadata-eval N/A

          \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \frac{\color{blue}{\mathsf{neg}\left(1\right)}}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       9. distribute-neg-frac N/A

          \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{b}\right)\right)} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       10. *-commutative N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(\mathsf{neg}\left(\frac{1}{b}\right)\right) + \color{blue}{{a}^{2} \cdot \frac{1}{2}}\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(\mathsf{neg}\left(\frac{1}{b}\right)\right) + \color{blue}{\left(a \cdot a\right)} \cdot \frac{1}{2}\right)\right)\right) \]

       12. associate-*l* N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(\mathsf{neg}\left(\frac{1}{b}\right)\right) + \color{blue}{a \cdot \left(a \cdot \frac{1}{2}\right)}\right)\right)\right) \]

       13. *-commutative N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(\mathsf{neg}\left(\frac{1}{b}\right)\right) + a \cdot \color{blue}{\left(\frac{1}{2} \cdot a\right)}\right)\right)\right) \]

       14. distribute-lft-out N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \color{blue}{\left(a \cdot \left(\left(\mathsf{neg}\left(\frac{1}{b}\right)\right) + \frac{1}{2} \cdot a\right)\right)}\right)\right) \]

       15. +-commutative N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \color{blue}{\left(\frac{1}{2} \cdot a + \left(\mathsf{neg}\left(\frac{1}{b}\right)\right)\right)}\right)\right)\right) \]

       16. sub-neg N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \color{blue}{\left(\frac{1}{2} \cdot a - \frac{1}{b}\right)}\right)\right)\right) \]

       17. *-lowering-*.f64 N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \color{blue}{\left(a \cdot \left(\frac{1}{2} \cdot a - \frac{1}{b}\right)\right)}\right)\right) \]

       18. sub-neg N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \color{blue}{\left(\frac{1}{2} \cdot a + \left(\mathsf{neg}\left(\frac{1}{b}\right)\right)\right)}\right)\right)\right) \]

       19. *-commutative N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(\color{blue}{a \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(\frac{1}{b}\right)\right)\right)\right)\right)\right) \]

       20. accelerator-lowering-fma.f64 N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \color{blue}{\mathsf{fma}\left(a, \frac{1}{2}, \mathsf{neg}\left(\frac{1}{b}\right)\right)}\right)\right)\right) \]

       21. distribute-neg-frac N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \mathsf{fma}\left(a, \frac{1}{2}, \color{blue}{\frac{\mathsf{neg}\left(1\right)}{b}}\right)\right)\right)\right) \]

       22. metadata-eval N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \mathsf{fma}\left(a, \frac{1}{2}, \frac{\color{blue}{-1}}{b}\right)\right)\right)\right) \]

       23. /-lowering-/.f64 34.0

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \mathsf{fma}\left(a, 0.5, \color{blue}{\frac{-1}{b}}\right)\right)\right)\right) \]

   11. Simplified 34.0%

       \[\leadsto x \cdot \color{blue}{\left(b \cdot \left(b \cdot \left(a \cdot \mathsf{fma}\left(a, 0.5, \frac{-1}{b}\right)\right)\right)\right)} \]

   12. Taylor expanded in b around inf

       \[\leadsto x \cdot \left(b \cdot \color{blue}{\left(\frac{1}{2} \cdot \left({a}^{2} \cdot b\right)\right)}\right) \]
   13. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \left(b \cdot \color{blue}{\left(\frac{1}{2} \cdot \left({a}^{2} \cdot b\right)\right)}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \left(b \cdot \left(\frac{1}{2} \cdot \color{blue}{\left({a}^{2} \cdot b\right)}\right)\right) \]

       3. unpow2 N/A

          \[\leadsto x \cdot \left(b \cdot \left(\frac{1}{2} \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot b\right)\right)\right) \]

       4. *-lowering-*.f64 51.1

          \[\leadsto x \cdot \left(b \cdot \left(0.5 \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot b\right)\right)\right) \]

   14. Simplified 51.1%

       \[\leadsto x \cdot \left(b \cdot \color{blue}{\left(0.5 \cdot \left(\left(a \cdot a\right) \cdot b\right)\right)}\right) \]

   if -2e10 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 1.99999999999999986e225

   1. Initial program 92.1%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      3. *-lowering-*.f64 69.2

         \[\leadsto x \cdot e^{-\color{blue}{a \cdot b}} \]

   5. Simplified 69.2%

      \[\leadsto x \cdot e^{\color{blue}{-a \cdot b}} \]

   6. Taylor expanded in b around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + b \cdot \left(-1 \cdot a + \frac{1}{2} \cdot \left({a}^{2} \cdot b\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(b \cdot \left(-1 \cdot a + \frac{1}{2} \cdot \left({a}^{2} \cdot b\right)\right) + 1\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(b, -1 \cdot a + \frac{1}{2} \cdot \left({a}^{2} \cdot b\right), 1\right)} \]

      3. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \color{blue}{\frac{1}{2} \cdot \left({a}^{2} \cdot b\right) + -1 \cdot a}, 1\right) \]

      4. *-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \color{blue}{\left({a}^{2} \cdot b\right) \cdot \frac{1}{2}} + -1 \cdot a, 1\right) \]

      5. *-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \color{blue}{\left(b \cdot {a}^{2}\right)} \cdot \frac{1}{2} + -1 \cdot a, 1\right) \]

      6. associate-*r* N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \color{blue}{b \cdot \left({a}^{2} \cdot \frac{1}{2}\right)} + -1 \cdot a, 1\right) \]

      7. *-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, b \cdot \color{blue}{\left(\frac{1}{2} \cdot {a}^{2}\right)} + -1 \cdot a, 1\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, \frac{1}{2} \cdot {a}^{2}, -1 \cdot a\right)}, 1\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\frac{1}{2} \cdot {a}^{2}}, -1 \cdot a\right), 1\right) \]

      10. unpow2 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{1}{2} \cdot \color{blue}{\left(a \cdot a\right)}, -1 \cdot a\right), 1\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{1}{2} \cdot \color{blue}{\left(a \cdot a\right)}, -1 \cdot a\right), 1\right) \]

      12. mul-1-neg N/A

          \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{1}{2} \cdot \left(a \cdot a\right), \color{blue}{\mathsf{neg}\left(a\right)}\right), 1\right) \]

      13. neg-lowering-neg.f64 65.7

          \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5 \cdot \left(a \cdot a\right), \color{blue}{-a}\right), 1\right) \]

   8. Simplified 65.7%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5 \cdot \left(a \cdot a\right), -a\right), 1\right)} \]

   9. Taylor expanded in b around inf

      \[\leadsto x \cdot \mathsf{fma}\left(b, \color{blue}{\frac{1}{2} \cdot \left({a}^{2} \cdot b\right)}, 1\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(b, \color{blue}{\left({a}^{2} \cdot b\right) \cdot \frac{1}{2}}, 1\right) \]

       2. *-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(b, \color{blue}{\left(b \cdot {a}^{2}\right)} \cdot \frac{1}{2}, 1\right) \]

       3. associate-*r* N/A

          \[\leadsto x \cdot \mathsf{fma}\left(b, \color{blue}{b \cdot \left({a}^{2} \cdot \frac{1}{2}\right)}, 1\right) \]

       4. *-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(b, b \cdot \color{blue}{\left(\frac{1}{2} \cdot {a}^{2}\right)}, 1\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(b, \color{blue}{b \cdot \left(\frac{1}{2} \cdot {a}^{2}\right)}, 1\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(b, b \cdot \color{blue}{\left(\frac{1}{2} \cdot {a}^{2}\right)}, 1\right) \]

       7. unpow2 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(b, b \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(a \cdot a\right)}\right), 1\right) \]

       8. *-lowering-*.f64 65.3

          \[\leadsto x \cdot \mathsf{fma}\left(b, b \cdot \left(0.5 \cdot \color{blue}{\left(a \cdot a\right)}\right), 1\right) \]

   11. Simplified 65.3%

       \[\leadsto x \cdot \mathsf{fma}\left(b, \color{blue}{b \cdot \left(0.5 \cdot \left(a \cdot a\right)\right)}, 1\right) \]

   if 1.99999999999999986e225 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 97.7%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(t \cdot y\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(t \cdot y\right)}} \]

      2. *-commutative N/A

         \[\leadsto x \cdot e^{\mathsf{neg}\left(\color{blue}{y \cdot t}\right)} \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      5. neg-lowering-neg.f64 62.6

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(-t\right)}} \]

   5. Simplified 62.6%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}} \]

   6. Taylor expanded in t around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + t \cdot \left(-1 \cdot y + \frac{1}{2} \cdot \left(t \cdot {y}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(t \cdot \left(-1 \cdot y + \frac{1}{2} \cdot \left(t \cdot {y}^{2}\right)\right) + 1\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(t, -1 \cdot y + \frac{1}{2} \cdot \left(t \cdot {y}^{2}\right), 1\right)} \]

      3. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(t, \color{blue}{\frac{1}{2} \cdot \left(t \cdot {y}^{2}\right) + -1 \cdot y}, 1\right) \]

      4. associate-*r* N/A

         \[\leadsto x \cdot \mathsf{fma}\left(t, \color{blue}{\left(\frac{1}{2} \cdot t\right) \cdot {y}^{2}} + -1 \cdot y, 1\right) \]

      5. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(t, \left(\frac{1}{2} \cdot t\right) \cdot \color{blue}{\left(y \cdot y\right)} + -1 \cdot y, 1\right) \]

      6. associate-*r* N/A

         \[\leadsto x \cdot \mathsf{fma}\left(t, \color{blue}{\left(\left(\frac{1}{2} \cdot t\right) \cdot y\right) \cdot y} + -1 \cdot y, 1\right) \]

      7. distribute-rgt-out N/A

         \[\leadsto x \cdot \mathsf{fma}\left(t, \color{blue}{y \cdot \left(\left(\frac{1}{2} \cdot t\right) \cdot y + -1\right)}, 1\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(t, \color{blue}{y \cdot \left(\left(\frac{1}{2} \cdot t\right) \cdot y + -1\right)}, 1\right) \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(t, y \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot t, y, -1\right)}, 1\right) \]

      10. *-lowering-*.f64 71.6

          \[\leadsto x \cdot \mathsf{fma}\left(t, y \cdot \mathsf{fma}\left(\color{blue}{0.5 \cdot t}, y, -1\right), 1\right) \]

   8. Simplified 71.6%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(t, y \cdot \mathsf{fma}\left(0.5 \cdot t, y, -1\right), 1\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -1.5 \cdot 10^{+286}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(t, y, 1\right)}\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -4 \cdot 10^{+244}:\\ \;\;\;\;a \cdot \left(a \cdot \frac{x}{a \cdot a}\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -20000000000:\\ \;\;\;\;x \cdot \left(b \cdot \left(0.5 \cdot \left(b \cdot \left(a \cdot a\right)\right)\right)\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 2 \cdot 10^{+225}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(b, b \cdot \left(0.5 \cdot \left(a \cdot a\right)\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(t, y \cdot \mathsf{fma}\left(t \cdot 0.5, y, -1\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 57.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+244}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(t \cdot t, y \cdot 0.5, t\right), 1\right)}\\ \mathbf{elif}\;t\_1 \leq -20000000000:\\ \;\;\;\;x \cdot \left(b \cdot \left(0.5 \cdot \left(b \cdot \left(a \cdot a\right)\right)\right)\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+225}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5 \cdot \left(a \cdot a\right), -a\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, x \cdot \left(y \cdot \mathsf{fma}\left(t \cdot 0.5, y, -1\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b)))))
   (if (<= t_1 -4e+244)
     (/ x (fma y (fma (* t t) (* y 0.5) t) 1.0))
     (if (<= t_1 -20000000000.0)
       (* x (* b (* 0.5 (* b (* a a)))))
       (if (<= t_1 2e+225)
         (* x (fma b (fma b (* 0.5 (* a a)) (- a)) 1.0))
         (fma t (* x (* y (fma (* t 0.5) y -1.0))) x))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	double tmp;
	if (t_1 <= -4e+244) {
		tmp = x / fma(y, fma((t * t), (y * 0.5), t), 1.0);
	} else if (t_1 <= -20000000000.0) {
		tmp = x * (b * (0.5 * (b * (a * a))));
	} else if (t_1 <= 2e+225) {
		tmp = x * fma(b, fma(b, (0.5 * (a * a)), -a), 1.0);
	} else {
		tmp = fma(t, (x * (y * fma((t * 0.5), y, -1.0))), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))
	tmp = 0.0
	if (t_1 <= -4e+244)
		tmp = Float64(x / fma(y, fma(Float64(t * t), Float64(y * 0.5), t), 1.0));
	elseif (t_1 <= -20000000000.0)
		tmp = Float64(x * Float64(b * Float64(0.5 * Float64(b * Float64(a * a)))));
	elseif (t_1 <= 2e+225)
		tmp = Float64(x * fma(b, fma(b, Float64(0.5 * Float64(a * a)), Float64(-a)), 1.0));
	else
		tmp = fma(t, Float64(x * Float64(y * fma(Float64(t * 0.5), y, -1.0))), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+244], N[(x / N[(y * N[(N[(t * t), $MachinePrecision] * N[(y * 0.5), $MachinePrecision] + t), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -20000000000.0], N[(x * N[(b * N[(0.5 * N[(b * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+225], N[(x * N[(b * N[(b * N[(0.5 * N[(a * a), $MachinePrecision]), $MachinePrecision] + (-a)), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(t * N[(x * N[(y * N[(N[(t * 0.5), $MachinePrecision] * y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -4.0000000000000003e244

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(t \cdot y\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(t \cdot y\right)}} \]

      2. *-commutative N/A

         \[\leadsto x \cdot e^{\mathsf{neg}\left(\color{blue}{y \cdot t}\right)} \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      5. neg-lowering-neg.f64 52.7

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(-t\right)}} \]

   5. Simplified 52.7%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}} \]
   6. Step-by-step derivation

      1. exp-prod N/A

         \[\leadsto x \cdot \color{blue}{{\left(e^{y}\right)}^{\left(\mathsf{neg}\left(t\right)\right)}} \]

      2. pow-neg N/A

         \[\leadsto x \cdot \color{blue}{\frac{1}{{\left(e^{y}\right)}^{t}}} \]

      3. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{x}{{\left(e^{y}\right)}^{t}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{{\left(e^{y}\right)}^{t}}} \]

      5. pow-exp N/A

         \[\leadsto \frac{x}{\color{blue}{e^{y \cdot t}}} \]

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{e^{y \cdot t}}} \]

      7. *-lowering-*.f64 52.7

         \[\leadsto \frac{x}{e^{\color{blue}{y \cdot t}}} \]

   7. Applied egg-rr 52.7%

      \[\leadsto \color{blue}{\frac{x}{e^{y \cdot t}}} \]

   8. Taylor expanded in y around 0

      \[\leadsto \frac{x}{\color{blue}{1 + y \cdot \left(t + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right)\right)}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot \left(t + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right)\right) + 1}} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(y, t + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right), 1\right)}} \]

      3. +-commutative N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(y, \color{blue}{\frac{1}{2} \cdot \left({t}^{2} \cdot y\right) + t}, 1\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(y, \color{blue}{\left({t}^{2} \cdot y\right) \cdot \frac{1}{2}} + t, 1\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(y, \color{blue}{{t}^{2} \cdot \left(y \cdot \frac{1}{2}\right)} + t, 1\right)} \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left({t}^{2}, y \cdot \frac{1}{2}, t\right)}, 1\right)} \]

      7. unpow2 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(\color{blue}{t \cdot t}, y \cdot \frac{1}{2}, t\right), 1\right)} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(\color{blue}{t \cdot t}, y \cdot \frac{1}{2}, t\right), 1\right)} \]

      9. *-lowering-*.f64 65.5

         \[\leadsto \frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(t \cdot t, \color{blue}{y \cdot 0.5}, t\right), 1\right)} \]

   10. Simplified 65.5%

       \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(t \cdot t, y \cdot 0.5, t\right), 1\right)}} \]

   if -4.0000000000000003e244 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -2e10

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      3. *-lowering-*.f64 42.4

         \[\leadsto x \cdot e^{-\color{blue}{a \cdot b}} \]

   5. Simplified 42.4%

      \[\leadsto x \cdot e^{\color{blue}{-a \cdot b}} \]

   6. Taylor expanded in a around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right) + 1\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, -1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right)} \]

      3. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right) \]

      4. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) + \left(\mathsf{neg}\left(b\right)\right)}, 1\right) \]

      5. associate-*r* N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\frac{1}{2} \cdot a\right) \cdot {b}^{2}} + \left(\mathsf{neg}\left(b\right)\right), 1\right) \]

      6. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \left(\frac{1}{2} \cdot a\right) \cdot \color{blue}{\left(b \cdot b\right)} + \left(\mathsf{neg}\left(b\right)\right), 1\right) \]

      7. associate-*r* N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\left(\frac{1}{2} \cdot a\right) \cdot b\right) \cdot b} + \left(\mathsf{neg}\left(b\right)\right), 1\right) \]

      8. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \left(\left(\frac{1}{2} \cdot a\right) \cdot b\right) \cdot b + \color{blue}{-1 \cdot b}, 1\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{b \cdot \left(\left(\frac{1}{2} \cdot a\right) \cdot b + -1\right)}, 1\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{b \cdot \left(\left(\frac{1}{2} \cdot a\right) \cdot b + -1\right)}, 1\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, b \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot a, b, -1\right)}, 1\right) \]

      12. *-lowering-*.f64 3.0

          \[\leadsto x \cdot \mathsf{fma}\left(a, b \cdot \mathsf{fma}\left(\color{blue}{0.5 \cdot a}, b, -1\right), 1\right) \]

   8. Simplified 3.0%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, b \cdot \mathsf{fma}\left(0.5 \cdot a, b, -1\right), 1\right)} \]

   9. Taylor expanded in b around inf

      \[\leadsto x \cdot \color{blue}{\left({b}^{2} \cdot \left(-1 \cdot \frac{a}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto x \cdot \left(\color{blue}{\left(b \cdot b\right)} \cdot \left(-1 \cdot \frac{a}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right) \]

       2. associate-*l* N/A

          \[\leadsto x \cdot \color{blue}{\left(b \cdot \left(b \cdot \left(-1 \cdot \frac{a}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \color{blue}{\left(b \cdot \left(b \cdot \left(-1 \cdot \frac{a}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \left(b \cdot \color{blue}{\left(b \cdot \left(-1 \cdot \frac{a}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)}\right) \]

       5. associate-*r/ N/A

          \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(\color{blue}{\frac{-1 \cdot a}{b}} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(\frac{\color{blue}{a \cdot -1}}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       7. associate-/l* N/A

          \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(\color{blue}{a \cdot \frac{-1}{b}} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       8. metadata-eval N/A

          \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \frac{\color{blue}{\mathsf{neg}\left(1\right)}}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       9. distribute-neg-frac N/A

          \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{b}\right)\right)} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       10. *-commutative N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(\mathsf{neg}\left(\frac{1}{b}\right)\right) + \color{blue}{{a}^{2} \cdot \frac{1}{2}}\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(\mathsf{neg}\left(\frac{1}{b}\right)\right) + \color{blue}{\left(a \cdot a\right)} \cdot \frac{1}{2}\right)\right)\right) \]

       12. associate-*l* N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(\mathsf{neg}\left(\frac{1}{b}\right)\right) + \color{blue}{a \cdot \left(a \cdot \frac{1}{2}\right)}\right)\right)\right) \]

       13. *-commutative N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(\mathsf{neg}\left(\frac{1}{b}\right)\right) + a \cdot \color{blue}{\left(\frac{1}{2} \cdot a\right)}\right)\right)\right) \]

       14. distribute-lft-out N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \color{blue}{\left(a \cdot \left(\left(\mathsf{neg}\left(\frac{1}{b}\right)\right) + \frac{1}{2} \cdot a\right)\right)}\right)\right) \]

       15. +-commutative N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \color{blue}{\left(\frac{1}{2} \cdot a + \left(\mathsf{neg}\left(\frac{1}{b}\right)\right)\right)}\right)\right)\right) \]

       16. sub-neg N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \color{blue}{\left(\frac{1}{2} \cdot a - \frac{1}{b}\right)}\right)\right)\right) \]

       17. *-lowering-*.f64 N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \color{blue}{\left(a \cdot \left(\frac{1}{2} \cdot a - \frac{1}{b}\right)\right)}\right)\right) \]

       18. sub-neg N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \color{blue}{\left(\frac{1}{2} \cdot a + \left(\mathsf{neg}\left(\frac{1}{b}\right)\right)\right)}\right)\right)\right) \]

       19. *-commutative N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(\color{blue}{a \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(\frac{1}{b}\right)\right)\right)\right)\right)\right) \]

       20. accelerator-lowering-fma.f64 N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \color{blue}{\mathsf{fma}\left(a, \frac{1}{2}, \mathsf{neg}\left(\frac{1}{b}\right)\right)}\right)\right)\right) \]

       21. distribute-neg-frac N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \mathsf{fma}\left(a, \frac{1}{2}, \color{blue}{\frac{\mathsf{neg}\left(1\right)}{b}}\right)\right)\right)\right) \]

       22. metadata-eval N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \mathsf{fma}\left(a, \frac{1}{2}, \frac{\color{blue}{-1}}{b}\right)\right)\right)\right) \]

       23. /-lowering-/.f64 34.0

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \mathsf{fma}\left(a, 0.5, \color{blue}{\frac{-1}{b}}\right)\right)\right)\right) \]

   11. Simplified 34.0%

       \[\leadsto x \cdot \color{blue}{\left(b \cdot \left(b \cdot \left(a \cdot \mathsf{fma}\left(a, 0.5, \frac{-1}{b}\right)\right)\right)\right)} \]

   12. Taylor expanded in b around inf

       \[\leadsto x \cdot \left(b \cdot \color{blue}{\left(\frac{1}{2} \cdot \left({a}^{2} \cdot b\right)\right)}\right) \]
   13. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \left(b \cdot \color{blue}{\left(\frac{1}{2} \cdot \left({a}^{2} \cdot b\right)\right)}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \left(b \cdot \left(\frac{1}{2} \cdot \color{blue}{\left({a}^{2} \cdot b\right)}\right)\right) \]

       3. unpow2 N/A

          \[\leadsto x \cdot \left(b \cdot \left(\frac{1}{2} \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot b\right)\right)\right) \]

       4. *-lowering-*.f64 51.1

          \[\leadsto x \cdot \left(b \cdot \left(0.5 \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot b\right)\right)\right) \]

   14. Simplified 51.1%

       \[\leadsto x \cdot \left(b \cdot \color{blue}{\left(0.5 \cdot \left(\left(a \cdot a\right) \cdot b\right)\right)}\right) \]

   if -2e10 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 1.99999999999999986e225

   1. Initial program 92.1%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      3. *-lowering-*.f64 69.2

         \[\leadsto x \cdot e^{-\color{blue}{a \cdot b}} \]

   5. Simplified 69.2%

      \[\leadsto x \cdot e^{\color{blue}{-a \cdot b}} \]

   6. Taylor expanded in b around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + b \cdot \left(-1 \cdot a + \frac{1}{2} \cdot \left({a}^{2} \cdot b\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(b \cdot \left(-1 \cdot a + \frac{1}{2} \cdot \left({a}^{2} \cdot b\right)\right) + 1\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(b, -1 \cdot a + \frac{1}{2} \cdot \left({a}^{2} \cdot b\right), 1\right)} \]

      3. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \color{blue}{\frac{1}{2} \cdot \left({a}^{2} \cdot b\right) + -1 \cdot a}, 1\right) \]

      4. *-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \color{blue}{\left({a}^{2} \cdot b\right) \cdot \frac{1}{2}} + -1 \cdot a, 1\right) \]

      5. *-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \color{blue}{\left(b \cdot {a}^{2}\right)} \cdot \frac{1}{2} + -1 \cdot a, 1\right) \]

      6. associate-*r* N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \color{blue}{b \cdot \left({a}^{2} \cdot \frac{1}{2}\right)} + -1 \cdot a, 1\right) \]

      7. *-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, b \cdot \color{blue}{\left(\frac{1}{2} \cdot {a}^{2}\right)} + -1 \cdot a, 1\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, \frac{1}{2} \cdot {a}^{2}, -1 \cdot a\right)}, 1\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\frac{1}{2} \cdot {a}^{2}}, -1 \cdot a\right), 1\right) \]

      10. unpow2 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{1}{2} \cdot \color{blue}{\left(a \cdot a\right)}, -1 \cdot a\right), 1\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{1}{2} \cdot \color{blue}{\left(a \cdot a\right)}, -1 \cdot a\right), 1\right) \]

      12. mul-1-neg N/A

          \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{1}{2} \cdot \left(a \cdot a\right), \color{blue}{\mathsf{neg}\left(a\right)}\right), 1\right) \]

      13. neg-lowering-neg.f64 65.7

          \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5 \cdot \left(a \cdot a\right), \color{blue}{-a}\right), 1\right) \]

   8. Simplified 65.7%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5 \cdot \left(a \cdot a\right), -a\right), 1\right)} \]

   if 1.99999999999999986e225 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 97.7%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(t \cdot y\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(t \cdot y\right)}} \]

      2. *-commutative N/A

         \[\leadsto x \cdot e^{\mathsf{neg}\left(\color{blue}{y \cdot t}\right)} \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      5. neg-lowering-neg.f64 62.6

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(-t\right)}} \]

   5. Simplified 62.6%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{x + t \cdot \left(-1 \cdot \left(x \cdot y\right) + \frac{1}{2} \cdot \left(t \cdot \left(x \cdot {y}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(x \cdot y\right) + \frac{1}{2} \cdot \left(t \cdot \left(x \cdot {y}^{2}\right)\right)\right) + x} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, -1 \cdot \left(x \cdot y\right) + \frac{1}{2} \cdot \left(t \cdot \left(x \cdot {y}^{2}\right)\right), x\right)} \]

   8. Simplified 73.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, x \cdot \left(y \cdot \mathsf{fma}\left(0.5 \cdot t, y, -1\right)\right), x\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -4 \cdot 10^{+244}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(t \cdot t, y \cdot 0.5, t\right), 1\right)}\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -20000000000:\\ \;\;\;\;x \cdot \left(b \cdot \left(0.5 \cdot \left(b \cdot \left(a \cdot a\right)\right)\right)\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 2 \cdot 10^{+225}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5 \cdot \left(a \cdot a\right), -a\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, x \cdot \left(y \cdot \mathsf{fma}\left(t \cdot 0.5, y, -1\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 54.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t\_1 \leq -1.5 \cdot 10^{+286}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(t, y, 1\right)}\\ \mathbf{elif}\;t\_1 \leq -4 \cdot 10^{+244}:\\ \;\;\;\;a \cdot \left(a \cdot \frac{x}{a \cdot a}\right)\\ \mathbf{elif}\;t\_1 \leq -20000000000:\\ \;\;\;\;x \cdot \left(b \cdot \left(0.5 \cdot \left(b \cdot \left(a \cdot a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(b, b \cdot \left(0.5 \cdot \left(a \cdot a\right)\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b)))))
   (if (<= t_1 -1.5e+286)
     (/ x (fma t y 1.0))
     (if (<= t_1 -4e+244)
       (* a (* a (/ x (* a a))))
       (if (<= t_1 -20000000000.0)
         (* x (* b (* 0.5 (* b (* a a)))))
         (* x (fma b (* b (* 0.5 (* a a))) 1.0)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	double tmp;
	if (t_1 <= -1.5e+286) {
		tmp = x / fma(t, y, 1.0);
	} else if (t_1 <= -4e+244) {
		tmp = a * (a * (x / (a * a)));
	} else if (t_1 <= -20000000000.0) {
		tmp = x * (b * (0.5 * (b * (a * a))));
	} else {
		tmp = x * fma(b, (b * (0.5 * (a * a))), 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))
	tmp = 0.0
	if (t_1 <= -1.5e+286)
		tmp = Float64(x / fma(t, y, 1.0));
	elseif (t_1 <= -4e+244)
		tmp = Float64(a * Float64(a * Float64(x / Float64(a * a))));
	elseif (t_1 <= -20000000000.0)
		tmp = Float64(x * Float64(b * Float64(0.5 * Float64(b * Float64(a * a)))));
	else
		tmp = Float64(x * fma(b, Float64(b * Float64(0.5 * Float64(a * a))), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1.5e+286], N[(x / N[(t * y + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -4e+244], N[(a * N[(a * N[(x / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -20000000000.0], N[(x * N[(b * N[(0.5 * N[(b * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(b * N[(b * N[(0.5 * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -1.4999999999999999e286

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(t \cdot y\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(t \cdot y\right)}} \]

      2. *-commutative N/A

         \[\leadsto x \cdot e^{\mathsf{neg}\left(\color{blue}{y \cdot t}\right)} \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      5. neg-lowering-neg.f64 66.3

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(-t\right)}} \]

   5. Simplified 66.3%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}} \]
   6. Step-by-step derivation

      1. exp-prod N/A

         \[\leadsto x \cdot \color{blue}{{\left(e^{y}\right)}^{\left(\mathsf{neg}\left(t\right)\right)}} \]

      2. pow-neg N/A

         \[\leadsto x \cdot \color{blue}{\frac{1}{{\left(e^{y}\right)}^{t}}} \]

      3. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{x}{{\left(e^{y}\right)}^{t}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{{\left(e^{y}\right)}^{t}}} \]

      5. pow-exp N/A

         \[\leadsto \frac{x}{\color{blue}{e^{y \cdot t}}} \]

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{e^{y \cdot t}}} \]

      7. *-lowering-*.f64 66.3

         \[\leadsto \frac{x}{e^{\color{blue}{y \cdot t}}} \]

   7. Applied egg-rr 66.3%

      \[\leadsto \color{blue}{\frac{x}{e^{y \cdot t}}} \]

   8. Taylor expanded in y around 0

      \[\leadsto \frac{x}{\color{blue}{1 + t \cdot y}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{t \cdot y + 1}} \]

      2. accelerator-lowering-fma.f64 59.4

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(t, y, 1\right)}} \]

   10. Simplified 59.4%

       \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(t, y, 1\right)}} \]

   if -1.4999999999999999e286 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -4.0000000000000003e244

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      3. *-lowering-*.f64 55.1

         \[\leadsto x \cdot e^{-\color{blue}{a \cdot b}} \]

   5. Simplified 55.1%

      \[\leadsto x \cdot e^{\color{blue}{-a \cdot b}} \]

   6. Taylor expanded in a around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right) + 1\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, -1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right)} \]

      3. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right) \]

      4. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) + \left(\mathsf{neg}\left(b\right)\right)}, 1\right) \]

      5. associate-*r* N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\frac{1}{2} \cdot a\right) \cdot {b}^{2}} + \left(\mathsf{neg}\left(b\right)\right), 1\right) \]

      6. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \left(\frac{1}{2} \cdot a\right) \cdot \color{blue}{\left(b \cdot b\right)} + \left(\mathsf{neg}\left(b\right)\right), 1\right) \]

      7. associate-*r* N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\left(\frac{1}{2} \cdot a\right) \cdot b\right) \cdot b} + \left(\mathsf{neg}\left(b\right)\right), 1\right) \]

      8. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \left(\left(\frac{1}{2} \cdot a\right) \cdot b\right) \cdot b + \color{blue}{-1 \cdot b}, 1\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{b \cdot \left(\left(\frac{1}{2} \cdot a\right) \cdot b + -1\right)}, 1\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{b \cdot \left(\left(\frac{1}{2} \cdot a\right) \cdot b + -1\right)}, 1\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, b \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot a, b, -1\right)}, 1\right) \]

      12. *-lowering-*.f64 2.3

          \[\leadsto x \cdot \mathsf{fma}\left(a, b \cdot \mathsf{fma}\left(\color{blue}{0.5 \cdot a}, b, -1\right), 1\right) \]

   8. Simplified 2.3%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, b \cdot \mathsf{fma}\left(0.5 \cdot a, b, -1\right), 1\right)} \]

   9. Taylor expanded in a around inf

      \[\leadsto \color{blue}{{a}^{2} \cdot \left(-1 \cdot \frac{b \cdot x}{a} + \left(\frac{1}{2} \cdot \left({b}^{2} \cdot x\right) + \frac{x}{{a}^{2}}\right)\right)} \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \color{blue}{\left(a \cdot a\right)} \cdot \left(-1 \cdot \frac{b \cdot x}{a} + \left(\frac{1}{2} \cdot \left({b}^{2} \cdot x\right) + \frac{x}{{a}^{2}}\right)\right) \]

       2. associate-*l* N/A

          \[\leadsto \color{blue}{a \cdot \left(a \cdot \left(-1 \cdot \frac{b \cdot x}{a} + \left(\frac{1}{2} \cdot \left({b}^{2} \cdot x\right) + \frac{x}{{a}^{2}}\right)\right)\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{a \cdot \left(a \cdot \left(-1 \cdot \frac{b \cdot x}{a} + \left(\frac{1}{2} \cdot \left({b}^{2} \cdot x\right) + \frac{x}{{a}^{2}}\right)\right)\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto a \cdot \color{blue}{\left(a \cdot \left(-1 \cdot \frac{b \cdot x}{a} + \left(\frac{1}{2} \cdot \left({b}^{2} \cdot x\right) + \frac{x}{{a}^{2}}\right)\right)\right)} \]

       5. associate-+r+ N/A

          \[\leadsto a \cdot \left(a \cdot \color{blue}{\left(\left(-1 \cdot \frac{b \cdot x}{a} + \frac{1}{2} \cdot \left({b}^{2} \cdot x\right)\right) + \frac{x}{{a}^{2}}\right)}\right) \]

   11. Simplified 9.4%

       \[\leadsto \color{blue}{a \cdot \left(a \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, x \cdot 0.5, \frac{x}{-a}\right), \frac{x}{a \cdot a}\right)\right)} \]

   12. Taylor expanded in b around 0

       \[\leadsto a \cdot \left(a \cdot \color{blue}{\frac{x}{{a}^{2}}}\right) \]
   13. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto a \cdot \left(a \cdot \color{blue}{\frac{x}{{a}^{2}}}\right) \]

       2. unpow2 N/A

          \[\leadsto a \cdot \left(a \cdot \frac{x}{\color{blue}{a \cdot a}}\right) \]

       3. *-lowering-*.f64 55.3

          \[\leadsto a \cdot \left(a \cdot \frac{x}{\color{blue}{a \cdot a}}\right) \]

   14. Simplified 55.3%

       \[\leadsto a \cdot \left(a \cdot \color{blue}{\frac{x}{a \cdot a}}\right) \]

   if -4.0000000000000003e244 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -2e10

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      3. *-lowering-*.f64 42.4

         \[\leadsto x \cdot e^{-\color{blue}{a \cdot b}} \]

   5. Simplified 42.4%

      \[\leadsto x \cdot e^{\color{blue}{-a \cdot b}} \]

   6. Taylor expanded in a around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right) + 1\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, -1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right)} \]

      3. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right) \]

      4. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) + \left(\mathsf{neg}\left(b\right)\right)}, 1\right) \]

      5. associate-*r* N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\frac{1}{2} \cdot a\right) \cdot {b}^{2}} + \left(\mathsf{neg}\left(b\right)\right), 1\right) \]

      6. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \left(\frac{1}{2} \cdot a\right) \cdot \color{blue}{\left(b \cdot b\right)} + \left(\mathsf{neg}\left(b\right)\right), 1\right) \]

      7. associate-*r* N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\left(\frac{1}{2} \cdot a\right) \cdot b\right) \cdot b} + \left(\mathsf{neg}\left(b\right)\right), 1\right) \]

      8. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \left(\left(\frac{1}{2} \cdot a\right) \cdot b\right) \cdot b + \color{blue}{-1 \cdot b}, 1\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{b \cdot \left(\left(\frac{1}{2} \cdot a\right) \cdot b + -1\right)}, 1\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{b \cdot \left(\left(\frac{1}{2} \cdot a\right) \cdot b + -1\right)}, 1\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, b \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot a, b, -1\right)}, 1\right) \]

      12. *-lowering-*.f64 3.0

          \[\leadsto x \cdot \mathsf{fma}\left(a, b \cdot \mathsf{fma}\left(\color{blue}{0.5 \cdot a}, b, -1\right), 1\right) \]

   8. Simplified 3.0%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, b \cdot \mathsf{fma}\left(0.5 \cdot a, b, -1\right), 1\right)} \]

   9. Taylor expanded in b around inf

      \[\leadsto x \cdot \color{blue}{\left({b}^{2} \cdot \left(-1 \cdot \frac{a}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto x \cdot \left(\color{blue}{\left(b \cdot b\right)} \cdot \left(-1 \cdot \frac{a}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right) \]

       2. associate-*l* N/A

          \[\leadsto x \cdot \color{blue}{\left(b \cdot \left(b \cdot \left(-1 \cdot \frac{a}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \color{blue}{\left(b \cdot \left(b \cdot \left(-1 \cdot \frac{a}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \left(b \cdot \color{blue}{\left(b \cdot \left(-1 \cdot \frac{a}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)}\right) \]

       5. associate-*r/ N/A

          \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(\color{blue}{\frac{-1 \cdot a}{b}} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(\frac{\color{blue}{a \cdot -1}}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       7. associate-/l* N/A

          \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(\color{blue}{a \cdot \frac{-1}{b}} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       8. metadata-eval N/A

          \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \frac{\color{blue}{\mathsf{neg}\left(1\right)}}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       9. distribute-neg-frac N/A

          \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{b}\right)\right)} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       10. *-commutative N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(\mathsf{neg}\left(\frac{1}{b}\right)\right) + \color{blue}{{a}^{2} \cdot \frac{1}{2}}\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(\mathsf{neg}\left(\frac{1}{b}\right)\right) + \color{blue}{\left(a \cdot a\right)} \cdot \frac{1}{2}\right)\right)\right) \]

       12. associate-*l* N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(\mathsf{neg}\left(\frac{1}{b}\right)\right) + \color{blue}{a \cdot \left(a \cdot \frac{1}{2}\right)}\right)\right)\right) \]

       13. *-commutative N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(\mathsf{neg}\left(\frac{1}{b}\right)\right) + a \cdot \color{blue}{\left(\frac{1}{2} \cdot a\right)}\right)\right)\right) \]

       14. distribute-lft-out N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \color{blue}{\left(a \cdot \left(\left(\mathsf{neg}\left(\frac{1}{b}\right)\right) + \frac{1}{2} \cdot a\right)\right)}\right)\right) \]

       15. +-commutative N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \color{blue}{\left(\frac{1}{2} \cdot a + \left(\mathsf{neg}\left(\frac{1}{b}\right)\right)\right)}\right)\right)\right) \]

       16. sub-neg N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \color{blue}{\left(\frac{1}{2} \cdot a - \frac{1}{b}\right)}\right)\right)\right) \]

       17. *-lowering-*.f64 N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \color{blue}{\left(a \cdot \left(\frac{1}{2} \cdot a - \frac{1}{b}\right)\right)}\right)\right) \]

       18. sub-neg N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \color{blue}{\left(\frac{1}{2} \cdot a + \left(\mathsf{neg}\left(\frac{1}{b}\right)\right)\right)}\right)\right)\right) \]

       19. *-commutative N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(\color{blue}{a \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(\frac{1}{b}\right)\right)\right)\right)\right)\right) \]

       20. accelerator-lowering-fma.f64 N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \color{blue}{\mathsf{fma}\left(a, \frac{1}{2}, \mathsf{neg}\left(\frac{1}{b}\right)\right)}\right)\right)\right) \]

       21. distribute-neg-frac N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \mathsf{fma}\left(a, \frac{1}{2}, \color{blue}{\frac{\mathsf{neg}\left(1\right)}{b}}\right)\right)\right)\right) \]

       22. metadata-eval N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \mathsf{fma}\left(a, \frac{1}{2}, \frac{\color{blue}{-1}}{b}\right)\right)\right)\right) \]

       23. /-lowering-/.f64 34.0

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \mathsf{fma}\left(a, 0.5, \color{blue}{\frac{-1}{b}}\right)\right)\right)\right) \]

   11. Simplified 34.0%

       \[\leadsto x \cdot \color{blue}{\left(b \cdot \left(b \cdot \left(a \cdot \mathsf{fma}\left(a, 0.5, \frac{-1}{b}\right)\right)\right)\right)} \]

   12. Taylor expanded in b around inf

       \[\leadsto x \cdot \left(b \cdot \color{blue}{\left(\frac{1}{2} \cdot \left({a}^{2} \cdot b\right)\right)}\right) \]
   13. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \left(b \cdot \color{blue}{\left(\frac{1}{2} \cdot \left({a}^{2} \cdot b\right)\right)}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \left(b \cdot \left(\frac{1}{2} \cdot \color{blue}{\left({a}^{2} \cdot b\right)}\right)\right) \]

       3. unpow2 N/A

          \[\leadsto x \cdot \left(b \cdot \left(\frac{1}{2} \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot b\right)\right)\right) \]

       4. *-lowering-*.f64 51.1

          \[\leadsto x \cdot \left(b \cdot \left(0.5 \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot b\right)\right)\right) \]

   14. Simplified 51.1%

       \[\leadsto x \cdot \left(b \cdot \color{blue}{\left(0.5 \cdot \left(\left(a \cdot a\right) \cdot b\right)\right)}\right) \]

   if -2e10 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 93.7%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      3. *-lowering-*.f64 62.3

         \[\leadsto x \cdot e^{-\color{blue}{a \cdot b}} \]

   5. Simplified 62.3%

      \[\leadsto x \cdot e^{\color{blue}{-a \cdot b}} \]

   6. Taylor expanded in b around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + b \cdot \left(-1 \cdot a + \frac{1}{2} \cdot \left({a}^{2} \cdot b\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(b \cdot \left(-1 \cdot a + \frac{1}{2} \cdot \left({a}^{2} \cdot b\right)\right) + 1\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(b, -1 \cdot a + \frac{1}{2} \cdot \left({a}^{2} \cdot b\right), 1\right)} \]

      3. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \color{blue}{\frac{1}{2} \cdot \left({a}^{2} \cdot b\right) + -1 \cdot a}, 1\right) \]

      4. *-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \color{blue}{\left({a}^{2} \cdot b\right) \cdot \frac{1}{2}} + -1 \cdot a, 1\right) \]

      5. *-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \color{blue}{\left(b \cdot {a}^{2}\right)} \cdot \frac{1}{2} + -1 \cdot a, 1\right) \]

      6. associate-*r* N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \color{blue}{b \cdot \left({a}^{2} \cdot \frac{1}{2}\right)} + -1 \cdot a, 1\right) \]

      7. *-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, b \cdot \color{blue}{\left(\frac{1}{2} \cdot {a}^{2}\right)} + -1 \cdot a, 1\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, \frac{1}{2} \cdot {a}^{2}, -1 \cdot a\right)}, 1\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\frac{1}{2} \cdot {a}^{2}}, -1 \cdot a\right), 1\right) \]

      10. unpow2 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{1}{2} \cdot \color{blue}{\left(a \cdot a\right)}, -1 \cdot a\right), 1\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{1}{2} \cdot \color{blue}{\left(a \cdot a\right)}, -1 \cdot a\right), 1\right) \]

      12. mul-1-neg N/A

          \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{1}{2} \cdot \left(a \cdot a\right), \color{blue}{\mathsf{neg}\left(a\right)}\right), 1\right) \]

      13. neg-lowering-neg.f64 61.1

          \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5 \cdot \left(a \cdot a\right), \color{blue}{-a}\right), 1\right) \]

   8. Simplified 61.1%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5 \cdot \left(a \cdot a\right), -a\right), 1\right)} \]

   9. Taylor expanded in b around inf

      \[\leadsto x \cdot \mathsf{fma}\left(b, \color{blue}{\frac{1}{2} \cdot \left({a}^{2} \cdot b\right)}, 1\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(b, \color{blue}{\left({a}^{2} \cdot b\right) \cdot \frac{1}{2}}, 1\right) \]

       2. *-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(b, \color{blue}{\left(b \cdot {a}^{2}\right)} \cdot \frac{1}{2}, 1\right) \]

       3. associate-*r* N/A

          \[\leadsto x \cdot \mathsf{fma}\left(b, \color{blue}{b \cdot \left({a}^{2} \cdot \frac{1}{2}\right)}, 1\right) \]

       4. *-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(b, b \cdot \color{blue}{\left(\frac{1}{2} \cdot {a}^{2}\right)}, 1\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(b, \color{blue}{b \cdot \left(\frac{1}{2} \cdot {a}^{2}\right)}, 1\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(b, b \cdot \color{blue}{\left(\frac{1}{2} \cdot {a}^{2}\right)}, 1\right) \]

       7. unpow2 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(b, b \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(a \cdot a\right)}\right), 1\right) \]

       8. *-lowering-*.f64 60.8

          \[\leadsto x \cdot \mathsf{fma}\left(b, b \cdot \left(0.5 \cdot \color{blue}{\left(a \cdot a\right)}\right), 1\right) \]

   11. Simplified 60.8%

       \[\leadsto x \cdot \mathsf{fma}\left(b, \color{blue}{b \cdot \left(0.5 \cdot \left(a \cdot a\right)\right)}, 1\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 58.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -1.5 \cdot 10^{+286}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(t, y, 1\right)}\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -4 \cdot 10^{+244}:\\ \;\;\;\;a \cdot \left(a \cdot \frac{x}{a \cdot a}\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -20000000000:\\ \;\;\;\;x \cdot \left(b \cdot \left(0.5 \cdot \left(b \cdot \left(a \cdot a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(b, b \cdot \left(0.5 \cdot \left(a \cdot a\right)\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 54.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(t, y, 1\right)}\\ \mathbf{elif}\;t\_1 \leq -20000000000:\\ \;\;\;\;x \cdot \left(b \cdot \left(0.5 \cdot \left(b \cdot \left(a \cdot a\right)\right)\right)\right)\\ \mathbf{elif}\;t\_1 \leq 0.0002:\\ \;\;\;\;\mathsf{fma}\left(\left(-z\right) - b, x \cdot a, x\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \left(x \cdot \left(0.5 \cdot \left(b \cdot b\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b)))))
   (if (<= t_1 (- INFINITY))
     (/ x (fma t y 1.0))
     (if (<= t_1 -20000000000.0)
       (* x (* b (* 0.5 (* b (* a a)))))
       (if (<= t_1 0.0002)
         (fma (- (- z) b) (* x a) x)
         (* a (* a (* x (* 0.5 (* b b))))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = x / fma(t, y, 1.0);
	} else if (t_1 <= -20000000000.0) {
		tmp = x * (b * (0.5 * (b * (a * a))));
	} else if (t_1 <= 0.0002) {
		tmp = fma((-z - b), (x * a), x);
	} else {
		tmp = a * (a * (x * (0.5 * (b * b))));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(x / fma(t, y, 1.0));
	elseif (t_1 <= -20000000000.0)
		tmp = Float64(x * Float64(b * Float64(0.5 * Float64(b * Float64(a * a)))));
	elseif (t_1 <= 0.0002)
		tmp = fma(Float64(Float64(-z) - b), Float64(x * a), x);
	else
		tmp = Float64(a * Float64(a * Float64(x * Float64(0.5 * Float64(b * b)))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(x / N[(t * y + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -20000000000.0], N[(x * N[(b * N[(0.5 * N[(b * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0002], N[(N[((-z) - b), $MachinePrecision] * N[(x * a), $MachinePrecision] + x), $MachinePrecision], N[(a * N[(a * N[(x * N[(0.5 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -inf.0

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(t \cdot y\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(t \cdot y\right)}} \]

      2. *-commutative N/A

         \[\leadsto x \cdot e^{\mathsf{neg}\left(\color{blue}{y \cdot t}\right)} \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      5. neg-lowering-neg.f64 76.9

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(-t\right)}} \]

   5. Simplified 76.9%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}} \]
   6. Step-by-step derivation

      1. exp-prod N/A

         \[\leadsto x \cdot \color{blue}{{\left(e^{y}\right)}^{\left(\mathsf{neg}\left(t\right)\right)}} \]

      2. pow-neg N/A

         \[\leadsto x \cdot \color{blue}{\frac{1}{{\left(e^{y}\right)}^{t}}} \]

      3. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{x}{{\left(e^{y}\right)}^{t}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{{\left(e^{y}\right)}^{t}}} \]

      5. pow-exp N/A

         \[\leadsto \frac{x}{\color{blue}{e^{y \cdot t}}} \]

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{e^{y \cdot t}}} \]

      7. *-lowering-*.f64 76.9

         \[\leadsto \frac{x}{e^{\color{blue}{y \cdot t}}} \]

   7. Applied egg-rr 76.9%

      \[\leadsto \color{blue}{\frac{x}{e^{y \cdot t}}} \]

   8. Taylor expanded in y around 0

      \[\leadsto \frac{x}{\color{blue}{1 + t \cdot y}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{t \cdot y + 1}} \]

      2. accelerator-lowering-fma.f64 63.5

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(t, y, 1\right)}} \]

   10. Simplified 63.5%

       \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(t, y, 1\right)}} \]

   if -inf.0 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -2e10

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      3. *-lowering-*.f64 42.0

         \[\leadsto x \cdot e^{-\color{blue}{a \cdot b}} \]

   5. Simplified 42.0%

      \[\leadsto x \cdot e^{\color{blue}{-a \cdot b}} \]

   6. Taylor expanded in a around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right) + 1\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, -1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right)} \]

      3. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right) \]

      4. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) + \left(\mathsf{neg}\left(b\right)\right)}, 1\right) \]

      5. associate-*r* N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\frac{1}{2} \cdot a\right) \cdot {b}^{2}} + \left(\mathsf{neg}\left(b\right)\right), 1\right) \]

      6. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \left(\frac{1}{2} \cdot a\right) \cdot \color{blue}{\left(b \cdot b\right)} + \left(\mathsf{neg}\left(b\right)\right), 1\right) \]

      7. associate-*r* N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\left(\frac{1}{2} \cdot a\right) \cdot b\right) \cdot b} + \left(\mathsf{neg}\left(b\right)\right), 1\right) \]

      8. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \left(\left(\frac{1}{2} \cdot a\right) \cdot b\right) \cdot b + \color{blue}{-1 \cdot b}, 1\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{b \cdot \left(\left(\frac{1}{2} \cdot a\right) \cdot b + -1\right)}, 1\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{b \cdot \left(\left(\frac{1}{2} \cdot a\right) \cdot b + -1\right)}, 1\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, b \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot a, b, -1\right)}, 1\right) \]

      12. *-lowering-*.f64 2.9

          \[\leadsto x \cdot \mathsf{fma}\left(a, b \cdot \mathsf{fma}\left(\color{blue}{0.5 \cdot a}, b, -1\right), 1\right) \]

   8. Simplified 2.9%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, b \cdot \mathsf{fma}\left(0.5 \cdot a, b, -1\right), 1\right)} \]

   9. Taylor expanded in b around inf

      \[\leadsto x \cdot \color{blue}{\left({b}^{2} \cdot \left(-1 \cdot \frac{a}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto x \cdot \left(\color{blue}{\left(b \cdot b\right)} \cdot \left(-1 \cdot \frac{a}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right) \]

       2. associate-*l* N/A

          \[\leadsto x \cdot \color{blue}{\left(b \cdot \left(b \cdot \left(-1 \cdot \frac{a}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \color{blue}{\left(b \cdot \left(b \cdot \left(-1 \cdot \frac{a}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \left(b \cdot \color{blue}{\left(b \cdot \left(-1 \cdot \frac{a}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)}\right) \]

       5. associate-*r/ N/A

          \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(\color{blue}{\frac{-1 \cdot a}{b}} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(\frac{\color{blue}{a \cdot -1}}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       7. associate-/l* N/A

          \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(\color{blue}{a \cdot \frac{-1}{b}} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       8. metadata-eval N/A

          \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \frac{\color{blue}{\mathsf{neg}\left(1\right)}}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       9. distribute-neg-frac N/A

          \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{b}\right)\right)} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       10. *-commutative N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(\mathsf{neg}\left(\frac{1}{b}\right)\right) + \color{blue}{{a}^{2} \cdot \frac{1}{2}}\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(\mathsf{neg}\left(\frac{1}{b}\right)\right) + \color{blue}{\left(a \cdot a\right)} \cdot \frac{1}{2}\right)\right)\right) \]

       12. associate-*l* N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(\mathsf{neg}\left(\frac{1}{b}\right)\right) + \color{blue}{a \cdot \left(a \cdot \frac{1}{2}\right)}\right)\right)\right) \]

       13. *-commutative N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(\mathsf{neg}\left(\frac{1}{b}\right)\right) + a \cdot \color{blue}{\left(\frac{1}{2} \cdot a\right)}\right)\right)\right) \]

       14. distribute-lft-out N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \color{blue}{\left(a \cdot \left(\left(\mathsf{neg}\left(\frac{1}{b}\right)\right) + \frac{1}{2} \cdot a\right)\right)}\right)\right) \]

       15. +-commutative N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \color{blue}{\left(\frac{1}{2} \cdot a + \left(\mathsf{neg}\left(\frac{1}{b}\right)\right)\right)}\right)\right)\right) \]

       16. sub-neg N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \color{blue}{\left(\frac{1}{2} \cdot a - \frac{1}{b}\right)}\right)\right)\right) \]

       17. *-lowering-*.f64 N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \color{blue}{\left(a \cdot \left(\frac{1}{2} \cdot a - \frac{1}{b}\right)\right)}\right)\right) \]

       18. sub-neg N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \color{blue}{\left(\frac{1}{2} \cdot a + \left(\mathsf{neg}\left(\frac{1}{b}\right)\right)\right)}\right)\right)\right) \]

       19. *-commutative N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(\color{blue}{a \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(\frac{1}{b}\right)\right)\right)\right)\right)\right) \]

       20. accelerator-lowering-fma.f64 N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \color{blue}{\mathsf{fma}\left(a, \frac{1}{2}, \mathsf{neg}\left(\frac{1}{b}\right)\right)}\right)\right)\right) \]

       21. distribute-neg-frac N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \mathsf{fma}\left(a, \frac{1}{2}, \color{blue}{\frac{\mathsf{neg}\left(1\right)}{b}}\right)\right)\right)\right) \]

       22. metadata-eval N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \mathsf{fma}\left(a, \frac{1}{2}, \frac{\color{blue}{-1}}{b}\right)\right)\right)\right) \]

       23. /-lowering-/.f64 31.8

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \mathsf{fma}\left(a, 0.5, \color{blue}{\frac{-1}{b}}\right)\right)\right)\right) \]

   11. Simplified 31.8%

       \[\leadsto x \cdot \color{blue}{\left(b \cdot \left(b \cdot \left(a \cdot \mathsf{fma}\left(a, 0.5, \frac{-1}{b}\right)\right)\right)\right)} \]

   12. Taylor expanded in b around inf

       \[\leadsto x \cdot \left(b \cdot \color{blue}{\left(\frac{1}{2} \cdot \left({a}^{2} \cdot b\right)\right)}\right) \]
   13. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \left(b \cdot \color{blue}{\left(\frac{1}{2} \cdot \left({a}^{2} \cdot b\right)\right)}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \left(b \cdot \left(\frac{1}{2} \cdot \color{blue}{\left({a}^{2} \cdot b\right)}\right)\right) \]

       3. unpow2 N/A

          \[\leadsto x \cdot \left(b \cdot \left(\frac{1}{2} \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot b\right)\right)\right) \]

       4. *-lowering-*.f64 46.1

          \[\leadsto x \cdot \left(b \cdot \left(0.5 \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot b\right)\right)\right) \]

   14. Simplified 46.1%

       \[\leadsto x \cdot \left(b \cdot \color{blue}{\left(0.5 \cdot \left(\left(a \cdot a\right) \cdot b\right)\right)}\right) \]

   if -2e10 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 2.0000000000000001e-4

   1. Initial program 88.6%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto x \cdot e^{\color{blue}{a \cdot \left(\log \left(1 - z\right) - b\right)}} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{a \cdot \left(\log \left(1 - z\right) - b\right)}} \]

      2. --lowering--.f64 N/A

         \[\leadsto x \cdot e^{a \cdot \color{blue}{\left(\log \left(1 - z\right) - b\right)}} \]

      3. sub-neg N/A

         \[\leadsto x \cdot e^{a \cdot \left(\log \color{blue}{\left(1 + \left(\mathsf{neg}\left(z\right)\right)\right)} - b\right)} \]

      4. accelerator-lowering-log1p.f64 N/A

         \[\leadsto x \cdot e^{a \cdot \left(\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(z\right)\right)} - b\right)} \]

      5. neg-lowering-neg.f64 97.1

         \[\leadsto x \cdot e^{a \cdot \left(\mathsf{log1p}\left(\color{blue}{-z}\right) - b\right)} \]

   5. Simplified 97.1%

      \[\leadsto x \cdot e^{\color{blue}{a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)}} \]

   6. Taylor expanded in z around 0

      \[\leadsto x \cdot e^{a \cdot \color{blue}{\left(-1 \cdot z - b\right)}} \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto x \cdot e^{a \cdot \color{blue}{\left(-1 \cdot z - b\right)}} \]

      2. mul-1-neg N/A

         \[\leadsto x \cdot e^{a \cdot \left(\color{blue}{\left(\mathsf{neg}\left(z\right)\right)} - b\right)} \]

      3. neg-lowering-neg.f64 97.1

         \[\leadsto x \cdot e^{a \cdot \left(\color{blue}{\left(-z\right)} - b\right)} \]

   8. Simplified 97.1%

      \[\leadsto x \cdot e^{a \cdot \color{blue}{\left(\left(-z\right) - b\right)}} \]

   9. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x + -1 \cdot \left(a \cdot \left(x \cdot \left(b + z\right)\right)\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{-1 \cdot \left(a \cdot \left(x \cdot \left(b + z\right)\right)\right) + x} \]

       2. mul-1-neg N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(a \cdot \left(x \cdot \left(b + z\right)\right)\right)\right)} + x \]

       3. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(x \cdot \left(b + z\right)\right) \cdot a}\right)\right) + x \]

       4. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\left(b + z\right) \cdot x\right)} \cdot a\right)\right) + x \]

       5. associate-*l* N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(b + z\right) \cdot \left(x \cdot a\right)}\right)\right) + x \]

       6. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(\left(b + z\right) \cdot \color{blue}{\left(a \cdot x\right)}\right)\right) + x \]

       7. distribute-lft-neg-in N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(b + z\right)\right)\right) \cdot \left(a \cdot x\right)} + x \]

       8. mul-1-neg N/A

          \[\leadsto \color{blue}{\left(-1 \cdot \left(b + z\right)\right)} \cdot \left(a \cdot x\right) + x \]

       9. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot \left(b + z\right), a \cdot x, x\right)} \]

       10. distribute-lft-in N/A

           \[\leadsto \mathsf{fma}\left(\color{blue}{-1 \cdot b + -1 \cdot z}, a \cdot x, x\right) \]

       11. +-commutative N/A

           \[\leadsto \mathsf{fma}\left(\color{blue}{-1 \cdot z + -1 \cdot b}, a \cdot x, x\right) \]

       12. mul-1-neg N/A

           \[\leadsto \mathsf{fma}\left(-1 \cdot z + \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}, a \cdot x, x\right) \]

       13. sub-neg N/A

           \[\leadsto \mathsf{fma}\left(\color{blue}{-1 \cdot z - b}, a \cdot x, x\right) \]

       14. --lowering--.f64 N/A

           \[\leadsto \mathsf{fma}\left(\color{blue}{-1 \cdot z - b}, a \cdot x, x\right) \]

       15. mul-1-neg N/A

           \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(z\right)\right)} - b, a \cdot x, x\right) \]

       16. neg-lowering-neg.f64 N/A

           \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(z\right)\right)} - b, a \cdot x, x\right) \]

       17. *-commutative N/A

           \[\leadsto \mathsf{fma}\left(\left(\mathsf{neg}\left(z\right)\right) - b, \color{blue}{x \cdot a}, x\right) \]

       18. *-lowering-*.f64 85.2

           \[\leadsto \mathsf{fma}\left(\left(-z\right) - b, \color{blue}{x \cdot a}, x\right) \]

   11. Simplified 85.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(-z\right) - b, x \cdot a, x\right)} \]

   if 2.0000000000000001e-4 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 96.2%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      3. *-lowering-*.f64 50.6

         \[\leadsto x \cdot e^{-\color{blue}{a \cdot b}} \]

   5. Simplified 50.6%

      \[\leadsto x \cdot e^{\color{blue}{-a \cdot b}} \]

   6. Taylor expanded in a around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right) + 1\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, -1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right)} \]

      3. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right) \]

      4. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) + \left(\mathsf{neg}\left(b\right)\right)}, 1\right) \]

      5. associate-*r* N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\frac{1}{2} \cdot a\right) \cdot {b}^{2}} + \left(\mathsf{neg}\left(b\right)\right), 1\right) \]

      6. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \left(\frac{1}{2} \cdot a\right) \cdot \color{blue}{\left(b \cdot b\right)} + \left(\mathsf{neg}\left(b\right)\right), 1\right) \]

      7. associate-*r* N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\left(\frac{1}{2} \cdot a\right) \cdot b\right) \cdot b} + \left(\mathsf{neg}\left(b\right)\right), 1\right) \]

      8. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \left(\left(\frac{1}{2} \cdot a\right) \cdot b\right) \cdot b + \color{blue}{-1 \cdot b}, 1\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{b \cdot \left(\left(\frac{1}{2} \cdot a\right) \cdot b + -1\right)}, 1\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{b \cdot \left(\left(\frac{1}{2} \cdot a\right) \cdot b + -1\right)}, 1\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, b \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot a, b, -1\right)}, 1\right) \]

      12. *-lowering-*.f64 43.5

          \[\leadsto x \cdot \mathsf{fma}\left(a, b \cdot \mathsf{fma}\left(\color{blue}{0.5 \cdot a}, b, -1\right), 1\right) \]

   8. Simplified 43.5%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, b \cdot \mathsf{fma}\left(0.5 \cdot a, b, -1\right), 1\right)} \]

   9. Taylor expanded in a around inf

      \[\leadsto \color{blue}{{a}^{2} \cdot \left(-1 \cdot \frac{b \cdot x}{a} + \left(\frac{1}{2} \cdot \left({b}^{2} \cdot x\right) + \frac{x}{{a}^{2}}\right)\right)} \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \color{blue}{\left(a \cdot a\right)} \cdot \left(-1 \cdot \frac{b \cdot x}{a} + \left(\frac{1}{2} \cdot \left({b}^{2} \cdot x\right) + \frac{x}{{a}^{2}}\right)\right) \]

       2. associate-*l* N/A

          \[\leadsto \color{blue}{a \cdot \left(a \cdot \left(-1 \cdot \frac{b \cdot x}{a} + \left(\frac{1}{2} \cdot \left({b}^{2} \cdot x\right) + \frac{x}{{a}^{2}}\right)\right)\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{a \cdot \left(a \cdot \left(-1 \cdot \frac{b \cdot x}{a} + \left(\frac{1}{2} \cdot \left({b}^{2} \cdot x\right) + \frac{x}{{a}^{2}}\right)\right)\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto a \cdot \color{blue}{\left(a \cdot \left(-1 \cdot \frac{b \cdot x}{a} + \left(\frac{1}{2} \cdot \left({b}^{2} \cdot x\right) + \frac{x}{{a}^{2}}\right)\right)\right)} \]

       5. associate-+r+ N/A

          \[\leadsto a \cdot \left(a \cdot \color{blue}{\left(\left(-1 \cdot \frac{b \cdot x}{a} + \frac{1}{2} \cdot \left({b}^{2} \cdot x\right)\right) + \frac{x}{{a}^{2}}\right)}\right) \]

   11. Simplified 53.2%

       \[\leadsto \color{blue}{a \cdot \left(a \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, x \cdot 0.5, \frac{x}{-a}\right), \frac{x}{a \cdot a}\right)\right)} \]

   12. Taylor expanded in b around inf

       \[\leadsto a \cdot \left(a \cdot \color{blue}{\left(\frac{1}{2} \cdot \left({b}^{2} \cdot x\right)\right)}\right) \]
   13. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto a \cdot \left(a \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot {b}^{2}\right) \cdot x\right)}\right) \]

       2. *-commutative N/A

          \[\leadsto a \cdot \left(a \cdot \color{blue}{\left(x \cdot \left(\frac{1}{2} \cdot {b}^{2}\right)\right)}\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto a \cdot \left(a \cdot \color{blue}{\left(x \cdot \left(\frac{1}{2} \cdot {b}^{2}\right)\right)}\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto a \cdot \left(a \cdot \left(x \cdot \color{blue}{\left(\frac{1}{2} \cdot {b}^{2}\right)}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto a \cdot \left(a \cdot \left(x \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(b \cdot b\right)}\right)\right)\right) \]

       6. *-lowering-*.f64 48.3

          \[\leadsto a \cdot \left(a \cdot \left(x \cdot \left(0.5 \cdot \color{blue}{\left(b \cdot b\right)}\right)\right)\right) \]

   14. Simplified 48.3%

       \[\leadsto a \cdot \left(a \cdot \color{blue}{\left(x \cdot \left(0.5 \cdot \left(b \cdot b\right)\right)\right)}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -\infty:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(t, y, 1\right)}\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -20000000000:\\ \;\;\;\;x \cdot \left(b \cdot \left(0.5 \cdot \left(b \cdot \left(a \cdot a\right)\right)\right)\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 0.0002:\\ \;\;\;\;\mathsf{fma}\left(\left(-z\right) - b, x \cdot a, x\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \left(x \cdot \left(0.5 \cdot \left(b \cdot b\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 54.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ t_2 := a \cdot \left(a \cdot \left(x \cdot \left(0.5 \cdot \left(b \cdot b\right)\right)\right)\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(t, y, 1\right)}\\ \mathbf{elif}\;t\_1 \leq -20000000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0.0002:\\ \;\;\;\;\mathsf{fma}\left(\left(-z\right) - b, x \cdot a, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b))))
        (t_2 (* a (* a (* x (* 0.5 (* b b)))))))
   (if (<= t_1 (- INFINITY))
     (/ x (fma t y 1.0))
     (if (<= t_1 -20000000000.0)
       t_2
       (if (<= t_1 0.0002) (fma (- (- z) b) (* x a) x) t_2)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	double t_2 = a * (a * (x * (0.5 * (b * b))));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = x / fma(t, y, 1.0);
	} else if (t_1 <= -20000000000.0) {
		tmp = t_2;
	} else if (t_1 <= 0.0002) {
		tmp = fma((-z - b), (x * a), x);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))
	t_2 = Float64(a * Float64(a * Float64(x * Float64(0.5 * Float64(b * b)))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(x / fma(t, y, 1.0));
	elseif (t_1 <= -20000000000.0)
		tmp = t_2;
	elseif (t_1 <= 0.0002)
		tmp = fma(Float64(Float64(-z) - b), Float64(x * a), x);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(a * N[(a * N[(x * N[(0.5 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(x / N[(t * y + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -20000000000.0], t$95$2, If[LessEqual[t$95$1, 0.0002], N[(N[((-z) - b), $MachinePrecision] * N[(x * a), $MachinePrecision] + x), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -inf.0

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(t \cdot y\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(t \cdot y\right)}} \]

      2. *-commutative N/A

         \[\leadsto x \cdot e^{\mathsf{neg}\left(\color{blue}{y \cdot t}\right)} \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      5. neg-lowering-neg.f64 76.9

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(-t\right)}} \]

   5. Simplified 76.9%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}} \]
   6. Step-by-step derivation

      1. exp-prod N/A

         \[\leadsto x \cdot \color{blue}{{\left(e^{y}\right)}^{\left(\mathsf{neg}\left(t\right)\right)}} \]

      2. pow-neg N/A

         \[\leadsto x \cdot \color{blue}{\frac{1}{{\left(e^{y}\right)}^{t}}} \]

      3. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{x}{{\left(e^{y}\right)}^{t}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{{\left(e^{y}\right)}^{t}}} \]

      5. pow-exp N/A

         \[\leadsto \frac{x}{\color{blue}{e^{y \cdot t}}} \]

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{e^{y \cdot t}}} \]

      7. *-lowering-*.f64 76.9

         \[\leadsto \frac{x}{e^{\color{blue}{y \cdot t}}} \]

   7. Applied egg-rr 76.9%

      \[\leadsto \color{blue}{\frac{x}{e^{y \cdot t}}} \]

   8. Taylor expanded in y around 0

      \[\leadsto \frac{x}{\color{blue}{1 + t \cdot y}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{t \cdot y + 1}} \]

      2. accelerator-lowering-fma.f64 63.5

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(t, y, 1\right)}} \]

   10. Simplified 63.5%

       \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(t, y, 1\right)}} \]

   if -inf.0 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -2e10 or 2.0000000000000001e-4 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 97.9%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      3. *-lowering-*.f64 46.9

         \[\leadsto x \cdot e^{-\color{blue}{a \cdot b}} \]

   5. Simplified 46.9%

      \[\leadsto x \cdot e^{\color{blue}{-a \cdot b}} \]

   6. Taylor expanded in a around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right) + 1\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, -1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right)} \]

      3. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right) \]

      4. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) + \left(\mathsf{neg}\left(b\right)\right)}, 1\right) \]

      5. associate-*r* N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\frac{1}{2} \cdot a\right) \cdot {b}^{2}} + \left(\mathsf{neg}\left(b\right)\right), 1\right) \]

      6. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \left(\frac{1}{2} \cdot a\right) \cdot \color{blue}{\left(b \cdot b\right)} + \left(\mathsf{neg}\left(b\right)\right), 1\right) \]

      7. associate-*r* N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\left(\frac{1}{2} \cdot a\right) \cdot b\right) \cdot b} + \left(\mathsf{neg}\left(b\right)\right), 1\right) \]

      8. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \left(\left(\frac{1}{2} \cdot a\right) \cdot b\right) \cdot b + \color{blue}{-1 \cdot b}, 1\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{b \cdot \left(\left(\frac{1}{2} \cdot a\right) \cdot b + -1\right)}, 1\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{b \cdot \left(\left(\frac{1}{2} \cdot a\right) \cdot b + -1\right)}, 1\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, b \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot a, b, -1\right)}, 1\right) \]

      12. *-lowering-*.f64 25.8

          \[\leadsto x \cdot \mathsf{fma}\left(a, b \cdot \mathsf{fma}\left(\color{blue}{0.5 \cdot a}, b, -1\right), 1\right) \]

   8. Simplified 25.8%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, b \cdot \mathsf{fma}\left(0.5 \cdot a, b, -1\right), 1\right)} \]

   9. Taylor expanded in a around inf

      \[\leadsto \color{blue}{{a}^{2} \cdot \left(-1 \cdot \frac{b \cdot x}{a} + \left(\frac{1}{2} \cdot \left({b}^{2} \cdot x\right) + \frac{x}{{a}^{2}}\right)\right)} \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \color{blue}{\left(a \cdot a\right)} \cdot \left(-1 \cdot \frac{b \cdot x}{a} + \left(\frac{1}{2} \cdot \left({b}^{2} \cdot x\right) + \frac{x}{{a}^{2}}\right)\right) \]

       2. associate-*l* N/A

          \[\leadsto \color{blue}{a \cdot \left(a \cdot \left(-1 \cdot \frac{b \cdot x}{a} + \left(\frac{1}{2} \cdot \left({b}^{2} \cdot x\right) + \frac{x}{{a}^{2}}\right)\right)\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{a \cdot \left(a \cdot \left(-1 \cdot \frac{b \cdot x}{a} + \left(\frac{1}{2} \cdot \left({b}^{2} \cdot x\right) + \frac{x}{{a}^{2}}\right)\right)\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto a \cdot \color{blue}{\left(a \cdot \left(-1 \cdot \frac{b \cdot x}{a} + \left(\frac{1}{2} \cdot \left({b}^{2} \cdot x\right) + \frac{x}{{a}^{2}}\right)\right)\right)} \]

       5. associate-+r+ N/A

          \[\leadsto a \cdot \left(a \cdot \color{blue}{\left(\left(-1 \cdot \frac{b \cdot x}{a} + \frac{1}{2} \cdot \left({b}^{2} \cdot x\right)\right) + \frac{x}{{a}^{2}}\right)}\right) \]

   11. Simplified 35.2%

       \[\leadsto \color{blue}{a \cdot \left(a \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, x \cdot 0.5, \frac{x}{-a}\right), \frac{x}{a \cdot a}\right)\right)} \]

   12. Taylor expanded in b around inf

       \[\leadsto a \cdot \left(a \cdot \color{blue}{\left(\frac{1}{2} \cdot \left({b}^{2} \cdot x\right)\right)}\right) \]
   13. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto a \cdot \left(a \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot {b}^{2}\right) \cdot x\right)}\right) \]

       2. *-commutative N/A

          \[\leadsto a \cdot \left(a \cdot \color{blue}{\left(x \cdot \left(\frac{1}{2} \cdot {b}^{2}\right)\right)}\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto a \cdot \left(a \cdot \color{blue}{\left(x \cdot \left(\frac{1}{2} \cdot {b}^{2}\right)\right)}\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto a \cdot \left(a \cdot \left(x \cdot \color{blue}{\left(\frac{1}{2} \cdot {b}^{2}\right)}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto a \cdot \left(a \cdot \left(x \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(b \cdot b\right)}\right)\right)\right) \]

       6. *-lowering-*.f64 46.8

          \[\leadsto a \cdot \left(a \cdot \left(x \cdot \left(0.5 \cdot \color{blue}{\left(b \cdot b\right)}\right)\right)\right) \]

   14. Simplified 46.8%

       \[\leadsto a \cdot \left(a \cdot \color{blue}{\left(x \cdot \left(0.5 \cdot \left(b \cdot b\right)\right)\right)}\right) \]

   if -2e10 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 2.0000000000000001e-4

   1. Initial program 88.6%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto x \cdot e^{\color{blue}{a \cdot \left(\log \left(1 - z\right) - b\right)}} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{a \cdot \left(\log \left(1 - z\right) - b\right)}} \]

      2. --lowering--.f64 N/A

         \[\leadsto x \cdot e^{a \cdot \color{blue}{\left(\log \left(1 - z\right) - b\right)}} \]

      3. sub-neg N/A

         \[\leadsto x \cdot e^{a \cdot \left(\log \color{blue}{\left(1 + \left(\mathsf{neg}\left(z\right)\right)\right)} - b\right)} \]

      4. accelerator-lowering-log1p.f64 N/A

         \[\leadsto x \cdot e^{a \cdot \left(\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(z\right)\right)} - b\right)} \]

      5. neg-lowering-neg.f64 97.1

         \[\leadsto x \cdot e^{a \cdot \left(\mathsf{log1p}\left(\color{blue}{-z}\right) - b\right)} \]

   5. Simplified 97.1%

      \[\leadsto x \cdot e^{\color{blue}{a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)}} \]

   6. Taylor expanded in z around 0

      \[\leadsto x \cdot e^{a \cdot \color{blue}{\left(-1 \cdot z - b\right)}} \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto x \cdot e^{a \cdot \color{blue}{\left(-1 \cdot z - b\right)}} \]

      2. mul-1-neg N/A

         \[\leadsto x \cdot e^{a \cdot \left(\color{blue}{\left(\mathsf{neg}\left(z\right)\right)} - b\right)} \]

      3. neg-lowering-neg.f64 97.1

         \[\leadsto x \cdot e^{a \cdot \left(\color{blue}{\left(-z\right)} - b\right)} \]

   8. Simplified 97.1%

      \[\leadsto x \cdot e^{a \cdot \color{blue}{\left(\left(-z\right) - b\right)}} \]

   9. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x + -1 \cdot \left(a \cdot \left(x \cdot \left(b + z\right)\right)\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{-1 \cdot \left(a \cdot \left(x \cdot \left(b + z\right)\right)\right) + x} \]

       2. mul-1-neg N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(a \cdot \left(x \cdot \left(b + z\right)\right)\right)\right)} + x \]

       3. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(x \cdot \left(b + z\right)\right) \cdot a}\right)\right) + x \]

       4. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\left(b + z\right) \cdot x\right)} \cdot a\right)\right) + x \]

       5. associate-*l* N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(b + z\right) \cdot \left(x \cdot a\right)}\right)\right) + x \]

       6. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(\left(b + z\right) \cdot \color{blue}{\left(a \cdot x\right)}\right)\right) + x \]

       7. distribute-lft-neg-in N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(b + z\right)\right)\right) \cdot \left(a \cdot x\right)} + x \]

       8. mul-1-neg N/A

          \[\leadsto \color{blue}{\left(-1 \cdot \left(b + z\right)\right)} \cdot \left(a \cdot x\right) + x \]

       9. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot \left(b + z\right), a \cdot x, x\right)} \]

       10. distribute-lft-in N/A

           \[\leadsto \mathsf{fma}\left(\color{blue}{-1 \cdot b + -1 \cdot z}, a \cdot x, x\right) \]

       11. +-commutative N/A

           \[\leadsto \mathsf{fma}\left(\color{blue}{-1 \cdot z + -1 \cdot b}, a \cdot x, x\right) \]

       12. mul-1-neg N/A

           \[\leadsto \mathsf{fma}\left(-1 \cdot z + \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}, a \cdot x, x\right) \]

       13. sub-neg N/A

           \[\leadsto \mathsf{fma}\left(\color{blue}{-1 \cdot z - b}, a \cdot x, x\right) \]

       14. --lowering--.f64 N/A

           \[\leadsto \mathsf{fma}\left(\color{blue}{-1 \cdot z - b}, a \cdot x, x\right) \]

       15. mul-1-neg N/A

           \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(z\right)\right)} - b, a \cdot x, x\right) \]

       16. neg-lowering-neg.f64 N/A

           \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(z\right)\right)} - b, a \cdot x, x\right) \]

       17. *-commutative N/A

           \[\leadsto \mathsf{fma}\left(\left(\mathsf{neg}\left(z\right)\right) - b, \color{blue}{x \cdot a}, x\right) \]

       18. *-lowering-*.f64 85.2

           \[\leadsto \mathsf{fma}\left(\left(-z\right) - b, \color{blue}{x \cdot a}, x\right) \]

   11. Simplified 85.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(-z\right) - b, x \cdot a, x\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 39.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(t, y, 1\right)}\\ \mathbf{elif}\;t\_1 \leq -20000000000:\\ \;\;\;\;-x \cdot \left(a \cdot b\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+152}:\\ \;\;\;\;\mathsf{fma}\left(\left(-z\right) - b, x \cdot a, x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - y \cdot t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b)))))
   (if (<= t_1 (- INFINITY))
     (/ x (fma t y 1.0))
     (if (<= t_1 -20000000000.0)
       (- (* x (* a b)))
       (if (<= t_1 1e+152)
         (fma (- (- z) b) (* x a) x)
         (* x (- 1.0 (* y t))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = x / fma(t, y, 1.0);
	} else if (t_1 <= -20000000000.0) {
		tmp = -(x * (a * b));
	} else if (t_1 <= 1e+152) {
		tmp = fma((-z - b), (x * a), x);
	} else {
		tmp = x * (1.0 - (y * t));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(x / fma(t, y, 1.0));
	elseif (t_1 <= -20000000000.0)
		tmp = Float64(-Float64(x * Float64(a * b)));
	elseif (t_1 <= 1e+152)
		tmp = fma(Float64(Float64(-z) - b), Float64(x * a), x);
	else
		tmp = Float64(x * Float64(1.0 - Float64(y * t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(x / N[(t * y + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -20000000000.0], (-N[(x * N[(a * b), $MachinePrecision]), $MachinePrecision]), If[LessEqual[t$95$1, 1e+152], N[(N[((-z) - b), $MachinePrecision] * N[(x * a), $MachinePrecision] + x), $MachinePrecision], N[(x * N[(1.0 - N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -inf.0

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(t \cdot y\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(t \cdot y\right)}} \]

      2. *-commutative N/A

         \[\leadsto x \cdot e^{\mathsf{neg}\left(\color{blue}{y \cdot t}\right)} \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      5. neg-lowering-neg.f64 76.9

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(-t\right)}} \]

   5. Simplified 76.9%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}} \]
   6. Step-by-step derivation

      1. exp-prod N/A

         \[\leadsto x \cdot \color{blue}{{\left(e^{y}\right)}^{\left(\mathsf{neg}\left(t\right)\right)}} \]

      2. pow-neg N/A

         \[\leadsto x \cdot \color{blue}{\frac{1}{{\left(e^{y}\right)}^{t}}} \]

      3. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{x}{{\left(e^{y}\right)}^{t}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{{\left(e^{y}\right)}^{t}}} \]

      5. pow-exp N/A

         \[\leadsto \frac{x}{\color{blue}{e^{y \cdot t}}} \]

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{e^{y \cdot t}}} \]

      7. *-lowering-*.f64 76.9

         \[\leadsto \frac{x}{e^{\color{blue}{y \cdot t}}} \]

   7. Applied egg-rr 76.9%

      \[\leadsto \color{blue}{\frac{x}{e^{y \cdot t}}} \]

   8. Taylor expanded in y around 0

      \[\leadsto \frac{x}{\color{blue}{1 + t \cdot y}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{t \cdot y + 1}} \]

      2. accelerator-lowering-fma.f64 63.5

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(t, y, 1\right)}} \]

   10. Simplified 63.5%

       \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(t, y, 1\right)}} \]

   if -inf.0 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -2e10

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      3. *-lowering-*.f64 42.0

         \[\leadsto x \cdot e^{-\color{blue}{a \cdot b}} \]

   5. Simplified 42.0%

      \[\leadsto x \cdot e^{\color{blue}{-a \cdot b}} \]

   6. Taylor expanded in a around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right) + 1\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, -1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right)} \]

      3. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right) \]

      4. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) + \left(\mathsf{neg}\left(b\right)\right)}, 1\right) \]

      5. associate-*r* N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\frac{1}{2} \cdot a\right) \cdot {b}^{2}} + \left(\mathsf{neg}\left(b\right)\right), 1\right) \]

      6. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \left(\frac{1}{2} \cdot a\right) \cdot \color{blue}{\left(b \cdot b\right)} + \left(\mathsf{neg}\left(b\right)\right), 1\right) \]

      7. associate-*r* N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\left(\frac{1}{2} \cdot a\right) \cdot b\right) \cdot b} + \left(\mathsf{neg}\left(b\right)\right), 1\right) \]

      8. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \left(\left(\frac{1}{2} \cdot a\right) \cdot b\right) \cdot b + \color{blue}{-1 \cdot b}, 1\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{b \cdot \left(\left(\frac{1}{2} \cdot a\right) \cdot b + -1\right)}, 1\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{b \cdot \left(\left(\frac{1}{2} \cdot a\right) \cdot b + -1\right)}, 1\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, b \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot a, b, -1\right)}, 1\right) \]

      12. *-lowering-*.f64 2.9

          \[\leadsto x \cdot \mathsf{fma}\left(a, b \cdot \mathsf{fma}\left(\color{blue}{0.5 \cdot a}, b, -1\right), 1\right) \]

   8. Simplified 2.9%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, b \cdot \mathsf{fma}\left(0.5 \cdot a, b, -1\right), 1\right)} \]

   9. Taylor expanded in b around inf

      \[\leadsto x \cdot \color{blue}{\left({b}^{2} \cdot \left(-1 \cdot \frac{a}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto x \cdot \left(\color{blue}{\left(b \cdot b\right)} \cdot \left(-1 \cdot \frac{a}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right) \]

       2. associate-*l* N/A

          \[\leadsto x \cdot \color{blue}{\left(b \cdot \left(b \cdot \left(-1 \cdot \frac{a}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \color{blue}{\left(b \cdot \left(b \cdot \left(-1 \cdot \frac{a}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \left(b \cdot \color{blue}{\left(b \cdot \left(-1 \cdot \frac{a}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)}\right) \]

       5. associate-*r/ N/A

          \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(\color{blue}{\frac{-1 \cdot a}{b}} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(\frac{\color{blue}{a \cdot -1}}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       7. associate-/l* N/A

          \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(\color{blue}{a \cdot \frac{-1}{b}} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       8. metadata-eval N/A

          \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \frac{\color{blue}{\mathsf{neg}\left(1\right)}}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       9. distribute-neg-frac N/A

          \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{b}\right)\right)} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       10. *-commutative N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(\mathsf{neg}\left(\frac{1}{b}\right)\right) + \color{blue}{{a}^{2} \cdot \frac{1}{2}}\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(\mathsf{neg}\left(\frac{1}{b}\right)\right) + \color{blue}{\left(a \cdot a\right)} \cdot \frac{1}{2}\right)\right)\right) \]

       12. associate-*l* N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(\mathsf{neg}\left(\frac{1}{b}\right)\right) + \color{blue}{a \cdot \left(a \cdot \frac{1}{2}\right)}\right)\right)\right) \]

       13. *-commutative N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(\mathsf{neg}\left(\frac{1}{b}\right)\right) + a \cdot \color{blue}{\left(\frac{1}{2} \cdot a\right)}\right)\right)\right) \]

       14. distribute-lft-out N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \color{blue}{\left(a \cdot \left(\left(\mathsf{neg}\left(\frac{1}{b}\right)\right) + \frac{1}{2} \cdot a\right)\right)}\right)\right) \]

       15. +-commutative N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \color{blue}{\left(\frac{1}{2} \cdot a + \left(\mathsf{neg}\left(\frac{1}{b}\right)\right)\right)}\right)\right)\right) \]

       16. sub-neg N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \color{blue}{\left(\frac{1}{2} \cdot a - \frac{1}{b}\right)}\right)\right)\right) \]

       17. *-lowering-*.f64 N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \color{blue}{\left(a \cdot \left(\frac{1}{2} \cdot a - \frac{1}{b}\right)\right)}\right)\right) \]

       18. sub-neg N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \color{blue}{\left(\frac{1}{2} \cdot a + \left(\mathsf{neg}\left(\frac{1}{b}\right)\right)\right)}\right)\right)\right) \]

       19. *-commutative N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(\color{blue}{a \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(\frac{1}{b}\right)\right)\right)\right)\right)\right) \]

       20. accelerator-lowering-fma.f64 N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \color{blue}{\mathsf{fma}\left(a, \frac{1}{2}, \mathsf{neg}\left(\frac{1}{b}\right)\right)}\right)\right)\right) \]

       21. distribute-neg-frac N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \mathsf{fma}\left(a, \frac{1}{2}, \color{blue}{\frac{\mathsf{neg}\left(1\right)}{b}}\right)\right)\right)\right) \]

       22. metadata-eval N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \mathsf{fma}\left(a, \frac{1}{2}, \frac{\color{blue}{-1}}{b}\right)\right)\right)\right) \]

       23. /-lowering-/.f64 31.8

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \mathsf{fma}\left(a, 0.5, \color{blue}{\frac{-1}{b}}\right)\right)\right)\right) \]

   11. Simplified 31.8%

       \[\leadsto x \cdot \color{blue}{\left(b \cdot \left(b \cdot \left(a \cdot \mathsf{fma}\left(a, 0.5, \frac{-1}{b}\right)\right)\right)\right)} \]

   12. Taylor expanded in b around 0

       \[\leadsto x \cdot \left(b \cdot \color{blue}{\left(-1 \cdot a\right)}\right) \]
   13. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto x \cdot \left(b \cdot \color{blue}{\left(\mathsf{neg}\left(a\right)\right)}\right) \]

       2. neg-lowering-neg.f64 21.3

          \[\leadsto x \cdot \left(b \cdot \color{blue}{\left(-a\right)}\right) \]

   14. Simplified 21.3%

       \[\leadsto x \cdot \left(b \cdot \color{blue}{\left(-a\right)}\right) \]

   if -2e10 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 1e152

   1. Initial program 91.2%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto x \cdot e^{\color{blue}{a \cdot \left(\log \left(1 - z\right) - b\right)}} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{a \cdot \left(\log \left(1 - z\right) - b\right)}} \]

      2. --lowering--.f64 N/A

         \[\leadsto x \cdot e^{a \cdot \color{blue}{\left(\log \left(1 - z\right) - b\right)}} \]

      3. sub-neg N/A

         \[\leadsto x \cdot e^{a \cdot \left(\log \color{blue}{\left(1 + \left(\mathsf{neg}\left(z\right)\right)\right)} - b\right)} \]

      4. accelerator-lowering-log1p.f64 N/A

         \[\leadsto x \cdot e^{a \cdot \left(\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(z\right)\right)} - b\right)} \]

      5. neg-lowering-neg.f64 86.2

         \[\leadsto x \cdot e^{a \cdot \left(\mathsf{log1p}\left(\color{blue}{-z}\right) - b\right)} \]

   5. Simplified 86.2%

      \[\leadsto x \cdot e^{\color{blue}{a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)}} \]

   6. Taylor expanded in z around 0

      \[\leadsto x \cdot e^{a \cdot \color{blue}{\left(-1 \cdot z - b\right)}} \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto x \cdot e^{a \cdot \color{blue}{\left(-1 \cdot z - b\right)}} \]

      2. mul-1-neg N/A

         \[\leadsto x \cdot e^{a \cdot \left(\color{blue}{\left(\mathsf{neg}\left(z\right)\right)} - b\right)} \]

      3. neg-lowering-neg.f64 86.2

         \[\leadsto x \cdot e^{a \cdot \left(\color{blue}{\left(-z\right)} - b\right)} \]

   8. Simplified 86.2%

      \[\leadsto x \cdot e^{a \cdot \color{blue}{\left(\left(-z\right) - b\right)}} \]

   9. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x + -1 \cdot \left(a \cdot \left(x \cdot \left(b + z\right)\right)\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{-1 \cdot \left(a \cdot \left(x \cdot \left(b + z\right)\right)\right) + x} \]

       2. mul-1-neg N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(a \cdot \left(x \cdot \left(b + z\right)\right)\right)\right)} + x \]

       3. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(x \cdot \left(b + z\right)\right) \cdot a}\right)\right) + x \]

       4. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\left(b + z\right) \cdot x\right)} \cdot a\right)\right) + x \]

       5. associate-*l* N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(b + z\right) \cdot \left(x \cdot a\right)}\right)\right) + x \]

       6. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(\left(b + z\right) \cdot \color{blue}{\left(a \cdot x\right)}\right)\right) + x \]

       7. distribute-lft-neg-in N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(b + z\right)\right)\right) \cdot \left(a \cdot x\right)} + x \]

       8. mul-1-neg N/A

          \[\leadsto \color{blue}{\left(-1 \cdot \left(b + z\right)\right)} \cdot \left(a \cdot x\right) + x \]

       9. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot \left(b + z\right), a \cdot x, x\right)} \]

       10. distribute-lft-in N/A

           \[\leadsto \mathsf{fma}\left(\color{blue}{-1 \cdot b + -1 \cdot z}, a \cdot x, x\right) \]

       11. +-commutative N/A

           \[\leadsto \mathsf{fma}\left(\color{blue}{-1 \cdot z + -1 \cdot b}, a \cdot x, x\right) \]

       12. mul-1-neg N/A

           \[\leadsto \mathsf{fma}\left(-1 \cdot z + \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}, a \cdot x, x\right) \]

       13. sub-neg N/A

           \[\leadsto \mathsf{fma}\left(\color{blue}{-1 \cdot z - b}, a \cdot x, x\right) \]

       14. --lowering--.f64 N/A

           \[\leadsto \mathsf{fma}\left(\color{blue}{-1 \cdot z - b}, a \cdot x, x\right) \]

       15. mul-1-neg N/A

           \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(z\right)\right)} - b, a \cdot x, x\right) \]

       16. neg-lowering-neg.f64 N/A

           \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(z\right)\right)} - b, a \cdot x, x\right) \]

       17. *-commutative N/A

           \[\leadsto \mathsf{fma}\left(\left(\mathsf{neg}\left(z\right)\right) - b, \color{blue}{x \cdot a}, x\right) \]

       18. *-lowering-*.f64 57.8

           \[\leadsto \mathsf{fma}\left(\left(-z\right) - b, \color{blue}{x \cdot a}, x\right) \]

   11. Simplified 57.8%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(-z\right) - b, x \cdot a, x\right)} \]

   if 1e152 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 97.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(t \cdot y\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(t \cdot y\right)}} \]

      2. *-commutative N/A

         \[\leadsto x \cdot e^{\mathsf{neg}\left(\color{blue}{y \cdot t}\right)} \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      5. neg-lowering-neg.f64 57.9

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(-t\right)}} \]

   5. Simplified 57.9%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}} \]

   6. Taylor expanded in y around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + -1 \cdot \left(t \cdot y\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot \left(1 + \color{blue}{\left(\mathsf{neg}\left(t \cdot y\right)\right)}\right) \]

      2. unsub-neg N/A

         \[\leadsto x \cdot \color{blue}{\left(1 - t \cdot y\right)} \]

      3. --lowering--.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(1 - t \cdot y\right)} \]

      4. *-lowering-*.f64 33.6

         \[\leadsto x \cdot \left(1 - \color{blue}{t \cdot y}\right) \]

   8. Simplified 33.6%

      \[\leadsto x \cdot \color{blue}{\left(1 - t \cdot y\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 40.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -\infty:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(t, y, 1\right)}\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -20000000000:\\ \;\;\;\;-x \cdot \left(a \cdot b\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 10^{+152}:\\ \;\;\;\;\mathsf{fma}\left(\left(-z\right) - b, x \cdot a, x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - y \cdot t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 34.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t\_1 \leq -20000000000:\\ \;\;\;\;-x \cdot \left(a \cdot b\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+152}:\\ \;\;\;\;\mathsf{fma}\left(\left(-z\right) - b, x \cdot a, x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - y \cdot t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b)))))
   (if (<= t_1 -20000000000.0)
     (- (* x (* a b)))
     (if (<= t_1 1e+152) (fma (- (- z) b) (* x a) x) (* x (- 1.0 (* y t)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	double tmp;
	if (t_1 <= -20000000000.0) {
		tmp = -(x * (a * b));
	} else if (t_1 <= 1e+152) {
		tmp = fma((-z - b), (x * a), x);
	} else {
		tmp = x * (1.0 - (y * t));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))
	tmp = 0.0
	if (t_1 <= -20000000000.0)
		tmp = Float64(-Float64(x * Float64(a * b)));
	elseif (t_1 <= 1e+152)
		tmp = fma(Float64(Float64(-z) - b), Float64(x * a), x);
	else
		tmp = Float64(x * Float64(1.0 - Float64(y * t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -20000000000.0], (-N[(x * N[(a * b), $MachinePrecision]), $MachinePrecision]), If[LessEqual[t$95$1, 1e+152], N[(N[((-z) - b), $MachinePrecision] * N[(x * a), $MachinePrecision] + x), $MachinePrecision], N[(x * N[(1.0 - N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -2e10

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      3. *-lowering-*.f64 44.5

         \[\leadsto x \cdot e^{-\color{blue}{a \cdot b}} \]

   5. Simplified 44.5%

      \[\leadsto x \cdot e^{\color{blue}{-a \cdot b}} \]

   6. Taylor expanded in a around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right) + 1\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, -1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right)} \]

      3. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right) \]

      4. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) + \left(\mathsf{neg}\left(b\right)\right)}, 1\right) \]

      5. associate-*r* N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\frac{1}{2} \cdot a\right) \cdot {b}^{2}} + \left(\mathsf{neg}\left(b\right)\right), 1\right) \]

      6. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \left(\frac{1}{2} \cdot a\right) \cdot \color{blue}{\left(b \cdot b\right)} + \left(\mathsf{neg}\left(b\right)\right), 1\right) \]

      7. associate-*r* N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\left(\frac{1}{2} \cdot a\right) \cdot b\right) \cdot b} + \left(\mathsf{neg}\left(b\right)\right), 1\right) \]

      8. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \left(\left(\frac{1}{2} \cdot a\right) \cdot b\right) \cdot b + \color{blue}{-1 \cdot b}, 1\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{b \cdot \left(\left(\frac{1}{2} \cdot a\right) \cdot b + -1\right)}, 1\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{b \cdot \left(\left(\frac{1}{2} \cdot a\right) \cdot b + -1\right)}, 1\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, b \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot a, b, -1\right)}, 1\right) \]

      12. *-lowering-*.f64 2.8

          \[\leadsto x \cdot \mathsf{fma}\left(a, b \cdot \mathsf{fma}\left(\color{blue}{0.5 \cdot a}, b, -1\right), 1\right) \]

   8. Simplified 2.8%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, b \cdot \mathsf{fma}\left(0.5 \cdot a, b, -1\right), 1\right)} \]

   9. Taylor expanded in b around inf

      \[\leadsto x \cdot \color{blue}{\left({b}^{2} \cdot \left(-1 \cdot \frac{a}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto x \cdot \left(\color{blue}{\left(b \cdot b\right)} \cdot \left(-1 \cdot \frac{a}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right) \]

       2. associate-*l* N/A

          \[\leadsto x \cdot \color{blue}{\left(b \cdot \left(b \cdot \left(-1 \cdot \frac{a}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \color{blue}{\left(b \cdot \left(b \cdot \left(-1 \cdot \frac{a}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \left(b \cdot \color{blue}{\left(b \cdot \left(-1 \cdot \frac{a}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)}\right) \]

       5. associate-*r/ N/A

          \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(\color{blue}{\frac{-1 \cdot a}{b}} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(\frac{\color{blue}{a \cdot -1}}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       7. associate-/l* N/A

          \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(\color{blue}{a \cdot \frac{-1}{b}} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       8. metadata-eval N/A

          \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \frac{\color{blue}{\mathsf{neg}\left(1\right)}}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       9. distribute-neg-frac N/A

          \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{b}\right)\right)} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       10. *-commutative N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(\mathsf{neg}\left(\frac{1}{b}\right)\right) + \color{blue}{{a}^{2} \cdot \frac{1}{2}}\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(\mathsf{neg}\left(\frac{1}{b}\right)\right) + \color{blue}{\left(a \cdot a\right)} \cdot \frac{1}{2}\right)\right)\right) \]

       12. associate-*l* N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(\mathsf{neg}\left(\frac{1}{b}\right)\right) + \color{blue}{a \cdot \left(a \cdot \frac{1}{2}\right)}\right)\right)\right) \]

       13. *-commutative N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(\mathsf{neg}\left(\frac{1}{b}\right)\right) + a \cdot \color{blue}{\left(\frac{1}{2} \cdot a\right)}\right)\right)\right) \]

       14. distribute-lft-out N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \color{blue}{\left(a \cdot \left(\left(\mathsf{neg}\left(\frac{1}{b}\right)\right) + \frac{1}{2} \cdot a\right)\right)}\right)\right) \]

       15. +-commutative N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \color{blue}{\left(\frac{1}{2} \cdot a + \left(\mathsf{neg}\left(\frac{1}{b}\right)\right)\right)}\right)\right)\right) \]

       16. sub-neg N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \color{blue}{\left(\frac{1}{2} \cdot a - \frac{1}{b}\right)}\right)\right)\right) \]

       17. *-lowering-*.f64 N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \color{blue}{\left(a \cdot \left(\frac{1}{2} \cdot a - \frac{1}{b}\right)\right)}\right)\right) \]

       18. sub-neg N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \color{blue}{\left(\frac{1}{2} \cdot a + \left(\mathsf{neg}\left(\frac{1}{b}\right)\right)\right)}\right)\right)\right) \]

       19. *-commutative N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(\color{blue}{a \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(\frac{1}{b}\right)\right)\right)\right)\right)\right) \]

       20. accelerator-lowering-fma.f64 N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \color{blue}{\mathsf{fma}\left(a, \frac{1}{2}, \mathsf{neg}\left(\frac{1}{b}\right)\right)}\right)\right)\right) \]

       21. distribute-neg-frac N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \mathsf{fma}\left(a, \frac{1}{2}, \color{blue}{\frac{\mathsf{neg}\left(1\right)}{b}}\right)\right)\right)\right) \]

       22. metadata-eval N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \mathsf{fma}\left(a, \frac{1}{2}, \frac{\color{blue}{-1}}{b}\right)\right)\right)\right) \]

       23. /-lowering-/.f64 29.7

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \mathsf{fma}\left(a, 0.5, \color{blue}{\frac{-1}{b}}\right)\right)\right)\right) \]

   11. Simplified 29.7%

       \[\leadsto x \cdot \color{blue}{\left(b \cdot \left(b \cdot \left(a \cdot \mathsf{fma}\left(a, 0.5, \frac{-1}{b}\right)\right)\right)\right)} \]

   12. Taylor expanded in b around 0

       \[\leadsto x \cdot \left(b \cdot \color{blue}{\left(-1 \cdot a\right)}\right) \]
   13. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto x \cdot \left(b \cdot \color{blue}{\left(\mathsf{neg}\left(a\right)\right)}\right) \]

       2. neg-lowering-neg.f64 18.5

          \[\leadsto x \cdot \left(b \cdot \color{blue}{\left(-a\right)}\right) \]

   14. Simplified 18.5%

       \[\leadsto x \cdot \left(b \cdot \color{blue}{\left(-a\right)}\right) \]

   if -2e10 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 1e152

   1. Initial program 91.2%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto x \cdot e^{\color{blue}{a \cdot \left(\log \left(1 - z\right) - b\right)}} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{a \cdot \left(\log \left(1 - z\right) - b\right)}} \]

      2. --lowering--.f64 N/A

         \[\leadsto x \cdot e^{a \cdot \color{blue}{\left(\log \left(1 - z\right) - b\right)}} \]

      3. sub-neg N/A

         \[\leadsto x \cdot e^{a \cdot \left(\log \color{blue}{\left(1 + \left(\mathsf{neg}\left(z\right)\right)\right)} - b\right)} \]

      4. accelerator-lowering-log1p.f64 N/A

         \[\leadsto x \cdot e^{a \cdot \left(\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(z\right)\right)} - b\right)} \]

      5. neg-lowering-neg.f64 86.2

         \[\leadsto x \cdot e^{a \cdot \left(\mathsf{log1p}\left(\color{blue}{-z}\right) - b\right)} \]

   5. Simplified 86.2%

      \[\leadsto x \cdot e^{\color{blue}{a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)}} \]

   6. Taylor expanded in z around 0

      \[\leadsto x \cdot e^{a \cdot \color{blue}{\left(-1 \cdot z - b\right)}} \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto x \cdot e^{a \cdot \color{blue}{\left(-1 \cdot z - b\right)}} \]

      2. mul-1-neg N/A

         \[\leadsto x \cdot e^{a \cdot \left(\color{blue}{\left(\mathsf{neg}\left(z\right)\right)} - b\right)} \]

      3. neg-lowering-neg.f64 86.2

         \[\leadsto x \cdot e^{a \cdot \left(\color{blue}{\left(-z\right)} - b\right)} \]

   8. Simplified 86.2%

      \[\leadsto x \cdot e^{a \cdot \color{blue}{\left(\left(-z\right) - b\right)}} \]

   9. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x + -1 \cdot \left(a \cdot \left(x \cdot \left(b + z\right)\right)\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{-1 \cdot \left(a \cdot \left(x \cdot \left(b + z\right)\right)\right) + x} \]

       2. mul-1-neg N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(a \cdot \left(x \cdot \left(b + z\right)\right)\right)\right)} + x \]

       3. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(x \cdot \left(b + z\right)\right) \cdot a}\right)\right) + x \]

       4. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\left(b + z\right) \cdot x\right)} \cdot a\right)\right) + x \]

       5. associate-*l* N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(b + z\right) \cdot \left(x \cdot a\right)}\right)\right) + x \]

       6. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(\left(b + z\right) \cdot \color{blue}{\left(a \cdot x\right)}\right)\right) + x \]

       7. distribute-lft-neg-in N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(b + z\right)\right)\right) \cdot \left(a \cdot x\right)} + x \]

       8. mul-1-neg N/A

          \[\leadsto \color{blue}{\left(-1 \cdot \left(b + z\right)\right)} \cdot \left(a \cdot x\right) + x \]

       9. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot \left(b + z\right), a \cdot x, x\right)} \]

       10. distribute-lft-in N/A

           \[\leadsto \mathsf{fma}\left(\color{blue}{-1 \cdot b + -1 \cdot z}, a \cdot x, x\right) \]

       11. +-commutative N/A

           \[\leadsto \mathsf{fma}\left(\color{blue}{-1 \cdot z + -1 \cdot b}, a \cdot x, x\right) \]

       12. mul-1-neg N/A

           \[\leadsto \mathsf{fma}\left(-1 \cdot z + \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}, a \cdot x, x\right) \]

       13. sub-neg N/A

           \[\leadsto \mathsf{fma}\left(\color{blue}{-1 \cdot z - b}, a \cdot x, x\right) \]

       14. --lowering--.f64 N/A

           \[\leadsto \mathsf{fma}\left(\color{blue}{-1 \cdot z - b}, a \cdot x, x\right) \]

       15. mul-1-neg N/A

           \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(z\right)\right)} - b, a \cdot x, x\right) \]

       16. neg-lowering-neg.f64 N/A

           \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(z\right)\right)} - b, a \cdot x, x\right) \]

       17. *-commutative N/A

           \[\leadsto \mathsf{fma}\left(\left(\mathsf{neg}\left(z\right)\right) - b, \color{blue}{x \cdot a}, x\right) \]

       18. *-lowering-*.f64 57.8

           \[\leadsto \mathsf{fma}\left(\left(-z\right) - b, \color{blue}{x \cdot a}, x\right) \]

   11. Simplified 57.8%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(-z\right) - b, x \cdot a, x\right)} \]

   if 1e152 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 97.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(t \cdot y\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(t \cdot y\right)}} \]

      2. *-commutative N/A

         \[\leadsto x \cdot e^{\mathsf{neg}\left(\color{blue}{y \cdot t}\right)} \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      5. neg-lowering-neg.f64 57.9

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(-t\right)}} \]

   5. Simplified 57.9%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}} \]

   6. Taylor expanded in y around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + -1 \cdot \left(t \cdot y\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot \left(1 + \color{blue}{\left(\mathsf{neg}\left(t \cdot y\right)\right)}\right) \]

      2. unsub-neg N/A

         \[\leadsto x \cdot \color{blue}{\left(1 - t \cdot y\right)} \]

      3. --lowering--.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(1 - t \cdot y\right)} \]

      4. *-lowering-*.f64 33.6

         \[\leadsto x \cdot \left(1 - \color{blue}{t \cdot y}\right) \]

   8. Simplified 33.6%

      \[\leadsto x \cdot \color{blue}{\left(1 - t \cdot y\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 36.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -20000000000:\\ \;\;\;\;-x \cdot \left(a \cdot b\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 10^{+152}:\\ \;\;\;\;\mathsf{fma}\left(\left(-z\right) - b, x \cdot a, x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - y \cdot t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 33.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t\_1 \leq -20000000000:\\ \;\;\;\;-x \cdot \left(a \cdot b\right)\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+65}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot t\right) \cdot \left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b)))))
   (if (<= t_1 -20000000000.0)
     (- (* x (* a b)))
     (if (<= t_1 4e+65) x (* (* y t) (- x))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	double tmp;
	if (t_1 <= -20000000000.0) {
		tmp = -(x * (a * b));
	} else if (t_1 <= 4e+65) {
		tmp = x;
	} else {
		tmp = (y * t) * -x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (y * (log(z) - t)) + (a * (log((1.0d0 - z)) - b))
    if (t_1 <= (-20000000000.0d0)) then
        tmp = -(x * (a * b))
    else if (t_1 <= 4d+65) then
        tmp = x
    else
        tmp = (y * t) * -x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (y * (Math.log(z) - t)) + (a * (Math.log((1.0 - z)) - b));
	double tmp;
	if (t_1 <= -20000000000.0) {
		tmp = -(x * (a * b));
	} else if (t_1 <= 4e+65) {
		tmp = x;
	} else {
		tmp = (y * t) * -x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (y * (math.log(z) - t)) + (a * (math.log((1.0 - z)) - b))
	tmp = 0
	if t_1 <= -20000000000.0:
		tmp = -(x * (a * b))
	elif t_1 <= 4e+65:
		tmp = x
	else:
		tmp = (y * t) * -x
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))
	tmp = 0.0
	if (t_1 <= -20000000000.0)
		tmp = Float64(-Float64(x * Float64(a * b)));
	elseif (t_1 <= 4e+65)
		tmp = x;
	else
		tmp = Float64(Float64(y * t) * Float64(-x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	tmp = 0.0;
	if (t_1 <= -20000000000.0)
		tmp = -(x * (a * b));
	elseif (t_1 <= 4e+65)
		tmp = x;
	else
		tmp = (y * t) * -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -20000000000.0], (-N[(x * N[(a * b), $MachinePrecision]), $MachinePrecision]), If[LessEqual[t$95$1, 4e+65], x, N[(N[(y * t), $MachinePrecision] * (-x)), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -2e10

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      3. *-lowering-*.f64 44.5

         \[\leadsto x \cdot e^{-\color{blue}{a \cdot b}} \]

   5. Simplified 44.5%

      \[\leadsto x \cdot e^{\color{blue}{-a \cdot b}} \]

   6. Taylor expanded in a around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right) + 1\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, -1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right)} \]

      3. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right) \]

      4. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) + \left(\mathsf{neg}\left(b\right)\right)}, 1\right) \]

      5. associate-*r* N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\frac{1}{2} \cdot a\right) \cdot {b}^{2}} + \left(\mathsf{neg}\left(b\right)\right), 1\right) \]

      6. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \left(\frac{1}{2} \cdot a\right) \cdot \color{blue}{\left(b \cdot b\right)} + \left(\mathsf{neg}\left(b\right)\right), 1\right) \]

      7. associate-*r* N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\left(\frac{1}{2} \cdot a\right) \cdot b\right) \cdot b} + \left(\mathsf{neg}\left(b\right)\right), 1\right) \]

      8. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \left(\left(\frac{1}{2} \cdot a\right) \cdot b\right) \cdot b + \color{blue}{-1 \cdot b}, 1\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{b \cdot \left(\left(\frac{1}{2} \cdot a\right) \cdot b + -1\right)}, 1\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{b \cdot \left(\left(\frac{1}{2} \cdot a\right) \cdot b + -1\right)}, 1\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, b \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot a, b, -1\right)}, 1\right) \]

      12. *-lowering-*.f64 2.8

          \[\leadsto x \cdot \mathsf{fma}\left(a, b \cdot \mathsf{fma}\left(\color{blue}{0.5 \cdot a}, b, -1\right), 1\right) \]

   8. Simplified 2.8%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, b \cdot \mathsf{fma}\left(0.5 \cdot a, b, -1\right), 1\right)} \]

   9. Taylor expanded in b around inf

      \[\leadsto x \cdot \color{blue}{\left({b}^{2} \cdot \left(-1 \cdot \frac{a}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto x \cdot \left(\color{blue}{\left(b \cdot b\right)} \cdot \left(-1 \cdot \frac{a}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right) \]

       2. associate-*l* N/A

          \[\leadsto x \cdot \color{blue}{\left(b \cdot \left(b \cdot \left(-1 \cdot \frac{a}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \color{blue}{\left(b \cdot \left(b \cdot \left(-1 \cdot \frac{a}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \left(b \cdot \color{blue}{\left(b \cdot \left(-1 \cdot \frac{a}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)}\right) \]

       5. associate-*r/ N/A

          \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(\color{blue}{\frac{-1 \cdot a}{b}} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(\frac{\color{blue}{a \cdot -1}}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       7. associate-/l* N/A

          \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(\color{blue}{a \cdot \frac{-1}{b}} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       8. metadata-eval N/A

          \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \frac{\color{blue}{\mathsf{neg}\left(1\right)}}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       9. distribute-neg-frac N/A

          \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{b}\right)\right)} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       10. *-commutative N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(\mathsf{neg}\left(\frac{1}{b}\right)\right) + \color{blue}{{a}^{2} \cdot \frac{1}{2}}\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(\mathsf{neg}\left(\frac{1}{b}\right)\right) + \color{blue}{\left(a \cdot a\right)} \cdot \frac{1}{2}\right)\right)\right) \]

       12. associate-*l* N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(\mathsf{neg}\left(\frac{1}{b}\right)\right) + \color{blue}{a \cdot \left(a \cdot \frac{1}{2}\right)}\right)\right)\right) \]

       13. *-commutative N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(\mathsf{neg}\left(\frac{1}{b}\right)\right) + a \cdot \color{blue}{\left(\frac{1}{2} \cdot a\right)}\right)\right)\right) \]

       14. distribute-lft-out N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \color{blue}{\left(a \cdot \left(\left(\mathsf{neg}\left(\frac{1}{b}\right)\right) + \frac{1}{2} \cdot a\right)\right)}\right)\right) \]

       15. +-commutative N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \color{blue}{\left(\frac{1}{2} \cdot a + \left(\mathsf{neg}\left(\frac{1}{b}\right)\right)\right)}\right)\right)\right) \]

       16. sub-neg N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \color{blue}{\left(\frac{1}{2} \cdot a - \frac{1}{b}\right)}\right)\right)\right) \]

       17. *-lowering-*.f64 N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \color{blue}{\left(a \cdot \left(\frac{1}{2} \cdot a - \frac{1}{b}\right)\right)}\right)\right) \]

       18. sub-neg N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \color{blue}{\left(\frac{1}{2} \cdot a + \left(\mathsf{neg}\left(\frac{1}{b}\right)\right)\right)}\right)\right)\right) \]

       19. *-commutative N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(\color{blue}{a \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(\frac{1}{b}\right)\right)\right)\right)\right)\right) \]

       20. accelerator-lowering-fma.f64 N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \color{blue}{\mathsf{fma}\left(a, \frac{1}{2}, \mathsf{neg}\left(\frac{1}{b}\right)\right)}\right)\right)\right) \]

       21. distribute-neg-frac N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \mathsf{fma}\left(a, \frac{1}{2}, \color{blue}{\frac{\mathsf{neg}\left(1\right)}{b}}\right)\right)\right)\right) \]

       22. metadata-eval N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \mathsf{fma}\left(a, \frac{1}{2}, \frac{\color{blue}{-1}}{b}\right)\right)\right)\right) \]

       23. /-lowering-/.f64 29.7

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \mathsf{fma}\left(a, 0.5, \color{blue}{\frac{-1}{b}}\right)\right)\right)\right) \]

   11. Simplified 29.7%

       \[\leadsto x \cdot \color{blue}{\left(b \cdot \left(b \cdot \left(a \cdot \mathsf{fma}\left(a, 0.5, \frac{-1}{b}\right)\right)\right)\right)} \]

   12. Taylor expanded in b around 0

       \[\leadsto x \cdot \left(b \cdot \color{blue}{\left(-1 \cdot a\right)}\right) \]
   13. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto x \cdot \left(b \cdot \color{blue}{\left(\mathsf{neg}\left(a\right)\right)}\right) \]

       2. neg-lowering-neg.f64 18.5

          \[\leadsto x \cdot \left(b \cdot \color{blue}{\left(-a\right)}\right) \]

   14. Simplified 18.5%

       \[\leadsto x \cdot \left(b \cdot \color{blue}{\left(-a\right)}\right) \]

   if -2e10 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 4e65

   1. Initial program 89.5%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      3. *-lowering-*.f64 78.6

         \[\leadsto x \cdot e^{-\color{blue}{a \cdot b}} \]

   5. Simplified 78.6%

      \[\leadsto x \cdot e^{\color{blue}{-a \cdot b}} \]

   6. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x} \]
   7. Step-by-step derivation

   8. Simplified 66.0%

      \[\leadsto \color{blue}{x} \]

   if 4e65 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 96.7%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(t \cdot y\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(t \cdot y\right)}} \]

      2. *-commutative N/A

         \[\leadsto x \cdot e^{\mathsf{neg}\left(\color{blue}{y \cdot t}\right)} \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      5. neg-lowering-neg.f64 53.7

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(-t\right)}} \]

   5. Simplified 53.7%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}} \]

   6. Taylor expanded in y around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + -1 \cdot \left(t \cdot y\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot \left(1 + \color{blue}{\left(\mathsf{neg}\left(t \cdot y\right)\right)}\right) \]

      2. unsub-neg N/A

         \[\leadsto x \cdot \color{blue}{\left(1 - t \cdot y\right)} \]

      3. --lowering--.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(1 - t \cdot y\right)} \]

      4. *-lowering-*.f64 27.2

         \[\leadsto x \cdot \left(1 - \color{blue}{t \cdot y}\right) \]

   8. Simplified 27.2%

      \[\leadsto x \cdot \color{blue}{\left(1 - t \cdot y\right)} \]

   9. Taylor expanded in t around inf

      \[\leadsto \color{blue}{-1 \cdot \left(t \cdot \left(x \cdot y\right)\right)} \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \color{blue}{\mathsf{neg}\left(t \cdot \left(x \cdot y\right)\right)} \]

       2. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\left(x \cdot y\right) \cdot t}\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{x \cdot \left(y \cdot t\right)}\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(x \cdot \color{blue}{\left(t \cdot y\right)}\right) \]

       5. distribute-rgt-neg-in N/A

          \[\leadsto \color{blue}{x \cdot \left(\mathsf{neg}\left(t \cdot y\right)\right)} \]

       6. mul-1-neg N/A

          \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \left(t \cdot y\right)\right)} \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \left(t \cdot y\right)\right)} \]

       8. mul-1-neg N/A

          \[\leadsto x \cdot \color{blue}{\left(\mathsf{neg}\left(t \cdot y\right)\right)} \]

       9. distribute-rgt-neg-in N/A

          \[\leadsto x \cdot \color{blue}{\left(t \cdot \left(\mathsf{neg}\left(y\right)\right)\right)} \]

       10. mul-1-neg N/A

           \[\leadsto x \cdot \left(t \cdot \color{blue}{\left(-1 \cdot y\right)}\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto x \cdot \color{blue}{\left(t \cdot \left(-1 \cdot y\right)\right)} \]

       12. mul-1-neg N/A

           \[\leadsto x \cdot \left(t \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right) \]

       13. neg-lowering-neg.f64 26.7

           \[\leadsto x \cdot \left(t \cdot \color{blue}{\left(-y\right)}\right) \]

   11. Simplified 26.7%

       \[\leadsto \color{blue}{x \cdot \left(t \cdot \left(-y\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 33.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -20000000000:\\ \;\;\;\;-x \cdot \left(a \cdot b\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 4 \cdot 10^{+65}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot t\right) \cdot \left(-x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 33.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -x \cdot \left(a \cdot b\right)\\ t_2 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t\_2 \leq -20000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+112}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- (* x (* a b))))
        (t_2 (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b)))))
   (if (<= t_2 -20000000000.0) t_1 (if (<= t_2 2e+112) x t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = -(x * (a * b));
	double t_2 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	double tmp;
	if (t_2 <= -20000000000.0) {
		tmp = t_1;
	} else if (t_2 <= 2e+112) {
		tmp = x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = -(x * (a * b))
    t_2 = (y * (log(z) - t)) + (a * (log((1.0d0 - z)) - b))
    if (t_2 <= (-20000000000.0d0)) then
        tmp = t_1
    else if (t_2 <= 2d+112) then
        tmp = x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = -(x * (a * b));
	double t_2 = (y * (Math.log(z) - t)) + (a * (Math.log((1.0 - z)) - b));
	double tmp;
	if (t_2 <= -20000000000.0) {
		tmp = t_1;
	} else if (t_2 <= 2e+112) {
		tmp = x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = -(x * (a * b))
	t_2 = (y * (math.log(z) - t)) + (a * (math.log((1.0 - z)) - b))
	tmp = 0
	if t_2 <= -20000000000.0:
		tmp = t_1
	elif t_2 <= 2e+112:
		tmp = x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(-Float64(x * Float64(a * b)))
	t_2 = Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))
	tmp = 0.0
	if (t_2 <= -20000000000.0)
		tmp = t_1;
	elseif (t_2 <= 2e+112)
		tmp = x;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = -(x * (a * b));
	t_2 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	tmp = 0.0;
	if (t_2 <= -20000000000.0)
		tmp = t_1;
	elseif (t_2 <= 2e+112)
		tmp = x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = (-N[(x * N[(a * b), $MachinePrecision]), $MachinePrecision])}, Block[{t$95$2 = N[(N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -20000000000.0], t$95$1, If[LessEqual[t$95$2, 2e+112], x, t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -2e10 or 1.9999999999999999e112 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 98.9%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      3. *-lowering-*.f64 47.3

         \[\leadsto x \cdot e^{-\color{blue}{a \cdot b}} \]

   5. Simplified 47.3%

      \[\leadsto x \cdot e^{\color{blue}{-a \cdot b}} \]

   6. Taylor expanded in a around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right) + 1\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, -1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right)} \]

      3. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right) \]

      4. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) + \left(\mathsf{neg}\left(b\right)\right)}, 1\right) \]

      5. associate-*r* N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\frac{1}{2} \cdot a\right) \cdot {b}^{2}} + \left(\mathsf{neg}\left(b\right)\right), 1\right) \]

      6. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \left(\frac{1}{2} \cdot a\right) \cdot \color{blue}{\left(b \cdot b\right)} + \left(\mathsf{neg}\left(b\right)\right), 1\right) \]

      7. associate-*r* N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\left(\frac{1}{2} \cdot a\right) \cdot b\right) \cdot b} + \left(\mathsf{neg}\left(b\right)\right), 1\right) \]

      8. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \left(\left(\frac{1}{2} \cdot a\right) \cdot b\right) \cdot b + \color{blue}{-1 \cdot b}, 1\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{b \cdot \left(\left(\frac{1}{2} \cdot a\right) \cdot b + -1\right)}, 1\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{b \cdot \left(\left(\frac{1}{2} \cdot a\right) \cdot b + -1\right)}, 1\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, b \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot a, b, -1\right)}, 1\right) \]

      12. *-lowering-*.f64 24.0

          \[\leadsto x \cdot \mathsf{fma}\left(a, b \cdot \mathsf{fma}\left(\color{blue}{0.5 \cdot a}, b, -1\right), 1\right) \]

   8. Simplified 24.0%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, b \cdot \mathsf{fma}\left(0.5 \cdot a, b, -1\right), 1\right)} \]

   9. Taylor expanded in b around inf

      \[\leadsto x \cdot \color{blue}{\left({b}^{2} \cdot \left(-1 \cdot \frac{a}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto x \cdot \left(\color{blue}{\left(b \cdot b\right)} \cdot \left(-1 \cdot \frac{a}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right) \]

       2. associate-*l* N/A

          \[\leadsto x \cdot \color{blue}{\left(b \cdot \left(b \cdot \left(-1 \cdot \frac{a}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \color{blue}{\left(b \cdot \left(b \cdot \left(-1 \cdot \frac{a}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \left(b \cdot \color{blue}{\left(b \cdot \left(-1 \cdot \frac{a}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)}\right) \]

       5. associate-*r/ N/A

          \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(\color{blue}{\frac{-1 \cdot a}{b}} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(\frac{\color{blue}{a \cdot -1}}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       7. associate-/l* N/A

          \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(\color{blue}{a \cdot \frac{-1}{b}} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       8. metadata-eval N/A

          \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \frac{\color{blue}{\mathsf{neg}\left(1\right)}}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       9. distribute-neg-frac N/A

          \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{b}\right)\right)} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       10. *-commutative N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(\mathsf{neg}\left(\frac{1}{b}\right)\right) + \color{blue}{{a}^{2} \cdot \frac{1}{2}}\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(\mathsf{neg}\left(\frac{1}{b}\right)\right) + \color{blue}{\left(a \cdot a\right)} \cdot \frac{1}{2}\right)\right)\right) \]

       12. associate-*l* N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(\mathsf{neg}\left(\frac{1}{b}\right)\right) + \color{blue}{a \cdot \left(a \cdot \frac{1}{2}\right)}\right)\right)\right) \]

       13. *-commutative N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(\mathsf{neg}\left(\frac{1}{b}\right)\right) + a \cdot \color{blue}{\left(\frac{1}{2} \cdot a\right)}\right)\right)\right) \]

       14. distribute-lft-out N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \color{blue}{\left(a \cdot \left(\left(\mathsf{neg}\left(\frac{1}{b}\right)\right) + \frac{1}{2} \cdot a\right)\right)}\right)\right) \]

       15. +-commutative N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \color{blue}{\left(\frac{1}{2} \cdot a + \left(\mathsf{neg}\left(\frac{1}{b}\right)\right)\right)}\right)\right)\right) \]

       16. sub-neg N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \color{blue}{\left(\frac{1}{2} \cdot a - \frac{1}{b}\right)}\right)\right)\right) \]

       17. *-lowering-*.f64 N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \color{blue}{\left(a \cdot \left(\frac{1}{2} \cdot a - \frac{1}{b}\right)\right)}\right)\right) \]

       18. sub-neg N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \color{blue}{\left(\frac{1}{2} \cdot a + \left(\mathsf{neg}\left(\frac{1}{b}\right)\right)\right)}\right)\right)\right) \]

       19. *-commutative N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(\color{blue}{a \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(\frac{1}{b}\right)\right)\right)\right)\right)\right) \]

       20. accelerator-lowering-fma.f64 N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \color{blue}{\mathsf{fma}\left(a, \frac{1}{2}, \mathsf{neg}\left(\frac{1}{b}\right)\right)}\right)\right)\right) \]

       21. distribute-neg-frac N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \mathsf{fma}\left(a, \frac{1}{2}, \color{blue}{\frac{\mathsf{neg}\left(1\right)}{b}}\right)\right)\right)\right) \]

       22. metadata-eval N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \mathsf{fma}\left(a, \frac{1}{2}, \frac{\color{blue}{-1}}{b}\right)\right)\right)\right) \]

       23. /-lowering-/.f64 41.0

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \mathsf{fma}\left(a, 0.5, \color{blue}{\frac{-1}{b}}\right)\right)\right)\right) \]

   11. Simplified 41.0%

       \[\leadsto x \cdot \color{blue}{\left(b \cdot \left(b \cdot \left(a \cdot \mathsf{fma}\left(a, 0.5, \frac{-1}{b}\right)\right)\right)\right)} \]

   12. Taylor expanded in b around 0

       \[\leadsto x \cdot \left(b \cdot \color{blue}{\left(-1 \cdot a\right)}\right) \]
   13. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto x \cdot \left(b \cdot \color{blue}{\left(\mathsf{neg}\left(a\right)\right)}\right) \]

       2. neg-lowering-neg.f64 21.8

          \[\leadsto x \cdot \left(b \cdot \color{blue}{\left(-a\right)}\right) \]

   14. Simplified 21.8%

       \[\leadsto x \cdot \left(b \cdot \color{blue}{\left(-a\right)}\right) \]

   if -2e10 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 1.9999999999999999e112

   1. Initial program 89.8%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      3. *-lowering-*.f64 74.1

         \[\leadsto x \cdot e^{-\color{blue}{a \cdot b}} \]

   5. Simplified 74.1%

      \[\leadsto x \cdot e^{\color{blue}{-a \cdot b}} \]

   6. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x} \]
   7. Step-by-step derivation

   8. Simplified 57.0%

      \[\leadsto \color{blue}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 32.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -20000000000:\\ \;\;\;\;-x \cdot \left(a \cdot b\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 2 \cdot 10^{+112}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-x \cdot \left(a \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 34.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -20000000000:\\ \;\;\;\;-x \cdot \left(a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - y \cdot t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b))) -20000000000.0)
   (- (* x (* a b)))
   (* x (- 1.0 (* y t)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((y * (log(z) - t)) + (a * (log((1.0 - z)) - b))) <= -20000000000.0) {
		tmp = -(x * (a * b));
	} else {
		tmp = x * (1.0 - (y * t));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (((y * (log(z) - t)) + (a * (log((1.0d0 - z)) - b))) <= (-20000000000.0d0)) then
        tmp = -(x * (a * b))
    else
        tmp = x * (1.0d0 - (y * t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((y * (Math.log(z) - t)) + (a * (Math.log((1.0 - z)) - b))) <= -20000000000.0) {
		tmp = -(x * (a * b));
	} else {
		tmp = x * (1.0 - (y * t));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if ((y * (math.log(z) - t)) + (a * (math.log((1.0 - z)) - b))) <= -20000000000.0:
		tmp = -(x * (a * b))
	else:
		tmp = x * (1.0 - (y * t))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b))) <= -20000000000.0)
		tmp = Float64(-Float64(x * Float64(a * b)));
	else
		tmp = Float64(x * Float64(1.0 - Float64(y * t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (((y * (log(z) - t)) + (a * (log((1.0 - z)) - b))) <= -20000000000.0)
		tmp = -(x * (a * b));
	else
		tmp = x * (1.0 - (y * t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -20000000000.0], (-N[(x * N[(a * b), $MachinePrecision]), $MachinePrecision]), N[(x * N[(1.0 - N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -2e10

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      3. *-lowering-*.f64 44.5

         \[\leadsto x \cdot e^{-\color{blue}{a \cdot b}} \]

   5. Simplified 44.5%

      \[\leadsto x \cdot e^{\color{blue}{-a \cdot b}} \]

   6. Taylor expanded in a around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right) + 1\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, -1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right)} \]

      3. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right) \]

      4. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) + \left(\mathsf{neg}\left(b\right)\right)}, 1\right) \]

      5. associate-*r* N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\frac{1}{2} \cdot a\right) \cdot {b}^{2}} + \left(\mathsf{neg}\left(b\right)\right), 1\right) \]

      6. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \left(\frac{1}{2} \cdot a\right) \cdot \color{blue}{\left(b \cdot b\right)} + \left(\mathsf{neg}\left(b\right)\right), 1\right) \]

      7. associate-*r* N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\left(\frac{1}{2} \cdot a\right) \cdot b\right) \cdot b} + \left(\mathsf{neg}\left(b\right)\right), 1\right) \]

      8. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \left(\left(\frac{1}{2} \cdot a\right) \cdot b\right) \cdot b + \color{blue}{-1 \cdot b}, 1\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{b \cdot \left(\left(\frac{1}{2} \cdot a\right) \cdot b + -1\right)}, 1\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{b \cdot \left(\left(\frac{1}{2} \cdot a\right) \cdot b + -1\right)}, 1\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, b \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot a, b, -1\right)}, 1\right) \]

      12. *-lowering-*.f64 2.8

          \[\leadsto x \cdot \mathsf{fma}\left(a, b \cdot \mathsf{fma}\left(\color{blue}{0.5 \cdot a}, b, -1\right), 1\right) \]

   8. Simplified 2.8%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, b \cdot \mathsf{fma}\left(0.5 \cdot a, b, -1\right), 1\right)} \]

   9. Taylor expanded in b around inf

      \[\leadsto x \cdot \color{blue}{\left({b}^{2} \cdot \left(-1 \cdot \frac{a}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto x \cdot \left(\color{blue}{\left(b \cdot b\right)} \cdot \left(-1 \cdot \frac{a}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right) \]

       2. associate-*l* N/A

          \[\leadsto x \cdot \color{blue}{\left(b \cdot \left(b \cdot \left(-1 \cdot \frac{a}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \color{blue}{\left(b \cdot \left(b \cdot \left(-1 \cdot \frac{a}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \left(b \cdot \color{blue}{\left(b \cdot \left(-1 \cdot \frac{a}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)}\right) \]

       5. associate-*r/ N/A

          \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(\color{blue}{\frac{-1 \cdot a}{b}} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(\frac{\color{blue}{a \cdot -1}}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       7. associate-/l* N/A

          \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(\color{blue}{a \cdot \frac{-1}{b}} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       8. metadata-eval N/A

          \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \frac{\color{blue}{\mathsf{neg}\left(1\right)}}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       9. distribute-neg-frac N/A

          \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{b}\right)\right)} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       10. *-commutative N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(\mathsf{neg}\left(\frac{1}{b}\right)\right) + \color{blue}{{a}^{2} \cdot \frac{1}{2}}\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(\mathsf{neg}\left(\frac{1}{b}\right)\right) + \color{blue}{\left(a \cdot a\right)} \cdot \frac{1}{2}\right)\right)\right) \]

       12. associate-*l* N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(\mathsf{neg}\left(\frac{1}{b}\right)\right) + \color{blue}{a \cdot \left(a \cdot \frac{1}{2}\right)}\right)\right)\right) \]

       13. *-commutative N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(\mathsf{neg}\left(\frac{1}{b}\right)\right) + a \cdot \color{blue}{\left(\frac{1}{2} \cdot a\right)}\right)\right)\right) \]

       14. distribute-lft-out N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \color{blue}{\left(a \cdot \left(\left(\mathsf{neg}\left(\frac{1}{b}\right)\right) + \frac{1}{2} \cdot a\right)\right)}\right)\right) \]

       15. +-commutative N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \color{blue}{\left(\frac{1}{2} \cdot a + \left(\mathsf{neg}\left(\frac{1}{b}\right)\right)\right)}\right)\right)\right) \]

       16. sub-neg N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \color{blue}{\left(\frac{1}{2} \cdot a - \frac{1}{b}\right)}\right)\right)\right) \]

       17. *-lowering-*.f64 N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \color{blue}{\left(a \cdot \left(\frac{1}{2} \cdot a - \frac{1}{b}\right)\right)}\right)\right) \]

       18. sub-neg N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \color{blue}{\left(\frac{1}{2} \cdot a + \left(\mathsf{neg}\left(\frac{1}{b}\right)\right)\right)}\right)\right)\right) \]

       19. *-commutative N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(\color{blue}{a \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(\frac{1}{b}\right)\right)\right)\right)\right)\right) \]

       20. accelerator-lowering-fma.f64 N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \color{blue}{\mathsf{fma}\left(a, \frac{1}{2}, \mathsf{neg}\left(\frac{1}{b}\right)\right)}\right)\right)\right) \]

       21. distribute-neg-frac N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \mathsf{fma}\left(a, \frac{1}{2}, \color{blue}{\frac{\mathsf{neg}\left(1\right)}{b}}\right)\right)\right)\right) \]

       22. metadata-eval N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \mathsf{fma}\left(a, \frac{1}{2}, \frac{\color{blue}{-1}}{b}\right)\right)\right)\right) \]

       23. /-lowering-/.f64 29.7

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \mathsf{fma}\left(a, 0.5, \color{blue}{\frac{-1}{b}}\right)\right)\right)\right) \]

   11. Simplified 29.7%

       \[\leadsto x \cdot \color{blue}{\left(b \cdot \left(b \cdot \left(a \cdot \mathsf{fma}\left(a, 0.5, \frac{-1}{b}\right)\right)\right)\right)} \]

   12. Taylor expanded in b around 0

       \[\leadsto x \cdot \left(b \cdot \color{blue}{\left(-1 \cdot a\right)}\right) \]
   13. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto x \cdot \left(b \cdot \color{blue}{\left(\mathsf{neg}\left(a\right)\right)}\right) \]

       2. neg-lowering-neg.f64 18.5

          \[\leadsto x \cdot \left(b \cdot \color{blue}{\left(-a\right)}\right) \]

   14. Simplified 18.5%

       \[\leadsto x \cdot \left(b \cdot \color{blue}{\left(-a\right)}\right) \]

   if -2e10 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 93.7%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(t \cdot y\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(t \cdot y\right)}} \]

      2. *-commutative N/A

         \[\leadsto x \cdot e^{\mathsf{neg}\left(\color{blue}{y \cdot t}\right)} \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      5. neg-lowering-neg.f64 62.6

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(-t\right)}} \]

   5. Simplified 62.6%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}} \]

   6. Taylor expanded in y around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + -1 \cdot \left(t \cdot y\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot \left(1 + \color{blue}{\left(\mathsf{neg}\left(t \cdot y\right)\right)}\right) \]

      2. unsub-neg N/A

         \[\leadsto x \cdot \color{blue}{\left(1 - t \cdot y\right)} \]

      3. --lowering--.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(1 - t \cdot y\right)} \]

      4. *-lowering-*.f64 43.8

         \[\leadsto x \cdot \left(1 - \color{blue}{t \cdot y}\right) \]

   8. Simplified 43.8%

      \[\leadsto x \cdot \color{blue}{\left(1 - t \cdot y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -20000000000:\\ \;\;\;\;-x \cdot \left(a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - y \cdot t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 34.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -20000000000:\\ \;\;\;\;-x \cdot \left(a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b))) -20000000000.0)
   (- (* x (* a b)))
   (* x (- 1.0 (* a b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((y * (log(z) - t)) + (a * (log((1.0 - z)) - b))) <= -20000000000.0) {
		tmp = -(x * (a * b));
	} else {
		tmp = x * (1.0 - (a * b));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (((y * (log(z) - t)) + (a * (log((1.0d0 - z)) - b))) <= (-20000000000.0d0)) then
        tmp = -(x * (a * b))
    else
        tmp = x * (1.0d0 - (a * b))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((y * (Math.log(z) - t)) + (a * (Math.log((1.0 - z)) - b))) <= -20000000000.0) {
		tmp = -(x * (a * b));
	} else {
		tmp = x * (1.0 - (a * b));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if ((y * (math.log(z) - t)) + (a * (math.log((1.0 - z)) - b))) <= -20000000000.0:
		tmp = -(x * (a * b))
	else:
		tmp = x * (1.0 - (a * b))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b))) <= -20000000000.0)
		tmp = Float64(-Float64(x * Float64(a * b)));
	else
		tmp = Float64(x * Float64(1.0 - Float64(a * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (((y * (log(z) - t)) + (a * (log((1.0 - z)) - b))) <= -20000000000.0)
		tmp = -(x * (a * b));
	else
		tmp = x * (1.0 - (a * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -20000000000.0], (-N[(x * N[(a * b), $MachinePrecision]), $MachinePrecision]), N[(x * N[(1.0 - N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -2e10

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      3. *-lowering-*.f64 44.5

         \[\leadsto x \cdot e^{-\color{blue}{a \cdot b}} \]

   5. Simplified 44.5%

      \[\leadsto x \cdot e^{\color{blue}{-a \cdot b}} \]

   6. Taylor expanded in a around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right) + 1\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, -1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right)} \]

      3. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right) \]

      4. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) + \left(\mathsf{neg}\left(b\right)\right)}, 1\right) \]

      5. associate-*r* N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\frac{1}{2} \cdot a\right) \cdot {b}^{2}} + \left(\mathsf{neg}\left(b\right)\right), 1\right) \]

      6. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \left(\frac{1}{2} \cdot a\right) \cdot \color{blue}{\left(b \cdot b\right)} + \left(\mathsf{neg}\left(b\right)\right), 1\right) \]

      7. associate-*r* N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\left(\frac{1}{2} \cdot a\right) \cdot b\right) \cdot b} + \left(\mathsf{neg}\left(b\right)\right), 1\right) \]

      8. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \left(\left(\frac{1}{2} \cdot a\right) \cdot b\right) \cdot b + \color{blue}{-1 \cdot b}, 1\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{b \cdot \left(\left(\frac{1}{2} \cdot a\right) \cdot b + -1\right)}, 1\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{b \cdot \left(\left(\frac{1}{2} \cdot a\right) \cdot b + -1\right)}, 1\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, b \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot a, b, -1\right)}, 1\right) \]

      12. *-lowering-*.f64 2.8

          \[\leadsto x \cdot \mathsf{fma}\left(a, b \cdot \mathsf{fma}\left(\color{blue}{0.5 \cdot a}, b, -1\right), 1\right) \]

   8. Simplified 2.8%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, b \cdot \mathsf{fma}\left(0.5 \cdot a, b, -1\right), 1\right)} \]

   9. Taylor expanded in b around inf

      \[\leadsto x \cdot \color{blue}{\left({b}^{2} \cdot \left(-1 \cdot \frac{a}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto x \cdot \left(\color{blue}{\left(b \cdot b\right)} \cdot \left(-1 \cdot \frac{a}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right) \]

       2. associate-*l* N/A

          \[\leadsto x \cdot \color{blue}{\left(b \cdot \left(b \cdot \left(-1 \cdot \frac{a}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \color{blue}{\left(b \cdot \left(b \cdot \left(-1 \cdot \frac{a}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \left(b \cdot \color{blue}{\left(b \cdot \left(-1 \cdot \frac{a}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)}\right) \]

       5. associate-*r/ N/A

          \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(\color{blue}{\frac{-1 \cdot a}{b}} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(\frac{\color{blue}{a \cdot -1}}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       7. associate-/l* N/A

          \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(\color{blue}{a \cdot \frac{-1}{b}} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       8. metadata-eval N/A

          \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \frac{\color{blue}{\mathsf{neg}\left(1\right)}}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       9. distribute-neg-frac N/A

          \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{b}\right)\right)} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       10. *-commutative N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(\mathsf{neg}\left(\frac{1}{b}\right)\right) + \color{blue}{{a}^{2} \cdot \frac{1}{2}}\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(\mathsf{neg}\left(\frac{1}{b}\right)\right) + \color{blue}{\left(a \cdot a\right)} \cdot \frac{1}{2}\right)\right)\right) \]

       12. associate-*l* N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(\mathsf{neg}\left(\frac{1}{b}\right)\right) + \color{blue}{a \cdot \left(a \cdot \frac{1}{2}\right)}\right)\right)\right) \]

       13. *-commutative N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(\mathsf{neg}\left(\frac{1}{b}\right)\right) + a \cdot \color{blue}{\left(\frac{1}{2} \cdot a\right)}\right)\right)\right) \]

       14. distribute-lft-out N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \color{blue}{\left(a \cdot \left(\left(\mathsf{neg}\left(\frac{1}{b}\right)\right) + \frac{1}{2} \cdot a\right)\right)}\right)\right) \]

       15. +-commutative N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \color{blue}{\left(\frac{1}{2} \cdot a + \left(\mathsf{neg}\left(\frac{1}{b}\right)\right)\right)}\right)\right)\right) \]

       16. sub-neg N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \color{blue}{\left(\frac{1}{2} \cdot a - \frac{1}{b}\right)}\right)\right)\right) \]

       17. *-lowering-*.f64 N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \color{blue}{\left(a \cdot \left(\frac{1}{2} \cdot a - \frac{1}{b}\right)\right)}\right)\right) \]

       18. sub-neg N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \color{blue}{\left(\frac{1}{2} \cdot a + \left(\mathsf{neg}\left(\frac{1}{b}\right)\right)\right)}\right)\right)\right) \]

       19. *-commutative N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(\color{blue}{a \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(\frac{1}{b}\right)\right)\right)\right)\right)\right) \]

       20. accelerator-lowering-fma.f64 N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \color{blue}{\mathsf{fma}\left(a, \frac{1}{2}, \mathsf{neg}\left(\frac{1}{b}\right)\right)}\right)\right)\right) \]

       21. distribute-neg-frac N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \mathsf{fma}\left(a, \frac{1}{2}, \color{blue}{\frac{\mathsf{neg}\left(1\right)}{b}}\right)\right)\right)\right) \]

       22. metadata-eval N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \mathsf{fma}\left(a, \frac{1}{2}, \frac{\color{blue}{-1}}{b}\right)\right)\right)\right) \]

       23. /-lowering-/.f64 29.7

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \mathsf{fma}\left(a, 0.5, \color{blue}{\frac{-1}{b}}\right)\right)\right)\right) \]

   11. Simplified 29.7%

       \[\leadsto x \cdot \color{blue}{\left(b \cdot \left(b \cdot \left(a \cdot \mathsf{fma}\left(a, 0.5, \frac{-1}{b}\right)\right)\right)\right)} \]

   12. Taylor expanded in b around 0

       \[\leadsto x \cdot \left(b \cdot \color{blue}{\left(-1 \cdot a\right)}\right) \]
   13. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto x \cdot \left(b \cdot \color{blue}{\left(\mathsf{neg}\left(a\right)\right)}\right) \]

       2. neg-lowering-neg.f64 18.5

          \[\leadsto x \cdot \left(b \cdot \color{blue}{\left(-a\right)}\right) \]

   14. Simplified 18.5%

       \[\leadsto x \cdot \left(b \cdot \color{blue}{\left(-a\right)}\right) \]

   if -2e10 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 93.7%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      3. *-lowering-*.f64 62.3

         \[\leadsto x \cdot e^{-\color{blue}{a \cdot b}} \]

   5. Simplified 62.3%

      \[\leadsto x \cdot e^{\color{blue}{-a \cdot b}} \]

   6. Taylor expanded in a around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + -1 \cdot \left(a \cdot b\right)\right)} \]
   7. Step-by-step derivation

      1. neg-mul-1 N/A

         \[\leadsto x \cdot \left(1 + \color{blue}{\left(\mathsf{neg}\left(a \cdot b\right)\right)}\right) \]

      2. unsub-neg N/A

         \[\leadsto x \cdot \color{blue}{\left(1 - a \cdot b\right)} \]

      3. --lowering--.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(1 - a \cdot b\right)} \]

      4. *-lowering-*.f64 41.7

         \[\leadsto x \cdot \left(1 - \color{blue}{a \cdot b}\right) \]

   8. Simplified 41.7%

      \[\leadsto x \cdot \color{blue}{\left(1 - a \cdot b\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 32.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -20000000000:\\ \;\;\;\;-x \cdot \left(a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 86.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot e^{y \cdot \left(\log z - t\right)}\\ \mathbf{if}\;y \leq -1 \cdot 10^{-43}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{-13}:\\ \;\;\;\;x \cdot e^{-a \cdot \left(z + b\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (exp (* y (- (log z) t))))))
   (if (<= y -1e-43)
     t_1
     (if (<= y 1.4e-13) (* x (exp (- (* a (+ z b))))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * exp((y * (log(z) - t)));
	double tmp;
	if (y <= -1e-43) {
		tmp = t_1;
	} else if (y <= 1.4e-13) {
		tmp = x * exp(-(a * (z + b)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * exp((y * (log(z) - t)))
    if (y <= (-1d-43)) then
        tmp = t_1
    else if (y <= 1.4d-13) then
        tmp = x * exp(-(a * (z + b)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * Math.exp((y * (Math.log(z) - t)));
	double tmp;
	if (y <= -1e-43) {
		tmp = t_1;
	} else if (y <= 1.4e-13) {
		tmp = x * Math.exp(-(a * (z + b)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x * math.exp((y * (math.log(z) - t)))
	tmp = 0
	if y <= -1e-43:
		tmp = t_1
	elif y <= 1.4e-13:
		tmp = x * math.exp(-(a * (z + b)))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x * exp(Float64(y * Float64(log(z) - t))))
	tmp = 0.0
	if (y <= -1e-43)
		tmp = t_1;
	elseif (y <= 1.4e-13)
		tmp = Float64(x * exp(Float64(-Float64(a * Float64(z + b)))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x * exp((y * (log(z) - t)));
	tmp = 0.0;
	if (y <= -1e-43)
		tmp = t_1;
	elseif (y <= 1.4e-13)
		tmp = x * exp(-(a * (z + b)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x * N[Exp[N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1e-43], t$95$1, If[LessEqual[y, 1.4e-13], N[(x * N[Exp[(-N[(a * N[(z + b), $MachinePrecision]), $MachinePrecision])], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.00000000000000008e-43 or 1.4000000000000001e-13 < y

   1. Initial program 97.9%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\log z - t\right)}} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\log z - t\right)}} \]

      2. --lowering--.f64 N/A

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(\log z - t\right)}} \]

      3. log-lowering-log.f64 90.4

         \[\leadsto x \cdot e^{y \cdot \left(\color{blue}{\log z} - t\right)} \]

   5. Simplified 90.4%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\log z - t\right)}} \]

   if -1.00000000000000008e-43 < y < 1.4000000000000001e-13

   1. Initial program 93.9%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto x \cdot e^{\color{blue}{a \cdot \left(\log \left(1 - z\right) - b\right)}} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{a \cdot \left(\log \left(1 - z\right) - b\right)}} \]

      2. --lowering--.f64 N/A

         \[\leadsto x \cdot e^{a \cdot \color{blue}{\left(\log \left(1 - z\right) - b\right)}} \]

      3. sub-neg N/A

         \[\leadsto x \cdot e^{a \cdot \left(\log \color{blue}{\left(1 + \left(\mathsf{neg}\left(z\right)\right)\right)} - b\right)} \]

      4. accelerator-lowering-log1p.f64 N/A

         \[\leadsto x \cdot e^{a \cdot \left(\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(z\right)\right)} - b\right)} \]

      5. neg-lowering-neg.f64 88.3

         \[\leadsto x \cdot e^{a \cdot \left(\mathsf{log1p}\left(\color{blue}{-z}\right) - b\right)} \]

   5. Simplified 88.3%

      \[\leadsto x \cdot e^{\color{blue}{a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)}} \]

   6. Taylor expanded in z around 0

      \[\leadsto x \cdot e^{a \cdot \color{blue}{\left(-1 \cdot z - b\right)}} \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto x \cdot e^{a \cdot \color{blue}{\left(-1 \cdot z - b\right)}} \]

      2. mul-1-neg N/A

         \[\leadsto x \cdot e^{a \cdot \left(\color{blue}{\left(\mathsf{neg}\left(z\right)\right)} - b\right)} \]

      3. neg-lowering-neg.f64 88.3

         \[\leadsto x \cdot e^{a \cdot \left(\color{blue}{\left(-z\right)} - b\right)} \]

   8. Simplified 88.3%

      \[\leadsto x \cdot e^{a \cdot \color{blue}{\left(\left(-z\right) - b\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-43}:\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{-13}:\\ \;\;\;\;x \cdot e^{-a \cdot \left(z + b\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 71.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot e^{-y \cdot t}\\ \mathbf{if}\;t \leq -400:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 3 \cdot 10^{-63}:\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{elif}\;t \leq 3 \cdot 10^{+22}:\\ \;\;\;\;x \cdot e^{b \cdot \left(-a\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (exp (- (* y t))))))
   (if (<= t -400.0)
     t_1
     (if (<= t 3e-63)
       (* x (pow z y))
       (if (<= t 3e+22) (* x (exp (* b (- a)))) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * exp(-(y * t));
	double tmp;
	if (t <= -400.0) {
		tmp = t_1;
	} else if (t <= 3e-63) {
		tmp = x * pow(z, y);
	} else if (t <= 3e+22) {
		tmp = x * exp((b * -a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * exp(-(y * t))
    if (t <= (-400.0d0)) then
        tmp = t_1
    else if (t <= 3d-63) then
        tmp = x * (z ** y)
    else if (t <= 3d+22) then
        tmp = x * exp((b * -a))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * Math.exp(-(y * t));
	double tmp;
	if (t <= -400.0) {
		tmp = t_1;
	} else if (t <= 3e-63) {
		tmp = x * Math.pow(z, y);
	} else if (t <= 3e+22) {
		tmp = x * Math.exp((b * -a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x * math.exp(-(y * t))
	tmp = 0
	if t <= -400.0:
		tmp = t_1
	elif t <= 3e-63:
		tmp = x * math.pow(z, y)
	elif t <= 3e+22:
		tmp = x * math.exp((b * -a))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x * exp(Float64(-Float64(y * t))))
	tmp = 0.0
	if (t <= -400.0)
		tmp = t_1;
	elseif (t <= 3e-63)
		tmp = Float64(x * (z ^ y));
	elseif (t <= 3e+22)
		tmp = Float64(x * exp(Float64(b * Float64(-a))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x * exp(-(y * t));
	tmp = 0.0;
	if (t <= -400.0)
		tmp = t_1;
	elseif (t <= 3e-63)
		tmp = x * (z ^ y);
	elseif (t <= 3e+22)
		tmp = x * exp((b * -a));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x * N[Exp[(-N[(y * t), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -400.0], t$95$1, If[LessEqual[t, 3e-63], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3e+22], N[(x * N[Exp[N[(b * (-a)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -400 or 3e22 < t

   1. Initial program 97.5%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(t \cdot y\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(t \cdot y\right)}} \]

      2. *-commutative N/A

         \[\leadsto x \cdot e^{\mathsf{neg}\left(\color{blue}{y \cdot t}\right)} \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      5. neg-lowering-neg.f64 79.4

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(-t\right)}} \]

   5. Simplified 79.4%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}} \]

   if -400 < t < 2.99999999999999979e-63

   1. Initial program 96.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\log z - t\right)}} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\log z - t\right)}} \]

      2. --lowering--.f64 N/A

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(\log z - t\right)}} \]

      3. log-lowering-log.f64 74.6

         \[\leadsto x \cdot e^{y \cdot \left(\color{blue}{\log z} - t\right)} \]

   5. Simplified 74.6%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\log z - t\right)}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{x \cdot {z}^{y}} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot {z}^{y}} \]

      2. pow-lowering-pow.f64 74.6

         \[\leadsto x \cdot \color{blue}{{z}^{y}} \]

   8. Simplified 74.6%

      \[\leadsto \color{blue}{x \cdot {z}^{y}} \]

   if 2.99999999999999979e-63 < t < 3e22

   1. Initial program 87.8%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      3. *-lowering-*.f64 87.8

         \[\leadsto x \cdot e^{-\color{blue}{a \cdot b}} \]

   5. Simplified 87.8%

      \[\leadsto x \cdot e^{\color{blue}{-a \cdot b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -400:\\ \;\;\;\;x \cdot e^{-y \cdot t}\\ \mathbf{elif}\;t \leq 3 \cdot 10^{-63}:\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{elif}\;t \leq 3 \cdot 10^{+22}:\\ \;\;\;\;x \cdot e^{b \cdot \left(-a\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{-y \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 76.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot {z}^{y}\\ \mathbf{if}\;y \leq -1.2 \cdot 10^{+40}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 17:\\ \;\;\;\;x \cdot e^{-a \cdot \left(z + b\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (pow z y))))
   (if (<= y -1.2e+40)
     t_1
     (if (<= y 17.0) (* x (exp (- (* a (+ z b))))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * pow(z, y);
	double tmp;
	if (y <= -1.2e+40) {
		tmp = t_1;
	} else if (y <= 17.0) {
		tmp = x * exp(-(a * (z + b)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * (z ** y)
    if (y <= (-1.2d+40)) then
        tmp = t_1
    else if (y <= 17.0d0) then
        tmp = x * exp(-(a * (z + b)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * Math.pow(z, y);
	double tmp;
	if (y <= -1.2e+40) {
		tmp = t_1;
	} else if (y <= 17.0) {
		tmp = x * Math.exp(-(a * (z + b)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x * math.pow(z, y)
	tmp = 0
	if y <= -1.2e+40:
		tmp = t_1
	elif y <= 17.0:
		tmp = x * math.exp(-(a * (z + b)))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x * (z ^ y))
	tmp = 0.0
	if (y <= -1.2e+40)
		tmp = t_1;
	elseif (y <= 17.0)
		tmp = Float64(x * exp(Float64(-Float64(a * Float64(z + b)))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x * (z ^ y);
	tmp = 0.0;
	if (y <= -1.2e+40)
		tmp = t_1;
	elseif (y <= 17.0)
		tmp = x * exp(-(a * (z + b)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.2e+40], t$95$1, If[LessEqual[y, 17.0], N[(x * N[Exp[(-N[(a * N[(z + b), $MachinePrecision]), $MachinePrecision])], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.2e40 or 17 < y

   1. Initial program 98.4%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\log z - t\right)}} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\log z - t\right)}} \]

      2. --lowering--.f64 N/A

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(\log z - t\right)}} \]

      3. log-lowering-log.f64 92.7

         \[\leadsto x \cdot e^{y \cdot \left(\color{blue}{\log z} - t\right)} \]

   5. Simplified 92.7%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\log z - t\right)}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{x \cdot {z}^{y}} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot {z}^{y}} \]

      2. pow-lowering-pow.f64 75.8

         \[\leadsto x \cdot \color{blue}{{z}^{y}} \]

   8. Simplified 75.8%

      \[\leadsto \color{blue}{x \cdot {z}^{y}} \]

   if -1.2e40 < y < 17

   1. Initial program 94.2%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto x \cdot e^{\color{blue}{a \cdot \left(\log \left(1 - z\right) - b\right)}} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{a \cdot \left(\log \left(1 - z\right) - b\right)}} \]

      2. --lowering--.f64 N/A

         \[\leadsto x \cdot e^{a \cdot \color{blue}{\left(\log \left(1 - z\right) - b\right)}} \]

      3. sub-neg N/A

         \[\leadsto x \cdot e^{a \cdot \left(\log \color{blue}{\left(1 + \left(\mathsf{neg}\left(z\right)\right)\right)} - b\right)} \]

      4. accelerator-lowering-log1p.f64 N/A

         \[\leadsto x \cdot e^{a \cdot \left(\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(z\right)\right)} - b\right)} \]

      5. neg-lowering-neg.f64 83.7

         \[\leadsto x \cdot e^{a \cdot \left(\mathsf{log1p}\left(\color{blue}{-z}\right) - b\right)} \]

   5. Simplified 83.7%

      \[\leadsto x \cdot e^{\color{blue}{a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)}} \]

   6. Taylor expanded in z around 0

      \[\leadsto x \cdot e^{a \cdot \color{blue}{\left(-1 \cdot z - b\right)}} \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto x \cdot e^{a \cdot \color{blue}{\left(-1 \cdot z - b\right)}} \]

      2. mul-1-neg N/A

         \[\leadsto x \cdot e^{a \cdot \left(\color{blue}{\left(\mathsf{neg}\left(z\right)\right)} - b\right)} \]

      3. neg-lowering-neg.f64 83.7

         \[\leadsto x \cdot e^{a \cdot \left(\color{blue}{\left(-z\right)} - b\right)} \]

   8. Simplified 83.7%

      \[\leadsto x \cdot e^{a \cdot \color{blue}{\left(\left(-z\right) - b\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{+40}:\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{elif}\;y \leq 17:\\ \;\;\;\;x \cdot e^{-a \cdot \left(z + b\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 73.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot {z}^{y}\\ \mathbf{if}\;y \leq -7.5 \cdot 10^{+39}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.8:\\ \;\;\;\;x \cdot e^{b \cdot \left(-a\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (pow z y))))
   (if (<= y -7.5e+39) t_1 (if (<= y 2.8) (* x (exp (* b (- a)))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * pow(z, y);
	double tmp;
	if (y <= -7.5e+39) {
		tmp = t_1;
	} else if (y <= 2.8) {
		tmp = x * exp((b * -a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * (z ** y)
    if (y <= (-7.5d+39)) then
        tmp = t_1
    else if (y <= 2.8d0) then
        tmp = x * exp((b * -a))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * Math.pow(z, y);
	double tmp;
	if (y <= -7.5e+39) {
		tmp = t_1;
	} else if (y <= 2.8) {
		tmp = x * Math.exp((b * -a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x * math.pow(z, y)
	tmp = 0
	if y <= -7.5e+39:
		tmp = t_1
	elif y <= 2.8:
		tmp = x * math.exp((b * -a))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x * (z ^ y))
	tmp = 0.0
	if (y <= -7.5e+39)
		tmp = t_1;
	elseif (y <= 2.8)
		tmp = Float64(x * exp(Float64(b * Float64(-a))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x * (z ^ y);
	tmp = 0.0;
	if (y <= -7.5e+39)
		tmp = t_1;
	elseif (y <= 2.8)
		tmp = x * exp((b * -a));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -7.5e+39], t$95$1, If[LessEqual[y, 2.8], N[(x * N[Exp[N[(b * (-a)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -7.5000000000000005e39 or 2.7999999999999998 < y

   1. Initial program 98.4%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\log z - t\right)}} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\log z - t\right)}} \]

      2. --lowering--.f64 N/A

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(\log z - t\right)}} \]

      3. log-lowering-log.f64 92.7

         \[\leadsto x \cdot e^{y \cdot \left(\color{blue}{\log z} - t\right)} \]

   5. Simplified 92.7%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\log z - t\right)}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{x \cdot {z}^{y}} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot {z}^{y}} \]

      2. pow-lowering-pow.f64 75.8

         \[\leadsto x \cdot \color{blue}{{z}^{y}} \]

   8. Simplified 75.8%

      \[\leadsto \color{blue}{x \cdot {z}^{y}} \]

   if -7.5000000000000005e39 < y < 2.7999999999999998

   1. Initial program 94.2%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      3. *-lowering-*.f64 75.2

         \[\leadsto x \cdot e^{-\color{blue}{a \cdot b}} \]

   5. Simplified 75.2%

      \[\leadsto x \cdot e^{\color{blue}{-a \cdot b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{+39}:\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{elif}\;y \leq 2.8:\\ \;\;\;\;x \cdot e^{b \cdot \left(-a\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 31.2% accurate, 9.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{+88}:\\ \;\;\;\;x \cdot \left(1 - y \cdot t\right)\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{-235}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(a, \left(-z\right) - b, 1\right)\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{+137}:\\ \;\;\;\;a \cdot \frac{x}{a}\\ \mathbf{else}:\\ \;\;\;\;-x \cdot \left(a \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -1.3e+88)
   (* x (- 1.0 (* y t)))
   (if (<= y 3.7e-235)
     (* x (fma a (- (- z) b) 1.0))
     (if (<= y 5.2e+137) (* a (/ x a)) (- (* x (* a b)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -1.3e+88) {
		tmp = x * (1.0 - (y * t));
	} else if (y <= 3.7e-235) {
		tmp = x * fma(a, (-z - b), 1.0);
	} else if (y <= 5.2e+137) {
		tmp = a * (x / a);
	} else {
		tmp = -(x * (a * b));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -1.3e+88)
		tmp = Float64(x * Float64(1.0 - Float64(y * t)));
	elseif (y <= 3.7e-235)
		tmp = Float64(x * fma(a, Float64(Float64(-z) - b), 1.0));
	elseif (y <= 5.2e+137)
		tmp = Float64(a * Float64(x / a));
	else
		tmp = Float64(-Float64(x * Float64(a * b)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -1.3e+88], N[(x * N[(1.0 - N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.7e-235], N[(x * N[(a * N[((-z) - b), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.2e+137], N[(a * N[(x / a), $MachinePrecision]), $MachinePrecision], (-N[(x * N[(a * b), $MachinePrecision]), $MachinePrecision])]]]

Derivation

1. Split input into 4 regimes

2. if y < -1.3e88

   1. Initial program 97.6%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(t \cdot y\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(t \cdot y\right)}} \]

      2. *-commutative N/A

         \[\leadsto x \cdot e^{\mathsf{neg}\left(\color{blue}{y \cdot t}\right)} \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      5. neg-lowering-neg.f64 64.4

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(-t\right)}} \]

   5. Simplified 64.4%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}} \]

   6. Taylor expanded in y around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + -1 \cdot \left(t \cdot y\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot \left(1 + \color{blue}{\left(\mathsf{neg}\left(t \cdot y\right)\right)}\right) \]

      2. unsub-neg N/A

         \[\leadsto x \cdot \color{blue}{\left(1 - t \cdot y\right)} \]

      3. --lowering--.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(1 - t \cdot y\right)} \]

      4. *-lowering-*.f64 26.7

         \[\leadsto x \cdot \left(1 - \color{blue}{t \cdot y}\right) \]

   8. Simplified 26.7%

      \[\leadsto x \cdot \color{blue}{\left(1 - t \cdot y\right)} \]

   if -1.3e88 < y < 3.7000000000000001e-235

   1. Initial program 93.9%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto x \cdot e^{\color{blue}{a \cdot \left(\log \left(1 - z\right) - b\right)}} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{a \cdot \left(\log \left(1 - z\right) - b\right)}} \]

      2. --lowering--.f64 N/A

         \[\leadsto x \cdot e^{a \cdot \color{blue}{\left(\log \left(1 - z\right) - b\right)}} \]

      3. sub-neg N/A

         \[\leadsto x \cdot e^{a \cdot \left(\log \color{blue}{\left(1 + \left(\mathsf{neg}\left(z\right)\right)\right)} - b\right)} \]

      4. accelerator-lowering-log1p.f64 N/A

         \[\leadsto x \cdot e^{a \cdot \left(\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(z\right)\right)} - b\right)} \]

      5. neg-lowering-neg.f64 82.2

         \[\leadsto x \cdot e^{a \cdot \left(\mathsf{log1p}\left(\color{blue}{-z}\right) - b\right)} \]

   5. Simplified 82.2%

      \[\leadsto x \cdot e^{\color{blue}{a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)}} \]

   6. Taylor expanded in z around 0

      \[\leadsto x \cdot e^{a \cdot \color{blue}{\left(-1 \cdot z - b\right)}} \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto x \cdot e^{a \cdot \color{blue}{\left(-1 \cdot z - b\right)}} \]

      2. mul-1-neg N/A

         \[\leadsto x \cdot e^{a \cdot \left(\color{blue}{\left(\mathsf{neg}\left(z\right)\right)} - b\right)} \]

      3. neg-lowering-neg.f64 82.2

         \[\leadsto x \cdot e^{a \cdot \left(\color{blue}{\left(-z\right)} - b\right)} \]

   8. Simplified 82.2%

      \[\leadsto x \cdot e^{a \cdot \color{blue}{\left(\left(-z\right) - b\right)}} \]

   9. Taylor expanded in a around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + -1 \cdot \left(a \cdot \left(b + z\right)\right)\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \left(a \cdot \left(b + z\right)\right) + 1\right)} \]

       2. *-commutative N/A

          \[\leadsto x \cdot \left(\color{blue}{\left(a \cdot \left(b + z\right)\right) \cdot -1} + 1\right) \]

       3. associate-*r* N/A

          \[\leadsto x \cdot \left(\color{blue}{a \cdot \left(\left(b + z\right) \cdot -1\right)} + 1\right) \]

       4. *-commutative N/A

          \[\leadsto x \cdot \left(a \cdot \color{blue}{\left(-1 \cdot \left(b + z\right)\right)} + 1\right) \]

       5. accelerator-lowering-fma.f64 N/A

          \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, -1 \cdot \left(b + z\right), 1\right)} \]

       6. distribute-lft-in N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{-1 \cdot b + -1 \cdot z}, 1\right) \]

       7. +-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{-1 \cdot z + -1 \cdot b}, 1\right) \]

       8. mul-1-neg N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, -1 \cdot z + \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}, 1\right) \]

       9. sub-neg N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{-1 \cdot z - b}, 1\right) \]

       10. --lowering--.f64 N/A

           \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{-1 \cdot z - b}, 1\right) \]

       11. mul-1-neg N/A

           \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\mathsf{neg}\left(z\right)\right)} - b, 1\right) \]

       12. neg-lowering-neg.f64 46.7

           \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(-z\right)} - b, 1\right) \]

   11. Simplified 46.7%

       \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, \left(-z\right) - b, 1\right)} \]

   if 3.7000000000000001e-235 < y < 5.1999999999999998e137

   1. Initial program 97.4%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      3. *-lowering-*.f64 59.7

         \[\leadsto x \cdot e^{-\color{blue}{a \cdot b}} \]

   5. Simplified 59.7%

      \[\leadsto x \cdot e^{\color{blue}{-a \cdot b}} \]

   6. Taylor expanded in a around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right) + 1\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, -1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right)} \]

      3. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right) \]

      4. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) + \left(\mathsf{neg}\left(b\right)\right)}, 1\right) \]

      5. associate-*r* N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\frac{1}{2} \cdot a\right) \cdot {b}^{2}} + \left(\mathsf{neg}\left(b\right)\right), 1\right) \]

      6. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \left(\frac{1}{2} \cdot a\right) \cdot \color{blue}{\left(b \cdot b\right)} + \left(\mathsf{neg}\left(b\right)\right), 1\right) \]

      7. associate-*r* N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\left(\frac{1}{2} \cdot a\right) \cdot b\right) \cdot b} + \left(\mathsf{neg}\left(b\right)\right), 1\right) \]

      8. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \left(\left(\frac{1}{2} \cdot a\right) \cdot b\right) \cdot b + \color{blue}{-1 \cdot b}, 1\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{b \cdot \left(\left(\frac{1}{2} \cdot a\right) \cdot b + -1\right)}, 1\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{b \cdot \left(\left(\frac{1}{2} \cdot a\right) \cdot b + -1\right)}, 1\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, b \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot a, b, -1\right)}, 1\right) \]

      12. *-lowering-*.f64 29.3

          \[\leadsto x \cdot \mathsf{fma}\left(a, b \cdot \mathsf{fma}\left(\color{blue}{0.5 \cdot a}, b, -1\right), 1\right) \]

   8. Simplified 29.3%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, b \cdot \mathsf{fma}\left(0.5 \cdot a, b, -1\right), 1\right)} \]

   9. Taylor expanded in a around inf

      \[\leadsto \color{blue}{{a}^{2} \cdot \left(-1 \cdot \frac{b \cdot x}{a} + \left(\frac{1}{2} \cdot \left({b}^{2} \cdot x\right) + \frac{x}{{a}^{2}}\right)\right)} \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \color{blue}{\left(a \cdot a\right)} \cdot \left(-1 \cdot \frac{b \cdot x}{a} + \left(\frac{1}{2} \cdot \left({b}^{2} \cdot x\right) + \frac{x}{{a}^{2}}\right)\right) \]

       2. associate-*l* N/A

          \[\leadsto \color{blue}{a \cdot \left(a \cdot \left(-1 \cdot \frac{b \cdot x}{a} + \left(\frac{1}{2} \cdot \left({b}^{2} \cdot x\right) + \frac{x}{{a}^{2}}\right)\right)\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{a \cdot \left(a \cdot \left(-1 \cdot \frac{b \cdot x}{a} + \left(\frac{1}{2} \cdot \left({b}^{2} \cdot x\right) + \frac{x}{{a}^{2}}\right)\right)\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto a \cdot \color{blue}{\left(a \cdot \left(-1 \cdot \frac{b \cdot x}{a} + \left(\frac{1}{2} \cdot \left({b}^{2} \cdot x\right) + \frac{x}{{a}^{2}}\right)\right)\right)} \]

       5. associate-+r+ N/A

          \[\leadsto a \cdot \left(a \cdot \color{blue}{\left(\left(-1 \cdot \frac{b \cdot x}{a} + \frac{1}{2} \cdot \left({b}^{2} \cdot x\right)\right) + \frac{x}{{a}^{2}}\right)}\right) \]

   11. Simplified 27.3%

       \[\leadsto \color{blue}{a \cdot \left(a \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, x \cdot 0.5, \frac{x}{-a}\right), \frac{x}{a \cdot a}\right)\right)} \]

   12. Taylor expanded in a around 0

       \[\leadsto a \cdot \color{blue}{\frac{x}{a}} \]
   13. Step-by-step derivation

       1. /-lowering-/.f64 34.9

          \[\leadsto a \cdot \color{blue}{\frac{x}{a}} \]

   14. Simplified 34.9%

       \[\leadsto a \cdot \color{blue}{\frac{x}{a}} \]

   if 5.1999999999999998e137 < y

   1. Initial program 97.8%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      3. *-lowering-*.f64 20.6

         \[\leadsto x \cdot e^{-\color{blue}{a \cdot b}} \]

   5. Simplified 20.6%

      \[\leadsto x \cdot e^{\color{blue}{-a \cdot b}} \]

   6. Taylor expanded in a around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right) + 1\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, -1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right)} \]

      3. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right) \]

      4. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) + \left(\mathsf{neg}\left(b\right)\right)}, 1\right) \]

      5. associate-*r* N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\frac{1}{2} \cdot a\right) \cdot {b}^{2}} + \left(\mathsf{neg}\left(b\right)\right), 1\right) \]

      6. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \left(\frac{1}{2} \cdot a\right) \cdot \color{blue}{\left(b \cdot b\right)} + \left(\mathsf{neg}\left(b\right)\right), 1\right) \]

      7. associate-*r* N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\left(\frac{1}{2} \cdot a\right) \cdot b\right) \cdot b} + \left(\mathsf{neg}\left(b\right)\right), 1\right) \]

      8. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \left(\left(\frac{1}{2} \cdot a\right) \cdot b\right) \cdot b + \color{blue}{-1 \cdot b}, 1\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{b \cdot \left(\left(\frac{1}{2} \cdot a\right) \cdot b + -1\right)}, 1\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{b \cdot \left(\left(\frac{1}{2} \cdot a\right) \cdot b + -1\right)}, 1\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, b \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot a, b, -1\right)}, 1\right) \]

      12. *-lowering-*.f64 12.0

          \[\leadsto x \cdot \mathsf{fma}\left(a, b \cdot \mathsf{fma}\left(\color{blue}{0.5 \cdot a}, b, -1\right), 1\right) \]

   8. Simplified 12.0%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, b \cdot \mathsf{fma}\left(0.5 \cdot a, b, -1\right), 1\right)} \]

   9. Taylor expanded in b around inf

      \[\leadsto x \cdot \color{blue}{\left({b}^{2} \cdot \left(-1 \cdot \frac{a}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto x \cdot \left(\color{blue}{\left(b \cdot b\right)} \cdot \left(-1 \cdot \frac{a}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right) \]

       2. associate-*l* N/A

          \[\leadsto x \cdot \color{blue}{\left(b \cdot \left(b \cdot \left(-1 \cdot \frac{a}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \color{blue}{\left(b \cdot \left(b \cdot \left(-1 \cdot \frac{a}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \left(b \cdot \color{blue}{\left(b \cdot \left(-1 \cdot \frac{a}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)}\right) \]

       5. associate-*r/ N/A

          \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(\color{blue}{\frac{-1 \cdot a}{b}} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(\frac{\color{blue}{a \cdot -1}}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       7. associate-/l* N/A

          \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(\color{blue}{a \cdot \frac{-1}{b}} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       8. metadata-eval N/A

          \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \frac{\color{blue}{\mathsf{neg}\left(1\right)}}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       9. distribute-neg-frac N/A

          \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{b}\right)\right)} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       10. *-commutative N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(\mathsf{neg}\left(\frac{1}{b}\right)\right) + \color{blue}{{a}^{2} \cdot \frac{1}{2}}\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(\mathsf{neg}\left(\frac{1}{b}\right)\right) + \color{blue}{\left(a \cdot a\right)} \cdot \frac{1}{2}\right)\right)\right) \]

       12. associate-*l* N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(\mathsf{neg}\left(\frac{1}{b}\right)\right) + \color{blue}{a \cdot \left(a \cdot \frac{1}{2}\right)}\right)\right)\right) \]

       13. *-commutative N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(\mathsf{neg}\left(\frac{1}{b}\right)\right) + a \cdot \color{blue}{\left(\frac{1}{2} \cdot a\right)}\right)\right)\right) \]

       14. distribute-lft-out N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \color{blue}{\left(a \cdot \left(\left(\mathsf{neg}\left(\frac{1}{b}\right)\right) + \frac{1}{2} \cdot a\right)\right)}\right)\right) \]

       15. +-commutative N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \color{blue}{\left(\frac{1}{2} \cdot a + \left(\mathsf{neg}\left(\frac{1}{b}\right)\right)\right)}\right)\right)\right) \]

       16. sub-neg N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \color{blue}{\left(\frac{1}{2} \cdot a - \frac{1}{b}\right)}\right)\right)\right) \]

       17. *-lowering-*.f64 N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \color{blue}{\left(a \cdot \left(\frac{1}{2} \cdot a - \frac{1}{b}\right)\right)}\right)\right) \]

       18. sub-neg N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \color{blue}{\left(\frac{1}{2} \cdot a + \left(\mathsf{neg}\left(\frac{1}{b}\right)\right)\right)}\right)\right)\right) \]

       19. *-commutative N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(\color{blue}{a \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(\frac{1}{b}\right)\right)\right)\right)\right)\right) \]

       20. accelerator-lowering-fma.f64 N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \color{blue}{\mathsf{fma}\left(a, \frac{1}{2}, \mathsf{neg}\left(\frac{1}{b}\right)\right)}\right)\right)\right) \]

       21. distribute-neg-frac N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \mathsf{fma}\left(a, \frac{1}{2}, \color{blue}{\frac{\mathsf{neg}\left(1\right)}{b}}\right)\right)\right)\right) \]

       22. metadata-eval N/A

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \mathsf{fma}\left(a, \frac{1}{2}, \frac{\color{blue}{-1}}{b}\right)\right)\right)\right) \]

       23. /-lowering-/.f64 44.1

           \[\leadsto x \cdot \left(b \cdot \left(b \cdot \left(a \cdot \mathsf{fma}\left(a, 0.5, \color{blue}{\frac{-1}{b}}\right)\right)\right)\right) \]

   11. Simplified 44.1%

       \[\leadsto x \cdot \color{blue}{\left(b \cdot \left(b \cdot \left(a \cdot \mathsf{fma}\left(a, 0.5, \frac{-1}{b}\right)\right)\right)\right)} \]

   12. Taylor expanded in b around 0

       \[\leadsto x \cdot \left(b \cdot \color{blue}{\left(-1 \cdot a\right)}\right) \]
   13. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto x \cdot \left(b \cdot \color{blue}{\left(\mathsf{neg}\left(a\right)\right)}\right) \]

       2. neg-lowering-neg.f64 29.1

          \[\leadsto x \cdot \left(b \cdot \color{blue}{\left(-a\right)}\right) \]

   14. Simplified 29.1%

       \[\leadsto x \cdot \left(b \cdot \color{blue}{\left(-a\right)}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 37.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{+88}:\\ \;\;\;\;x \cdot \left(1 - y \cdot t\right)\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{-235}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(a, \left(-z\right) - b, 1\right)\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{+137}:\\ \;\;\;\;a \cdot \frac{x}{a}\\ \mathbf{else}:\\ \;\;\;\;-x \cdot \left(a \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 19.6% accurate, 328.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 x)

C

double code(double x, double y, double z, double t, double a, double b) {
	return x;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = x
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return x;
}

Python

def code(x, y, z, t, a, b):
	return x

Julia

function code(x, y, z, t, a, b)
	return x
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = x;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := x

Derivation

1. Initial program 96.2%

   \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
2. Add Preprocessing

3. Taylor expanded in b around inf

   \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

   2. neg-lowering-neg.f64 N/A

      \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

   3. *-lowering-*.f64 55.2

      \[\leadsto x \cdot e^{-\color{blue}{a \cdot b}} \]

5. Simplified 55.2%

   \[\leadsto x \cdot e^{\color{blue}{-a \cdot b}} \]

6. Taylor expanded in a around 0

   \[\leadsto \color{blue}{x} \]
7. Step-by-step derivation

8. Simplified 19.6%

   \[\leadsto \color{blue}{x} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024205 
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, B"
  :precision binary64
  (* x (exp (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b))))))