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

Percentage Accurate: 96.3% → 96.3%

Time: 16.2s

Alternatives: 19

Speedup: 1.0×

• 42.3% 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: 96.3% 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: 96.3% 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 95.8%

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

Alternative 2: 51.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+269}:\\ \;\;\;\;t \cdot \frac{x}{t}\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{+211}:\\ \;\;\;\;\left(x \cdot \left(y \cdot y\right)\right) \cdot \left(t \cdot \left(t \cdot 0.5\right)\right)\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{+135}:\\ \;\;\;\;\left(0.5 \cdot \left(a \cdot a\right)\right) \cdot \left(x \cdot \left(b \cdot b\right)\right)\\ \mathbf{elif}\;t\_1 \leq -4000000000:\\ \;\;\;\;x \cdot \left(\left(y \cdot \left(y \cdot y\right)\right) \cdot \left(-0.16666666666666666 \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)\right)\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-10}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(0.5, b \cdot \left(a \cdot a\right), -a\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(t \cdot \mathsf{fma}\left(x, \left(y \cdot y\right) \cdot 0.5, \frac{\mathsf{fma}\left(x, -y, \frac{x}{t}\right)}{t}\right)\right)\\ \end{array} \end{array} \]

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 -5e+269)
     (* t (/ x t))
     (if (<= t_1 -5e+211)
       (* (* x (* y y)) (* t (* t 0.5)))
       (if (<= t_1 -5e+135)
         (* (* 0.5 (* a a)) (* x (* b b)))
         (if (<= t_1 -4000000000.0)
           (* x (* (* y (* y y)) (* -0.16666666666666666 (* t (* t t)))))
           (if (<= t_1 4e-10)
             (* x (fma b (fma 0.5 (* b (* a a)) (- a)) 1.0))
             (*
              t
              (* t (fma x (* (* y y) 0.5) (/ (fma x (- y) (/ x t)) 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 <= -5e+269) {
		tmp = t * (x / t);
	} else if (t_1 <= -5e+211) {
		tmp = (x * (y * y)) * (t * (t * 0.5));
	} else if (t_1 <= -5e+135) {
		tmp = (0.5 * (a * a)) * (x * (b * b));
	} else if (t_1 <= -4000000000.0) {
		tmp = x * ((y * (y * y)) * (-0.16666666666666666 * (t * (t * t))));
	} else if (t_1 <= 4e-10) {
		tmp = x * fma(b, fma(0.5, (b * (a * a)), -a), 1.0);
	} else {
		tmp = t * (t * fma(x, ((y * y) * 0.5), (fma(x, -y, (x / t)) / 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 <= -5e+269)
		tmp = Float64(t * Float64(x / t));
	elseif (t_1 <= -5e+211)
		tmp = Float64(Float64(x * Float64(y * y)) * Float64(t * Float64(t * 0.5)));
	elseif (t_1 <= -5e+135)
		tmp = Float64(Float64(0.5 * Float64(a * a)) * Float64(x * Float64(b * b)));
	elseif (t_1 <= -4000000000.0)
		tmp = Float64(x * Float64(Float64(y * Float64(y * y)) * Float64(-0.16666666666666666 * Float64(t * Float64(t * t)))));
	elseif (t_1 <= 4e-10)
		tmp = Float64(x * fma(b, fma(0.5, Float64(b * Float64(a * a)), Float64(-a)), 1.0));
	else
		tmp = Float64(t * Float64(t * fma(x, Float64(Float64(y * y) * 0.5), Float64(fma(x, Float64(-y), Float64(x / t)) / 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, -5e+269], N[(t * N[(x / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -5e+211], N[(N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(t * N[(t * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -5e+135], N[(N[(0.5 * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(x * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -4000000000.0], N[(x * N[(N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(-0.16666666666666666 * N[(t * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e-10], N[(x * N[(b * N[(0.5 * N[(b * N[(a * a), $MachinePrecision]), $MachinePrecision] + (-a)), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(t * N[(t * N[(x * N[(N[(y * y), $MachinePrecision] * 0.5), $MachinePrecision] + N[(N[(x * (-y) + N[(x / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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))) < -5.0000000000000002e269

   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. lower-*.f64 N/A

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

      5. lower-neg.f64 65.7

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

   5. Simplified 65.7%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

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

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

      8. mul-1-neg N/A

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

      9. lower-*.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 2.8

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

   8. Simplified 2.8%

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

   9. Taylor expanded in t around inf

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

       1. lower-*.f64 N/A

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

       2. +-commutative N/A

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

       3. mul-1-neg N/A

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

       4. unsub-neg N/A

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

       5. lower--.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. lower-*.f64 11.6

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

   11. Simplified 11.6%

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

   12. Taylor expanded in t around 0

       \[\leadsto t \cdot \color{blue}{\frac{x}{t}} \]
   13. Step-by-step derivation

       1. lower-/.f64 37.8

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

   14. Simplified 37.8%

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

   if -5.0000000000000002e269 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -4.9999999999999995e211

   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. lower-*.f64 N/A

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

      5. lower-neg.f64 28.5

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

   5. Simplified 28.5%

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

   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. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. mul-1-neg N/A

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

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

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

      12. mul-1-neg N/A

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

      13. lower-*.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 2.6

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

   8. Simplified 2.6%

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

   9. Taylor expanded in t around inf

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. associate-*l* N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 N/A

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

       9. unpow2 N/A

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

       10. associate-*r* N/A

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

       11. lower-*.f64 N/A

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

       12. lower-*.f64 37.7

           \[\leadsto \left(x \cdot \left(y \cdot y\right)\right) \cdot \left(\color{blue}{\left(0.5 \cdot t\right)} \cdot t\right) \]

   11. Simplified 37.7%

       \[\leadsto \color{blue}{\left(x \cdot \left(y \cdot y\right)\right) \cdot \left(\left(0.5 \cdot t\right) \cdot t\right)} \]

   if -4.9999999999999995e211 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -5.00000000000000029e135

   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. lower-neg.f64 N/A

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

      3. lower-*.f64 29.7

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

   5. Simplified 29.7%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 2.8

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

   8. Simplified 2.8%

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

   9. Taylor expanded in a around inf

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

       1. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 46.3

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

   11. Simplified 46.3%

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

   if -5.00000000000000029e135 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -4e9

   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. lower-*.f64 N/A

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

      5. lower-neg.f64 25.3

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

   5. Simplified 25.3%

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

   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. lower-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. lower-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. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. cube-mult N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. mul-1-neg N/A

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

      17. lower-neg.f64 3.1

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

   8. Simplified 3.1%

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

   9. Taylor expanded in t around inf

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. associate-*r* N/A

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

       5. lower-*.f64 N/A

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

       6. cube-mult N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 N/A

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

       11. lower-*.f64 N/A

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

       12. cube-mult N/A

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

       13. unpow2 N/A

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

       14. lower-*.f64 N/A

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

       15. unpow2 N/A

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

       16. lower-*.f64 59.6

           \[\leadsto x \cdot \left(\left(y \cdot \left(y \cdot y\right)\right) \cdot \left(-0.16666666666666666 \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)\right) \]

   11. Simplified 59.6%

       \[\leadsto x \cdot \color{blue}{\left(\left(y \cdot \left(y \cdot y\right)\right) \cdot \left(-0.16666666666666666 \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)\right)} \]

   if -4e9 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 4.00000000000000015e-10

   1. Initial program 87.2%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   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. lower-neg.f64 N/A

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

      3. lower-*.f64 84.9

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

   5. Simplified 84.9%

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

   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. lower-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. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 90.4

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

   8. Simplified 90.4%

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

   if 4.00000000000000015e-10 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   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 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. lower-*.f64 N/A

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

      5. lower-neg.f64 59.4

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

   5. Simplified 59.4%

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

   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. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. mul-1-neg N/A

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

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

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

      12. mul-1-neg N/A

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

      13. lower-*.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 63.4

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

   8. Simplified 63.4%

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

   9. Taylor expanded in t around -inf

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

       1. unpow2 N/A

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

       2. associate-*l* N/A

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

       3. lower-*.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. +-commutative N/A

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

       6. *-commutative N/A

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

       7. associate-*l* N/A

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

       8. lower-fma.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

       12. associate-*r/ N/A

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

   11. Simplified 74.3%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -5 \cdot 10^{+269}:\\ \;\;\;\;t \cdot \frac{x}{t}\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -5 \cdot 10^{+211}:\\ \;\;\;\;\left(x \cdot \left(y \cdot y\right)\right) \cdot \left(t \cdot \left(t \cdot 0.5\right)\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -5 \cdot 10^{+135}:\\ \;\;\;\;\left(0.5 \cdot \left(a \cdot a\right)\right) \cdot \left(x \cdot \left(b \cdot b\right)\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -4000000000:\\ \;\;\;\;x \cdot \left(\left(y \cdot \left(y \cdot y\right)\right) \cdot \left(-0.16666666666666666 \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 4 \cdot 10^{-10}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(0.5, b \cdot \left(a \cdot a\right), -a\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(t \cdot \mathsf{fma}\left(x, \left(y \cdot y\right) \cdot 0.5, \frac{\mathsf{fma}\left(x, -y, \frac{x}{t}\right)}{t}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 48.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(y \cdot y\right)\\ t_2 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+269}:\\ \;\;\;\;t \cdot \frac{x}{t}\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{+211}:\\ \;\;\;\;\left(x \cdot \left(y \cdot y\right)\right) \cdot \left(t \cdot \left(t \cdot 0.5\right)\right)\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{+135}:\\ \;\;\;\;\left(0.5 \cdot \left(a \cdot a\right)\right) \cdot \left(x \cdot \left(b \cdot b\right)\right)\\ \mathbf{elif}\;t\_2 \leq -4000000000:\\ \;\;\;\;x \cdot \left(t\_1 \cdot \left(-0.16666666666666666 \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)\right)\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+203}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(t, \mathsf{fma}\left(t, t\_1 \cdot \mathsf{fma}\left(-0.16666666666666666, t, \frac{0.5}{y}\right), -y\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, t \cdot \left(x \cdot \left(\left(y \cdot y\right) \cdot 0.5\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* y (* y y)))
        (t_2 (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b)))))
   (if (<= t_2 -5e+269)
     (* t (/ x t))
     (if (<= t_2 -5e+211)
       (* (* x (* y y)) (* t (* t 0.5)))
       (if (<= t_2 -5e+135)
         (* (* 0.5 (* a a)) (* x (* b b)))
         (if (<= t_2 -4000000000.0)
           (* x (* t_1 (* -0.16666666666666666 (* t (* t t)))))
           (if (<= t_2 4e+203)
             (*
              x
              (fma
               t
               (fma t (* t_1 (fma -0.16666666666666666 t (/ 0.5 y))) (- y))
               1.0))
             (fma t (* t (* x (* (* y y) 0.5))) x))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = y * (y * y);
	double t_2 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	double tmp;
	if (t_2 <= -5e+269) {
		tmp = t * (x / t);
	} else if (t_2 <= -5e+211) {
		tmp = (x * (y * y)) * (t * (t * 0.5));
	} else if (t_2 <= -5e+135) {
		tmp = (0.5 * (a * a)) * (x * (b * b));
	} else if (t_2 <= -4000000000.0) {
		tmp = x * (t_1 * (-0.16666666666666666 * (t * (t * t))));
	} else if (t_2 <= 4e+203) {
		tmp = x * fma(t, fma(t, (t_1 * fma(-0.16666666666666666, t, (0.5 / y))), -y), 1.0);
	} else {
		tmp = fma(t, (t * (x * ((y * y) * 0.5))), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(y * Float64(y * y))
	t_2 = Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))
	tmp = 0.0
	if (t_2 <= -5e+269)
		tmp = Float64(t * Float64(x / t));
	elseif (t_2 <= -5e+211)
		tmp = Float64(Float64(x * Float64(y * y)) * Float64(t * Float64(t * 0.5)));
	elseif (t_2 <= -5e+135)
		tmp = Float64(Float64(0.5 * Float64(a * a)) * Float64(x * Float64(b * b)));
	elseif (t_2 <= -4000000000.0)
		tmp = Float64(x * Float64(t_1 * Float64(-0.16666666666666666 * Float64(t * Float64(t * t)))));
	elseif (t_2 <= 4e+203)
		tmp = Float64(x * fma(t, fma(t, Float64(t_1 * fma(-0.16666666666666666, t, Float64(0.5 / y))), Float64(-y)), 1.0));
	else
		tmp = fma(t, Float64(t * Float64(x * Float64(Float64(y * y) * 0.5))), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(y * N[(y * y), $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, -5e+269], N[(t * N[(x / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -5e+211], N[(N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(t * N[(t * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -5e+135], N[(N[(0.5 * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(x * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -4000000000.0], N[(x * N[(t$95$1 * N[(-0.16666666666666666 * N[(t * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 4e+203], N[(x * N[(t * N[(t * N[(t$95$1 * N[(-0.16666666666666666 * t + N[(0.5 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + (-y)), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(t * N[(t * N[(x * N[(N[(y * y), $MachinePrecision] * 0.5), $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))) < -5.0000000000000002e269

   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. lower-*.f64 N/A

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

      5. lower-neg.f64 65.7

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

   5. Simplified 65.7%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

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

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

      8. mul-1-neg N/A

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

      9. lower-*.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 2.8

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

   8. Simplified 2.8%

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

   9. Taylor expanded in t around inf

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

       1. lower-*.f64 N/A

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

       2. +-commutative N/A

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

       3. mul-1-neg N/A

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

       4. unsub-neg N/A

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

       5. lower--.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. lower-*.f64 11.6

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

   11. Simplified 11.6%

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

   12. Taylor expanded in t around 0

       \[\leadsto t \cdot \color{blue}{\frac{x}{t}} \]
   13. Step-by-step derivation

       1. lower-/.f64 37.8

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

   14. Simplified 37.8%

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

   if -5.0000000000000002e269 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -4.9999999999999995e211

   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. lower-*.f64 N/A

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

      5. lower-neg.f64 28.5

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

   5. Simplified 28.5%

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

   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. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. mul-1-neg N/A

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

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

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

      12. mul-1-neg N/A

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

      13. lower-*.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 2.6

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

   8. Simplified 2.6%

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

   9. Taylor expanded in t around inf

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. associate-*l* N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 N/A

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

       9. unpow2 N/A

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

       10. associate-*r* N/A

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

       11. lower-*.f64 N/A

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

       12. lower-*.f64 37.7

           \[\leadsto \left(x \cdot \left(y \cdot y\right)\right) \cdot \left(\color{blue}{\left(0.5 \cdot t\right)} \cdot t\right) \]

   11. Simplified 37.7%

       \[\leadsto \color{blue}{\left(x \cdot \left(y \cdot y\right)\right) \cdot \left(\left(0.5 \cdot t\right) \cdot t\right)} \]

   if -4.9999999999999995e211 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -5.00000000000000029e135

   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. lower-neg.f64 N/A

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

      3. lower-*.f64 29.7

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

   5. Simplified 29.7%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 2.8

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

   8. Simplified 2.8%

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

   9. Taylor expanded in a around inf

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

       1. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 46.3

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

   11. Simplified 46.3%

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

   if -5.00000000000000029e135 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -4e9

   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. lower-*.f64 N/A

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

      5. lower-neg.f64 25.3

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

   5. Simplified 25.3%

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

   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. lower-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. lower-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. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. cube-mult N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. mul-1-neg N/A

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

      17. lower-neg.f64 3.1

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

   8. Simplified 3.1%

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

   9. Taylor expanded in t around inf

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. associate-*r* N/A

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

       5. lower-*.f64 N/A

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

       6. cube-mult N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 N/A

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

       11. lower-*.f64 N/A

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

       12. cube-mult N/A

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

       13. unpow2 N/A

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

       14. lower-*.f64 N/A

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

       15. unpow2 N/A

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

       16. lower-*.f64 59.6

           \[\leadsto x \cdot \left(\left(y \cdot \left(y \cdot y\right)\right) \cdot \left(-0.16666666666666666 \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)\right) \]

   11. Simplified 59.6%

       \[\leadsto x \cdot \color{blue}{\left(\left(y \cdot \left(y \cdot y\right)\right) \cdot \left(-0.16666666666666666 \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)\right)} \]

   if -4e9 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 4e203

   1. Initial program 89.6%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   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. lower-*.f64 N/A

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

      5. lower-neg.f64 72.2

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

   5. Simplified 72.2%

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

   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. lower-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. lower-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. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. cube-mult N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. mul-1-neg N/A

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

      17. lower-neg.f64 71.4

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

   8. Simplified 71.4%

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

   9. Taylor expanded in y around inf

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

       1. lower-*.f64 N/A

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

       2. cube-mult N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-fma.f64 N/A

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

       8. associate-*r/ N/A

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

       9. metadata-eval N/A

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

       10. lower-/.f64 73.8

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

   11. Simplified 73.8%

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

   if 4e203 < (+.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. lower-*.f64 N/A

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

      5. lower-neg.f64 63.7

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

   5. Simplified 63.7%

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

   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. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. mul-1-neg N/A

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

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

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

      12. mul-1-neg N/A

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

      13. lower-*.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 72.4

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

   8. Simplified 72.4%

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

   9. Taylor expanded in t around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. lower-*.f64 N/A

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

       5. *-commutative N/A

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

       6. associate-*l* N/A

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

       7. lower-*.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 81.4

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

   11. Simplified 81.4%

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

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -5 \cdot 10^{+269}:\\ \;\;\;\;t \cdot \frac{x}{t}\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -5 \cdot 10^{+211}:\\ \;\;\;\;\left(x \cdot \left(y \cdot y\right)\right) \cdot \left(t \cdot \left(t \cdot 0.5\right)\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -5 \cdot 10^{+135}:\\ \;\;\;\;\left(0.5 \cdot \left(a \cdot a\right)\right) \cdot \left(x \cdot \left(b \cdot b\right)\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -4000000000:\\ \;\;\;\;x \cdot \left(\left(y \cdot \left(y \cdot y\right)\right) \cdot \left(-0.16666666666666666 \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 4 \cdot 10^{+203}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(t, \mathsf{fma}\left(t, \left(y \cdot \left(y \cdot y\right)\right) \cdot \mathsf{fma}\left(-0.16666666666666666, t, \frac{0.5}{y}\right), -y\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, t \cdot \left(x \cdot \left(\left(y \cdot y\right) \cdot 0.5\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 47.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+269}:\\ \;\;\;\;t \cdot \frac{x}{t}\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{+211}:\\ \;\;\;\;\left(x \cdot \left(y \cdot y\right)\right) \cdot \left(t \cdot \left(t \cdot 0.5\right)\right)\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{+135}:\\ \;\;\;\;\left(0.5 \cdot \left(a \cdot a\right)\right) \cdot \left(x \cdot \left(b \cdot b\right)\right)\\ \mathbf{elif}\;t\_1 \leq -4000000000:\\ \;\;\;\;x \cdot \left(\left(y \cdot \left(y \cdot y\right)\right) \cdot \left(-0.16666666666666666 \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, t \cdot \left(x \cdot \left(\left(y \cdot y\right) \cdot 0.5\right)\right), x\right)\\ \end{array} \end{array} \]

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 -5e+269)
     (* t (/ x t))
     (if (<= t_1 -5e+211)
       (* (* x (* y y)) (* t (* t 0.5)))
       (if (<= t_1 -5e+135)
         (* (* 0.5 (* a a)) (* x (* b b)))
         (if (<= t_1 -4000000000.0)
           (* x (* (* y (* y y)) (* -0.16666666666666666 (* t (* t t)))))
           (fma t (* t (* x (* (* y y) 0.5))) 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 <= -5e+269) {
		tmp = t * (x / t);
	} else if (t_1 <= -5e+211) {
		tmp = (x * (y * y)) * (t * (t * 0.5));
	} else if (t_1 <= -5e+135) {
		tmp = (0.5 * (a * a)) * (x * (b * b));
	} else if (t_1 <= -4000000000.0) {
		tmp = x * ((y * (y * y)) * (-0.16666666666666666 * (t * (t * t))));
	} else {
		tmp = fma(t, (t * (x * ((y * y) * 0.5))), 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 <= -5e+269)
		tmp = Float64(t * Float64(x / t));
	elseif (t_1 <= -5e+211)
		tmp = Float64(Float64(x * Float64(y * y)) * Float64(t * Float64(t * 0.5)));
	elseif (t_1 <= -5e+135)
		tmp = Float64(Float64(0.5 * Float64(a * a)) * Float64(x * Float64(b * b)));
	elseif (t_1 <= -4000000000.0)
		tmp = Float64(x * Float64(Float64(y * Float64(y * y)) * Float64(-0.16666666666666666 * Float64(t * Float64(t * t)))));
	else
		tmp = fma(t, Float64(t * Float64(x * Float64(Float64(y * y) * 0.5))), 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, -5e+269], N[(t * N[(x / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -5e+211], N[(N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(t * N[(t * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -5e+135], N[(N[(0.5 * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(x * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -4000000000.0], N[(x * N[(N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(-0.16666666666666666 * N[(t * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t * N[(t * N[(x * N[(N[(y * y), $MachinePrecision] * 0.5), $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))) < -5.0000000000000002e269

   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. lower-*.f64 N/A

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

      5. lower-neg.f64 65.7

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

   5. Simplified 65.7%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

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

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

      8. mul-1-neg N/A

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

      9. lower-*.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 2.8

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

   8. Simplified 2.8%

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

   9. Taylor expanded in t around inf

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

       1. lower-*.f64 N/A

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

       2. +-commutative N/A

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

       3. mul-1-neg N/A

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

       4. unsub-neg N/A

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

       5. lower--.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. lower-*.f64 11.6

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

   11. Simplified 11.6%

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

   12. Taylor expanded in t around 0

       \[\leadsto t \cdot \color{blue}{\frac{x}{t}} \]
   13. Step-by-step derivation

       1. lower-/.f64 37.8

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

   14. Simplified 37.8%

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

   if -5.0000000000000002e269 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -4.9999999999999995e211

   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. lower-*.f64 N/A

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

      5. lower-neg.f64 28.5

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

   5. Simplified 28.5%

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

   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. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. mul-1-neg N/A

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

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

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

      12. mul-1-neg N/A

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

      13. lower-*.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 2.6

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

   8. Simplified 2.6%

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

   9. Taylor expanded in t around inf

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. associate-*l* N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 N/A

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

       9. unpow2 N/A

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

       10. associate-*r* N/A

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

       11. lower-*.f64 N/A

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

       12. lower-*.f64 37.7

           \[\leadsto \left(x \cdot \left(y \cdot y\right)\right) \cdot \left(\color{blue}{\left(0.5 \cdot t\right)} \cdot t\right) \]

   11. Simplified 37.7%

       \[\leadsto \color{blue}{\left(x \cdot \left(y \cdot y\right)\right) \cdot \left(\left(0.5 \cdot t\right) \cdot t\right)} \]

   if -4.9999999999999995e211 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -5.00000000000000029e135

   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. lower-neg.f64 N/A

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

      3. lower-*.f64 29.7

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

   5. Simplified 29.7%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 2.8

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

   8. Simplified 2.8%

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

   9. Taylor expanded in a around inf

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

       1. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 46.3

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

   11. Simplified 46.3%

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

   if -5.00000000000000029e135 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -4e9

   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. lower-*.f64 N/A

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

      5. lower-neg.f64 25.3

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

   5. Simplified 25.3%

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

   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. lower-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. lower-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. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. cube-mult N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. mul-1-neg N/A

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

      17. lower-neg.f64 3.1

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

   8. Simplified 3.1%

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

   9. Taylor expanded in t around inf

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. associate-*r* N/A

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

       5. lower-*.f64 N/A

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

       6. cube-mult N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 N/A

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

       11. lower-*.f64 N/A

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

       12. cube-mult N/A

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

       13. unpow2 N/A

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

       14. lower-*.f64 N/A

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

       15. unpow2 N/A

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

       16. lower-*.f64 59.6

           \[\leadsto x \cdot \left(\left(y \cdot \left(y \cdot y\right)\right) \cdot \left(-0.16666666666666666 \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)\right) \]

   11. Simplified 59.6%

       \[\leadsto x \cdot \color{blue}{\left(\left(y \cdot \left(y \cdot y\right)\right) \cdot \left(-0.16666666666666666 \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)\right)} \]

   if -4e9 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 92.9%

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

   3. Taylor expanded in 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. lower-*.f64 N/A

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

      5. lower-neg.f64 68.5

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

   5. Simplified 68.5%

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

   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. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. mul-1-neg N/A

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

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

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

      12. mul-1-neg N/A

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

      13. lower-*.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 70.4

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

   8. Simplified 70.4%

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

   9. Taylor expanded in t around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. lower-*.f64 N/A

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

       5. *-commutative N/A

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

       6. associate-*l* N/A

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

       7. lower-*.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 74.4

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

   11. Simplified 74.4%

       \[\leadsto \mathsf{fma}\left(t, \color{blue}{t \cdot \left(x \cdot \left(\left(y \cdot y\right) \cdot 0.5\right)\right)}, x\right) \]
3. Recombined 5 regimes into one program.

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -5 \cdot 10^{+269}:\\ \;\;\;\;t \cdot \frac{x}{t}\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -5 \cdot 10^{+211}:\\ \;\;\;\;\left(x \cdot \left(y \cdot y\right)\right) \cdot \left(t \cdot \left(t \cdot 0.5\right)\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -5 \cdot 10^{+135}:\\ \;\;\;\;\left(0.5 \cdot \left(a \cdot a\right)\right) \cdot \left(x \cdot \left(b \cdot b\right)\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -4000000000:\\ \;\;\;\;x \cdot \left(\left(y \cdot \left(y \cdot y\right)\right) \cdot \left(-0.16666666666666666 \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, t \cdot \left(x \cdot \left(\left(y \cdot y\right) \cdot 0.5\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 44.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ t_2 := \left(x \cdot \left(y \cdot y\right)\right) \cdot \left(t \cdot \left(t \cdot 0.5\right)\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+269}:\\ \;\;\;\;t \cdot \frac{x}{t}\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{+211}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -4 \cdot 10^{+16}:\\ \;\;\;\;\left(0.5 \cdot \left(a \cdot a\right)\right) \cdot \left(x \cdot \left(b \cdot b\right)\right)\\ \mathbf{elif}\;t\_1 \leq 5000000000:\\ \;\;\;\;x \cdot \mathsf{fma}\left(a, \left(-z\right) - b, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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 (* (* x (* y y)) (* t (* t 0.5)))))
   (if (<= t_1 -5e+269)
     (* t (/ x t))
     (if (<= t_1 -5e+211)
       t_2
       (if (<= t_1 -4e+16)
         (* (* 0.5 (* a a)) (* x (* b b)))
         (if (<= t_1 5000000000.0) (* x (fma a (- (- z) b) 1.0)) 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 = (x * (y * y)) * (t * (t * 0.5));
	double tmp;
	if (t_1 <= -5e+269) {
		tmp = t * (x / t);
	} else if (t_1 <= -5e+211) {
		tmp = t_2;
	} else if (t_1 <= -4e+16) {
		tmp = (0.5 * (a * a)) * (x * (b * b));
	} else if (t_1 <= 5000000000.0) {
		tmp = x * fma(a, (-z - b), 1.0);
	} 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(Float64(x * Float64(y * y)) * Float64(t * Float64(t * 0.5)))
	tmp = 0.0
	if (t_1 <= -5e+269)
		tmp = Float64(t * Float64(x / t));
	elseif (t_1 <= -5e+211)
		tmp = t_2;
	elseif (t_1 <= -4e+16)
		tmp = Float64(Float64(0.5 * Float64(a * a)) * Float64(x * Float64(b * b)));
	elseif (t_1 <= 5000000000.0)
		tmp = Float64(x * fma(a, Float64(Float64(-z) - b), 1.0));
	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[(N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(t * N[(t * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+269], N[(t * N[(x / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -5e+211], t$95$2, If[LessEqual[t$95$1, -4e+16], N[(N[(0.5 * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(x * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5000000000.0], N[(x * N[(a * N[((-z) - b), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]

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))) < -5.0000000000000002e269

   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. lower-*.f64 N/A

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

      5. lower-neg.f64 65.7

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

   5. Simplified 65.7%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

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

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

      8. mul-1-neg N/A

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

      9. lower-*.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 2.8

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

   8. Simplified 2.8%

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

   9. Taylor expanded in t around inf

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

       1. lower-*.f64 N/A

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

       2. +-commutative N/A

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

       3. mul-1-neg N/A

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

       4. unsub-neg N/A

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

       5. lower--.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. lower-*.f64 11.6

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

   11. Simplified 11.6%

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

   12. Taylor expanded in t around 0

       \[\leadsto t \cdot \color{blue}{\frac{x}{t}} \]
   13. Step-by-step derivation

       1. lower-/.f64 37.8

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

   14. Simplified 37.8%

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

   if -5.0000000000000002e269 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -4.9999999999999995e211 or 5e9 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 96.6%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   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. lower-*.f64 N/A

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

      5. lower-neg.f64 54.8

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

   5. Simplified 54.8%

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

   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. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. mul-1-neg N/A

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

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

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

      12. mul-1-neg N/A

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

      13. lower-*.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 54.0

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

   8. Simplified 54.0%

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

   9. Taylor expanded in t around inf

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. associate-*l* N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 N/A

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

       9. unpow2 N/A

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

       10. associate-*r* N/A

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

       11. lower-*.f64 N/A

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

       12. lower-*.f64 57.4

           \[\leadsto \left(x \cdot \left(y \cdot y\right)\right) \cdot \left(\color{blue}{\left(0.5 \cdot t\right)} \cdot t\right) \]

   11. Simplified 57.4%

       \[\leadsto \color{blue}{\left(x \cdot \left(y \cdot y\right)\right) \cdot \left(\left(0.5 \cdot t\right) \cdot t\right)} \]

   if -4.9999999999999995e211 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -4e16

   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. lower-neg.f64 N/A

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

      3. lower-*.f64 35.5

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

   5. Simplified 35.5%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 2.7

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

   8. Simplified 2.7%

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

   9. Taylor expanded in a around inf

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

       1. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 43.8

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

   11. Simplified 43.8%

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

   if -4e16 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 5e9

   1. Initial program 87.9%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   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. lower-*.f64 N/A

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

      2. lower--.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. lower-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. lower-neg.f64 94.1

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

   5. Simplified 94.1%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. distribute-lft-out N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-+.f64 94.1

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

   8. Simplified 94.1%

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

   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. mul-1-neg N/A

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

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

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

       4. mul-1-neg N/A

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

       5. lower-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. unsub-neg N/A

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

       10. lower--.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. lower-neg.f64 81.2

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

   11. Simplified 81.2%

       \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, \left(-z\right) - b, 1\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -5 \cdot 10^{+269}:\\ \;\;\;\;t \cdot \frac{x}{t}\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -5 \cdot 10^{+211}:\\ \;\;\;\;\left(x \cdot \left(y \cdot y\right)\right) \cdot \left(t \cdot \left(t \cdot 0.5\right)\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -4 \cdot 10^{+16}:\\ \;\;\;\;\left(0.5 \cdot \left(a \cdot a\right)\right) \cdot \left(x \cdot \left(b \cdot b\right)\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 5000000000:\\ \;\;\;\;x \cdot \mathsf{fma}\left(a, \left(-z\right) - b, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot y\right)\right) \cdot \left(t \cdot \left(t \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 41.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ t_2 := \left(0.5 \cdot \left(a \cdot a\right)\right) \cdot \left(x \cdot \left(b \cdot b\right)\right)\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+215}:\\ \;\;\;\;t \cdot \frac{x}{t}\\ \mathbf{elif}\;t\_1 \leq -4 \cdot 10^{+16}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-10}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(a, \left(-z\right) - b, 1\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+242}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(x, -t, \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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 (* (* 0.5 (* a a)) (* x (* b b)))))
   (if (<= t_1 -4e+215)
     (* t (/ x t))
     (if (<= t_1 -4e+16)
       t_2
       (if (<= t_1 4e-10)
         (* x (fma a (- (- z) b) 1.0))
         (if (<= t_1 5e+242) (* y (fma x (- t) (/ x y))) 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 = (0.5 * (a * a)) * (x * (b * b));
	double tmp;
	if (t_1 <= -4e+215) {
		tmp = t * (x / t);
	} else if (t_1 <= -4e+16) {
		tmp = t_2;
	} else if (t_1 <= 4e-10) {
		tmp = x * fma(a, (-z - b), 1.0);
	} else if (t_1 <= 5e+242) {
		tmp = y * fma(x, -t, (x / y));
	} 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(Float64(0.5 * Float64(a * a)) * Float64(x * Float64(b * b)))
	tmp = 0.0
	if (t_1 <= -4e+215)
		tmp = Float64(t * Float64(x / t));
	elseif (t_1 <= -4e+16)
		tmp = t_2;
	elseif (t_1 <= 4e-10)
		tmp = Float64(x * fma(a, Float64(Float64(-z) - b), 1.0));
	elseif (t_1 <= 5e+242)
		tmp = Float64(y * fma(x, Float64(-t), Float64(x / y)));
	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[(N[(0.5 * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(x * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+215], N[(t * N[(x / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -4e+16], t$95$2, If[LessEqual[t$95$1, 4e-10], N[(x * N[(a * N[((-z) - b), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+242], N[(y * N[(x * (-t) + N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]

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))) < -3.99999999999999963e215

   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. lower-*.f64 N/A

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

      5. lower-neg.f64 55.1

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

   5. Simplified 55.1%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

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

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

      8. mul-1-neg N/A

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

      9. lower-*.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 3.0

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

   8. Simplified 3.0%

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

   9. Taylor expanded in t around inf

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

       1. lower-*.f64 N/A

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

       2. +-commutative N/A

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

       3. mul-1-neg N/A

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

       4. unsub-neg N/A

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

       5. lower--.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. lower-*.f64 12.5

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

   11. Simplified 12.5%

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

   12. Taylor expanded in t around 0

       \[\leadsto t \cdot \color{blue}{\frac{x}{t}} \]
   13. Step-by-step derivation

       1. lower-/.f64 31.6

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

   14. Simplified 31.6%

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

   if -3.99999999999999963e215 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -4e16 or 5.0000000000000004e242 < (+.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 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. lower-neg.f64 N/A

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

      3. lower-*.f64 48.1

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

   5. Simplified 48.1%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 38.2

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

   8. Simplified 38.2%

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

   9. Taylor expanded in a around inf

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

       1. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 53.4

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

   11. Simplified 53.4%

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

   if -4e16 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 4.00000000000000015e-10

   1. Initial program 87.7%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   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. lower-*.f64 N/A

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

      2. lower--.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. lower-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. lower-neg.f64 93.9

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

   5. Simplified 93.9%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. distribute-lft-out N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-+.f64 93.9

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

   8. Simplified 93.9%

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

   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. mul-1-neg N/A

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

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

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

       4. mul-1-neg N/A

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

       5. lower-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. unsub-neg N/A

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

       10. lower--.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. lower-neg.f64 82.6

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

   11. Simplified 82.6%

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

   if 4.00000000000000015e-10 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 5.0000000000000004e242

   1. Initial program 95.5%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   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. lower-*.f64 N/A

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

      5. lower-neg.f64 53.9

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

   5. Simplified 53.9%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

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

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

      8. mul-1-neg N/A

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

      9. lower-*.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 25.8

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

   8. Simplified 25.8%

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

   9. Taylor expanded in y around inf

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

       1. lower-*.f64 N/A

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

       2. mul-1-neg N/A

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

       3. *-commutative N/A

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

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

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

       5. mul-1-neg N/A

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

       6. lower-fma.f64 N/A

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

       7. mul-1-neg N/A

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

       8. lower-neg.f64 N/A

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

       9. lower-/.f64 34.6

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

   11. Simplified 34.6%

       \[\leadsto \color{blue}{y \cdot \mathsf{fma}\left(x, -t, \frac{x}{y}\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 7: 46.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+269}:\\ \;\;\;\;t \cdot \frac{x}{t}\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{+211}:\\ \;\;\;\;\left(x \cdot \left(y \cdot y\right)\right) \cdot \left(t \cdot \left(t \cdot 0.5\right)\right)\\ \mathbf{elif}\;t\_1 \leq -4 \cdot 10^{+16}:\\ \;\;\;\;\left(0.5 \cdot \left(a \cdot a\right)\right) \cdot \left(x \cdot \left(b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, t \cdot \left(x \cdot \left(\left(y \cdot y\right) \cdot 0.5\right)\right), x\right)\\ \end{array} \end{array} \]

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 -5e+269)
     (* t (/ x t))
     (if (<= t_1 -5e+211)
       (* (* x (* y y)) (* t (* t 0.5)))
       (if (<= t_1 -4e+16)
         (* (* 0.5 (* a a)) (* x (* b b)))
         (fma t (* t (* x (* (* y y) 0.5))) 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 <= -5e+269) {
		tmp = t * (x / t);
	} else if (t_1 <= -5e+211) {
		tmp = (x * (y * y)) * (t * (t * 0.5));
	} else if (t_1 <= -4e+16) {
		tmp = (0.5 * (a * a)) * (x * (b * b));
	} else {
		tmp = fma(t, (t * (x * ((y * y) * 0.5))), 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 <= -5e+269)
		tmp = Float64(t * Float64(x / t));
	elseif (t_1 <= -5e+211)
		tmp = Float64(Float64(x * Float64(y * y)) * Float64(t * Float64(t * 0.5)));
	elseif (t_1 <= -4e+16)
		tmp = Float64(Float64(0.5 * Float64(a * a)) * Float64(x * Float64(b * b)));
	else
		tmp = fma(t, Float64(t * Float64(x * Float64(Float64(y * y) * 0.5))), 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, -5e+269], N[(t * N[(x / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -5e+211], N[(N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(t * N[(t * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -4e+16], N[(N[(0.5 * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(x * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t * N[(t * N[(x * N[(N[(y * y), $MachinePrecision] * 0.5), $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))) < -5.0000000000000002e269

   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. lower-*.f64 N/A

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

      5. lower-neg.f64 65.7

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

   5. Simplified 65.7%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

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

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

      8. mul-1-neg N/A

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

      9. lower-*.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 2.8

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

   8. Simplified 2.8%

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

   9. Taylor expanded in t around inf

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

       1. lower-*.f64 N/A

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

       2. +-commutative N/A

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

       3. mul-1-neg N/A

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

       4. unsub-neg N/A

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

       5. lower--.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. lower-*.f64 11.6

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

   11. Simplified 11.6%

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

   12. Taylor expanded in t around 0

       \[\leadsto t \cdot \color{blue}{\frac{x}{t}} \]
   13. Step-by-step derivation

       1. lower-/.f64 37.8

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

   14. Simplified 37.8%

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

   if -5.0000000000000002e269 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -4.9999999999999995e211

   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. lower-*.f64 N/A

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

      5. lower-neg.f64 28.5

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

   5. Simplified 28.5%

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

   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. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. mul-1-neg N/A

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

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

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

      12. mul-1-neg N/A

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

      13. lower-*.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 2.6

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

   8. Simplified 2.6%

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

   9. Taylor expanded in t around inf

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. associate-*l* N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 N/A

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

       9. unpow2 N/A

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

       10. associate-*r* N/A

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

       11. lower-*.f64 N/A

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

       12. lower-*.f64 37.7

           \[\leadsto \left(x \cdot \left(y \cdot y\right)\right) \cdot \left(\color{blue}{\left(0.5 \cdot t\right)} \cdot t\right) \]

   11. Simplified 37.7%

       \[\leadsto \color{blue}{\left(x \cdot \left(y \cdot y\right)\right) \cdot \left(\left(0.5 \cdot t\right) \cdot t\right)} \]

   if -4.9999999999999995e211 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -4e16

   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. lower-neg.f64 N/A

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

      3. lower-*.f64 35.5

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

   5. Simplified 35.5%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 2.7

          \[\leadsto \mathsf{fma}\left(a, 0.5 \cdot \left(a \cdot \left(x \cdot \left(b \cdot b\right)\right)\right) - \color{blue}{x \cdot b}, x\right) \]

   8. Simplified 2.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, 0.5 \cdot \left(a \cdot \left(x \cdot \left(b \cdot b\right)\right)\right) - x \cdot b, x\right)} \]

   9. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left({a}^{2} \cdot \left({b}^{2} \cdot x\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot {a}^{2}\right) \cdot \left({b}^{2} \cdot x\right)} \]

       2. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot {a}^{2}\right) \cdot \left({b}^{2} \cdot x\right)} \]

       3. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot {a}^{2}\right)} \cdot \left({b}^{2} \cdot x\right) \]

       4. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\left(a \cdot a\right)}\right) \cdot \left({b}^{2} \cdot x\right) \]

       5. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\left(a \cdot a\right)}\right) \cdot \left({b}^{2} \cdot x\right) \]

       6. *-commutative N/A

          \[\leadsto \left(\frac{1}{2} \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(x \cdot {b}^{2}\right)} \]

       7. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(x \cdot {b}^{2}\right)} \]

       8. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \left(a \cdot a\right)\right) \cdot \left(x \cdot \color{blue}{\left(b \cdot b\right)}\right) \]

       9. lower-*.f64 43.8

          \[\leadsto \left(0.5 \cdot \left(a \cdot a\right)\right) \cdot \left(x \cdot \color{blue}{\left(b \cdot b\right)}\right) \]

   11. Simplified 43.8%

       \[\leadsto \color{blue}{\left(0.5 \cdot \left(a \cdot a\right)\right) \cdot \left(x \cdot \left(b \cdot b\right)\right)} \]

   if -4e16 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 93.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   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. lower-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      5. lower-neg.f64 67.6

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(-t\right)}} \]

   5. Simplified 67.6%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}} \]

   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. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, -1 \cdot \left(x \cdot y\right) + \frac{1}{2} \cdot \left(t \cdot \left(x \cdot {y}^{2}\right)\right), x\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{\frac{1}{2} \cdot \left(t \cdot \left(x \cdot {y}^{2}\right)\right) + -1 \cdot \left(x \cdot y\right)}, x\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{\left(\frac{1}{2} \cdot t\right) \cdot \left(x \cdot {y}^{2}\right)} + -1 \cdot \left(x \cdot y\right), x\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot t, x \cdot {y}^{2}, -1 \cdot \left(x \cdot y\right)\right)}, x\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(\color{blue}{\frac{1}{2} \cdot t}, x \cdot {y}^{2}, -1 \cdot \left(x \cdot y\right)\right), x\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(\frac{1}{2} \cdot t, \color{blue}{x \cdot {y}^{2}}, -1 \cdot \left(x \cdot y\right)\right), x\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(\frac{1}{2} \cdot t, x \cdot \color{blue}{\left(y \cdot y\right)}, -1 \cdot \left(x \cdot y\right)\right), x\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(\frac{1}{2} \cdot t, x \cdot \color{blue}{\left(y \cdot y\right)}, -1 \cdot \left(x \cdot y\right)\right), x\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(\frac{1}{2} \cdot t, x \cdot \left(y \cdot y\right), \color{blue}{\mathsf{neg}\left(x \cdot y\right)}\right), x\right) \]

      11. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(\frac{1}{2} \cdot t, x \cdot \left(y \cdot y\right), \color{blue}{x \cdot \left(\mathsf{neg}\left(y\right)\right)}\right), x\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(\frac{1}{2} \cdot t, x \cdot \left(y \cdot y\right), x \cdot \color{blue}{\left(-1 \cdot y\right)}\right), x\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(\frac{1}{2} \cdot t, x \cdot \left(y \cdot y\right), \color{blue}{x \cdot \left(-1 \cdot y\right)}\right), x\right) \]

      14. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(\frac{1}{2} \cdot t, x \cdot \left(y \cdot y\right), x \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right), x\right) \]

      15. lower-neg.f64 69.5

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(0.5 \cdot t, x \cdot \left(y \cdot y\right), x \cdot \color{blue}{\left(-y\right)}\right), x\right) \]

   8. Simplified 69.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, \mathsf{fma}\left(0.5 \cdot t, x \cdot \left(y \cdot y\right), x \cdot \left(-y\right)\right), x\right)} \]

   9. Taylor expanded in t around inf

      \[\leadsto \mathsf{fma}\left(t, \color{blue}{\frac{1}{2} \cdot \left(t \cdot \left(x \cdot {y}^{2}\right)\right)}, x\right) \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(t, \color{blue}{\left(\frac{1}{2} \cdot t\right) \cdot \left(x \cdot {y}^{2}\right)}, x\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(t, \color{blue}{\left(t \cdot \frac{1}{2}\right)} \cdot \left(x \cdot {y}^{2}\right), x\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(t, \color{blue}{t \cdot \left(\frac{1}{2} \cdot \left(x \cdot {y}^{2}\right)\right)}, x\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \color{blue}{t \cdot \left(\frac{1}{2} \cdot \left(x \cdot {y}^{2}\right)\right)}, x\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(t, t \cdot \color{blue}{\left(\left(x \cdot {y}^{2}\right) \cdot \frac{1}{2}\right)}, x\right) \]

       6. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(t, t \cdot \color{blue}{\left(x \cdot \left({y}^{2} \cdot \frac{1}{2}\right)\right)}, x\right) \]

       7. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, t \cdot \color{blue}{\left(x \cdot \left({y}^{2} \cdot \frac{1}{2}\right)\right)}, x\right) \]

       8. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, t \cdot \left(x \cdot \color{blue}{\left({y}^{2} \cdot \frac{1}{2}\right)}\right), x\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(t, t \cdot \left(x \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot \frac{1}{2}\right)\right), x\right) \]

       10. lower-*.f64 73.4

           \[\leadsto \mathsf{fma}\left(t, t \cdot \left(x \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot 0.5\right)\right), x\right) \]

   11. Simplified 73.4%

       \[\leadsto \mathsf{fma}\left(t, \color{blue}{t \cdot \left(x \cdot \left(\left(y \cdot y\right) \cdot 0.5\right)\right)}, x\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -5 \cdot 10^{+269}:\\ \;\;\;\;t \cdot \frac{x}{t}\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -5 \cdot 10^{+211}:\\ \;\;\;\;\left(x \cdot \left(y \cdot y\right)\right) \cdot \left(t \cdot \left(t \cdot 0.5\right)\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -4 \cdot 10^{+16}:\\ \;\;\;\;\left(0.5 \cdot \left(a \cdot a\right)\right) \cdot \left(x \cdot \left(b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, t \cdot \left(x \cdot \left(\left(y \cdot y\right) \cdot 0.5\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 33.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+269}:\\ \;\;\;\;t \cdot \frac{x}{t}\\ \mathbf{elif}\;t\_1 \leq -4000000000:\\ \;\;\;\;t \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - y \cdot t\right)\\ \end{array} \end{array} \]

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 -5e+269)
     (* t (/ x t))
     (if (<= t_1 -4000000000.0) (* t (* x (- y))) (* 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 <= -5e+269) {
		tmp = t * (x / t);
	} else if (t_1 <= -4000000000.0) {
		tmp = t * (x * -y);
	} 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) :: t_1
    real(8) :: tmp
    t_1 = (y * (log(z) - t)) + (a * (log((1.0d0 - z)) - b))
    if (t_1 <= (-5d+269)) then
        tmp = t * (x / t)
    else if (t_1 <= (-4000000000.0d0)) then
        tmp = t * (x * -y)
    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 t_1 = (y * (Math.log(z) - t)) + (a * (Math.log((1.0 - z)) - b));
	double tmp;
	if (t_1 <= -5e+269) {
		tmp = t * (x / t);
	} else if (t_1 <= -4000000000.0) {
		tmp = t * (x * -y);
	} else {
		tmp = x * (1.0 - (y * t));
	}
	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 <= -5e+269:
		tmp = t * (x / t)
	elif t_1 <= -4000000000.0:
		tmp = t * (x * -y)
	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 <= -5e+269)
		tmp = Float64(t * Float64(x / t));
	elseif (t_1 <= -4000000000.0)
		tmp = Float64(t * Float64(x * Float64(-y)));
	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)
	t_1 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	tmp = 0.0;
	if (t_1 <= -5e+269)
		tmp = t * (x / t);
	elseif (t_1 <= -4000000000.0)
		tmp = t * (x * -y);
	else
		tmp = x * (1.0 - (y * t));
	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, -5e+269], N[(t * N[(x / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -4000000000.0], N[(t * N[(x * (-y)), $MachinePrecision]), $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))) < -5.0000000000000002e269

   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. lower-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      5. lower-neg.f64 65.7

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(-t\right)}} \]

   5. Simplified 65.7%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + -1 \cdot \left(t \cdot \left(x \cdot y\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{-1 \cdot \left(t \cdot \left(x \cdot y\right)\right) + x} \]

      2. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(t \cdot \left(x \cdot y\right)\right)\right)} + x \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{t \cdot \left(\mathsf{neg}\left(x \cdot y\right)\right)} + x \]

      4. mul-1-neg N/A

         \[\leadsto t \cdot \color{blue}{\left(-1 \cdot \left(x \cdot y\right)\right)} + x \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, -1 \cdot \left(x \cdot y\right), x\right)} \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(x \cdot y\right)}, x\right) \]

      7. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{x \cdot \left(\mathsf{neg}\left(y\right)\right)}, x\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(t, x \cdot \color{blue}{\left(-1 \cdot y\right)}, x\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{x \cdot \left(-1 \cdot y\right)}, x\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(t, x \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}, x\right) \]

      11. lower-neg.f64 2.8

          \[\leadsto \mathsf{fma}\left(t, x \cdot \color{blue}{\left(-y\right)}, x\right) \]

   8. Simplified 2.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, x \cdot \left(-y\right), x\right)} \]

   9. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(x \cdot y\right) + \frac{x}{t}\right)} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(x \cdot y\right) + \frac{x}{t}\right)} \]

       2. +-commutative N/A

          \[\leadsto t \cdot \color{blue}{\left(\frac{x}{t} + -1 \cdot \left(x \cdot y\right)\right)} \]

       3. mul-1-neg N/A

          \[\leadsto t \cdot \left(\frac{x}{t} + \color{blue}{\left(\mathsf{neg}\left(x \cdot y\right)\right)}\right) \]

       4. unsub-neg N/A

          \[\leadsto t \cdot \color{blue}{\left(\frac{x}{t} - x \cdot y\right)} \]

       5. lower--.f64 N/A

          \[\leadsto t \cdot \color{blue}{\left(\frac{x}{t} - x \cdot y\right)} \]

       6. lower-/.f64 N/A

          \[\leadsto t \cdot \left(\color{blue}{\frac{x}{t}} - x \cdot y\right) \]

       7. lower-*.f64 11.6

          \[\leadsto t \cdot \left(\frac{x}{t} - \color{blue}{x \cdot y}\right) \]

   11. Simplified 11.6%

       \[\leadsto \color{blue}{t \cdot \left(\frac{x}{t} - x \cdot y\right)} \]

   12. Taylor expanded in t around 0

       \[\leadsto t \cdot \color{blue}{\frac{x}{t}} \]
   13. Step-by-step derivation

       1. lower-/.f64 37.8

          \[\leadsto t \cdot \color{blue}{\frac{x}{t}} \]

   14. Simplified 37.8%

       \[\leadsto t \cdot \color{blue}{\frac{x}{t}} \]

   if -5.0000000000000002e269 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -4e9

   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. lower-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      5. lower-neg.f64 41.7

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(-t\right)}} \]

   5. Simplified 41.7%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + -1 \cdot \left(t \cdot \left(x \cdot y\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{-1 \cdot \left(t \cdot \left(x \cdot y\right)\right) + x} \]

      2. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(t \cdot \left(x \cdot y\right)\right)\right)} + x \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{t \cdot \left(\mathsf{neg}\left(x \cdot y\right)\right)} + x \]

      4. mul-1-neg N/A

         \[\leadsto t \cdot \color{blue}{\left(-1 \cdot \left(x \cdot y\right)\right)} + x \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, -1 \cdot \left(x \cdot y\right), x\right)} \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(x \cdot y\right)}, x\right) \]

      7. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{x \cdot \left(\mathsf{neg}\left(y\right)\right)}, x\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(t, x \cdot \color{blue}{\left(-1 \cdot y\right)}, x\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{x \cdot \left(-1 \cdot y\right)}, x\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(t, x \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}, x\right) \]

      11. lower-neg.f64 3.2

          \[\leadsto \mathsf{fma}\left(t, x \cdot \color{blue}{\left(-y\right)}, x\right) \]

   8. Simplified 3.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, x \cdot \left(-y\right), x\right)} \]

   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. distribute-rgt-neg-in N/A

          \[\leadsto \color{blue}{t \cdot \left(\mathsf{neg}\left(x \cdot y\right)\right)} \]

       3. mul-1-neg N/A

          \[\leadsto t \cdot \color{blue}{\left(-1 \cdot \left(x \cdot y\right)\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(x \cdot y\right)\right)} \]

       5. mul-1-neg N/A

          \[\leadsto t \cdot \color{blue}{\left(\mathsf{neg}\left(x \cdot y\right)\right)} \]

       6. *-commutative N/A

          \[\leadsto t \cdot \left(\mathsf{neg}\left(\color{blue}{y \cdot x}\right)\right) \]

       7. distribute-rgt-neg-in N/A

          \[\leadsto t \cdot \color{blue}{\left(y \cdot \left(\mathsf{neg}\left(x\right)\right)\right)} \]

       8. neg-mul-1 N/A

          \[\leadsto t \cdot \left(y \cdot \color{blue}{\left(-1 \cdot x\right)}\right) \]

       9. lower-*.f64 N/A

          \[\leadsto t \cdot \color{blue}{\left(y \cdot \left(-1 \cdot x\right)\right)} \]

       10. neg-mul-1 N/A

           \[\leadsto t \cdot \left(y \cdot \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right) \]

       11. lower-neg.f64 22.4

           \[\leadsto t \cdot \left(y \cdot \color{blue}{\left(-x\right)}\right) \]

   11. Simplified 22.4%

       \[\leadsto \color{blue}{t \cdot \left(y \cdot \left(-x\right)\right)} \]

   if -4e9 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 92.9%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in 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. lower-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      5. lower-neg.f64 68.5

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(-t\right)}} \]

   5. Simplified 68.5%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}} \]

   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. lower--.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(1 - t \cdot y\right)} \]

      4. *-commutative N/A

         \[\leadsto x \cdot \left(1 - \color{blue}{y \cdot t}\right) \]

      5. lower-*.f64 51.6

         \[\leadsto x \cdot \left(1 - \color{blue}{y \cdot t}\right) \]

   8. Simplified 51.6%

      \[\leadsto x \cdot \color{blue}{\left(1 - y \cdot t\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 42.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -5 \cdot 10^{+269}:\\ \;\;\;\;t \cdot \frac{x}{t}\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -4000000000:\\ \;\;\;\;t \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - y \cdot t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 31.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot \left(x \cdot \left(-y\right)\right)\\ t_2 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t\_2 \leq -4000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5000000000:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* t (* x (- y))))
        (t_2 (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b)))))
   (if (<= t_2 -4000000000.0)
     t_1
     (if (<= t_2 5000000000.0) (* x (- 1.0 (* a b))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = t * (x * -y);
	double t_2 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	double tmp;
	if (t_2 <= -4000000000.0) {
		tmp = t_1;
	} else if (t_2 <= 5000000000.0) {
		tmp = x * (1.0 - (a * 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) :: t_2
    real(8) :: tmp
    t_1 = t * (x * -y)
    t_2 = (y * (log(z) - t)) + (a * (log((1.0d0 - z)) - b))
    if (t_2 <= (-4000000000.0d0)) then
        tmp = t_1
    else if (t_2 <= 5000000000.0d0) then
        tmp = x * (1.0d0 - (a * 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 = t * (x * -y);
	double t_2 = (y * (Math.log(z) - t)) + (a * (Math.log((1.0 - z)) - b));
	double tmp;
	if (t_2 <= -4000000000.0) {
		tmp = t_1;
	} else if (t_2 <= 5000000000.0) {
		tmp = x * (1.0 - (a * b));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = t * (x * -y)
	t_2 = (y * (math.log(z) - t)) + (a * (math.log((1.0 - z)) - b))
	tmp = 0
	if t_2 <= -4000000000.0:
		tmp = t_1
	elif t_2 <= 5000000000.0:
		tmp = x * (1.0 - (a * b))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(t * Float64(x * Float64(-y)))
	t_2 = Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))
	tmp = 0.0
	if (t_2 <= -4000000000.0)
		tmp = t_1;
	elseif (t_2 <= 5000000000.0)
		tmp = Float64(x * Float64(1.0 - Float64(a * b)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = t * (x * -y);
	t_2 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	tmp = 0.0;
	if (t_2 <= -4000000000.0)
		tmp = t_1;
	elseif (t_2 <= 5000000000.0)
		tmp = x * (1.0 - (a * b));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(t * N[(x * (-y)), $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, -4000000000.0], t$95$1, If[LessEqual[t$95$2, 5000000000.0], N[(x * N[(1.0 - N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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))) < -4e9 or 5e9 < (+.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. lower-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      5. lower-neg.f64 55.4

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(-t\right)}} \]

   5. Simplified 55.4%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + -1 \cdot \left(t \cdot \left(x \cdot y\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{-1 \cdot \left(t \cdot \left(x \cdot y\right)\right) + x} \]

      2. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(t \cdot \left(x \cdot y\right)\right)\right)} + x \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{t \cdot \left(\mathsf{neg}\left(x \cdot y\right)\right)} + x \]

      4. mul-1-neg N/A

         \[\leadsto t \cdot \color{blue}{\left(-1 \cdot \left(x \cdot y\right)\right)} + x \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, -1 \cdot \left(x \cdot y\right), x\right)} \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(x \cdot y\right)}, x\right) \]

      7. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{x \cdot \left(\mathsf{neg}\left(y\right)\right)}, x\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(t, x \cdot \color{blue}{\left(-1 \cdot y\right)}, x\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{x \cdot \left(-1 \cdot y\right)}, x\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(t, x \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}, x\right) \]

      11. lower-neg.f64 17.6

          \[\leadsto \mathsf{fma}\left(t, x \cdot \color{blue}{\left(-y\right)}, x\right) \]

   8. Simplified 17.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, x \cdot \left(-y\right), x\right)} \]

   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. distribute-rgt-neg-in N/A

          \[\leadsto \color{blue}{t \cdot \left(\mathsf{neg}\left(x \cdot y\right)\right)} \]

       3. mul-1-neg N/A

          \[\leadsto t \cdot \color{blue}{\left(-1 \cdot \left(x \cdot y\right)\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(x \cdot y\right)\right)} \]

       5. mul-1-neg N/A

          \[\leadsto t \cdot \color{blue}{\left(\mathsf{neg}\left(x \cdot y\right)\right)} \]

       6. *-commutative N/A

          \[\leadsto t \cdot \left(\mathsf{neg}\left(\color{blue}{y \cdot x}\right)\right) \]

       7. distribute-rgt-neg-in N/A

          \[\leadsto t \cdot \color{blue}{\left(y \cdot \left(\mathsf{neg}\left(x\right)\right)\right)} \]

       8. neg-mul-1 N/A

          \[\leadsto t \cdot \left(y \cdot \color{blue}{\left(-1 \cdot x\right)}\right) \]

       9. lower-*.f64 N/A

          \[\leadsto t \cdot \color{blue}{\left(y \cdot \left(-1 \cdot x\right)\right)} \]

       10. neg-mul-1 N/A

           \[\leadsto t \cdot \left(y \cdot \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right) \]

       11. lower-neg.f64 26.8

           \[\leadsto t \cdot \left(y \cdot \color{blue}{\left(-x\right)}\right) \]

   11. Simplified 26.8%

       \[\leadsto \color{blue}{t \cdot \left(y \cdot \left(-x\right)\right)} \]

   if -4e9 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 5e9

   1. Initial program 87.5%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   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. lower-neg.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      3. lower-*.f64 85.1

         \[\leadsto x \cdot e^{-\color{blue}{a \cdot b}} \]

   5. Simplified 85.1%

      \[\leadsto x \cdot e^{\color{blue}{-a \cdot b}} \]

   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. lower--.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(1 - a \cdot b\right)} \]

      4. lower-*.f64 83.4

         \[\leadsto x \cdot \left(1 - \color{blue}{a \cdot b}\right) \]

   8. Simplified 83.4%

      \[\leadsto x \cdot \color{blue}{\left(1 - a \cdot b\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -4000000000:\\ \;\;\;\;t \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 5000000000:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(x \cdot \left(-y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 31.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot \left(x \cdot \left(-y\right)\right)\\ t_2 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t\_2 \leq -4000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5000000000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* t (* x (- y))))
        (t_2 (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b)))))
   (if (<= t_2 -4000000000.0) t_1 (if (<= t_2 5000000000.0) x t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = t * (x * -y);
	double t_2 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	double tmp;
	if (t_2 <= -4000000000.0) {
		tmp = t_1;
	} else if (t_2 <= 5000000000.0) {
		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 = t * (x * -y)
    t_2 = (y * (log(z) - t)) + (a * (log((1.0d0 - z)) - b))
    if (t_2 <= (-4000000000.0d0)) then
        tmp = t_1
    else if (t_2 <= 5000000000.0d0) 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 = t * (x * -y);
	double t_2 = (y * (Math.log(z) - t)) + (a * (Math.log((1.0 - z)) - b));
	double tmp;
	if (t_2 <= -4000000000.0) {
		tmp = t_1;
	} else if (t_2 <= 5000000000.0) {
		tmp = x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = t * (x * -y)
	t_2 = (y * (math.log(z) - t)) + (a * (math.log((1.0 - z)) - b))
	tmp = 0
	if t_2 <= -4000000000.0:
		tmp = t_1
	elif t_2 <= 5000000000.0:
		tmp = x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(t * Float64(x * Float64(-y)))
	t_2 = Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))
	tmp = 0.0
	if (t_2 <= -4000000000.0)
		tmp = t_1;
	elseif (t_2 <= 5000000000.0)
		tmp = x;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = t * (x * -y);
	t_2 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	tmp = 0.0;
	if (t_2 <= -4000000000.0)
		tmp = t_1;
	elseif (t_2 <= 5000000000.0)
		tmp = x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(t * N[(x * (-y)), $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, -4000000000.0], t$95$1, If[LessEqual[t$95$2, 5000000000.0], 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))) < -4e9 or 5e9 < (+.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. lower-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      5. lower-neg.f64 55.4

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(-t\right)}} \]

   5. Simplified 55.4%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + -1 \cdot \left(t \cdot \left(x \cdot y\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{-1 \cdot \left(t \cdot \left(x \cdot y\right)\right) + x} \]

      2. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(t \cdot \left(x \cdot y\right)\right)\right)} + x \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{t \cdot \left(\mathsf{neg}\left(x \cdot y\right)\right)} + x \]

      4. mul-1-neg N/A

         \[\leadsto t \cdot \color{blue}{\left(-1 \cdot \left(x \cdot y\right)\right)} + x \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, -1 \cdot \left(x \cdot y\right), x\right)} \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(x \cdot y\right)}, x\right) \]

      7. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{x \cdot \left(\mathsf{neg}\left(y\right)\right)}, x\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(t, x \cdot \color{blue}{\left(-1 \cdot y\right)}, x\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{x \cdot \left(-1 \cdot y\right)}, x\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(t, x \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}, x\right) \]

      11. lower-neg.f64 17.6

          \[\leadsto \mathsf{fma}\left(t, x \cdot \color{blue}{\left(-y\right)}, x\right) \]

   8. Simplified 17.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, x \cdot \left(-y\right), x\right)} \]

   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. distribute-rgt-neg-in N/A

          \[\leadsto \color{blue}{t \cdot \left(\mathsf{neg}\left(x \cdot y\right)\right)} \]

       3. mul-1-neg N/A

          \[\leadsto t \cdot \color{blue}{\left(-1 \cdot \left(x \cdot y\right)\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(x \cdot y\right)\right)} \]

       5. mul-1-neg N/A

          \[\leadsto t \cdot \color{blue}{\left(\mathsf{neg}\left(x \cdot y\right)\right)} \]

       6. *-commutative N/A

          \[\leadsto t \cdot \left(\mathsf{neg}\left(\color{blue}{y \cdot x}\right)\right) \]

       7. distribute-rgt-neg-in N/A

          \[\leadsto t \cdot \color{blue}{\left(y \cdot \left(\mathsf{neg}\left(x\right)\right)\right)} \]

       8. neg-mul-1 N/A

          \[\leadsto t \cdot \left(y \cdot \color{blue}{\left(-1 \cdot x\right)}\right) \]

       9. lower-*.f64 N/A

          \[\leadsto t \cdot \color{blue}{\left(y \cdot \left(-1 \cdot x\right)\right)} \]

       10. neg-mul-1 N/A

           \[\leadsto t \cdot \left(y \cdot \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right) \]

       11. lower-neg.f64 26.8

           \[\leadsto t \cdot \left(y \cdot \color{blue}{\left(-x\right)}\right) \]

   11. Simplified 26.8%

       \[\leadsto \color{blue}{t \cdot \left(y \cdot \left(-x\right)\right)} \]

   if -4e9 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 5e9

   1. Initial program 87.5%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\log z - t\right)}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\log z - t\right)}} \]

      2. lower--.f64 N/A

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(\log z - t\right)}} \]

      3. lower-log.f64 84.1

         \[\leadsto x \cdot e^{y \cdot \left(\color{blue}{\log z} - t\right)} \]

   5. Simplified 84.1%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\log z - t\right)}} \]

   6. Taylor expanded in y around 0

      \[\leadsto x \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Simplified 81.8%

      \[\leadsto x \cdot \color{blue}{1} \]
   9. Step-by-step derivation

      1. *-rgt-identity 81.8

         \[\leadsto \color{blue}{x} \]

   10. Applied egg-rr 81.8%

       \[\leadsto \color{blue}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -4000000000:\\ \;\;\;\;t \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 5000000000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(x \cdot \left(-y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 32.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \leq 0:\\ \;\;\;\;t \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - y \cdot t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (exp (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b)))) 0.0)
   (* t (* x (- y)))
   (* x (- 1.0 (* y t)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (exp(((y * (log(z) - t)) + (a * (log((1.0 - z)) - b)))) <= 0.0) {
		tmp = t * (x * -y);
	} 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 (exp(((y * (log(z) - t)) + (a * (log((1.0d0 - z)) - b)))) <= 0.0d0) then
        tmp = t * (x * -y)
    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 (Math.exp(((y * (Math.log(z) - t)) + (a * (Math.log((1.0 - z)) - b)))) <= 0.0) {
		tmp = t * (x * -y);
	} else {
		tmp = x * (1.0 - (y * t));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if math.exp(((y * (math.log(z) - t)) + (a * (math.log((1.0 - z)) - b)))) <= 0.0:
		tmp = t * (x * -y)
	else:
		tmp = x * (1.0 - (y * t))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (exp(Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))) <= 0.0)
		tmp = Float64(t * Float64(x * Float64(-y)));
	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 (exp(((y * (log(z) - t)) + (a * (log((1.0 - z)) - b)))) <= 0.0)
		tmp = t * (x * -y);
	else
		tmp = x * (1.0 - (y * t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[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], 0.0], N[(t * N[(x * (-y)), $MachinePrecision]), $MachinePrecision], N[(x * N[(1.0 - N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))) < 0.0

   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. lower-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      5. lower-neg.f64 51.3

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(-t\right)}} \]

   5. Simplified 51.3%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + -1 \cdot \left(t \cdot \left(x \cdot y\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{-1 \cdot \left(t \cdot \left(x \cdot y\right)\right) + x} \]

      2. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(t \cdot \left(x \cdot y\right)\right)\right)} + x \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{t \cdot \left(\mathsf{neg}\left(x \cdot y\right)\right)} + x \]

      4. mul-1-neg N/A

         \[\leadsto t \cdot \color{blue}{\left(-1 \cdot \left(x \cdot y\right)\right)} + x \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, -1 \cdot \left(x \cdot y\right), x\right)} \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(x \cdot y\right)}, x\right) \]

      7. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{x \cdot \left(\mathsf{neg}\left(y\right)\right)}, x\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(t, x \cdot \color{blue}{\left(-1 \cdot y\right)}, x\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{x \cdot \left(-1 \cdot y\right)}, x\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(t, x \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}, x\right) \]

      11. lower-neg.f64 3.0

          \[\leadsto \mathsf{fma}\left(t, x \cdot \color{blue}{\left(-y\right)}, x\right) \]

   8. Simplified 3.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, x \cdot \left(-y\right), x\right)} \]

   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. distribute-rgt-neg-in N/A

          \[\leadsto \color{blue}{t \cdot \left(\mathsf{neg}\left(x \cdot y\right)\right)} \]

       3. mul-1-neg N/A

          \[\leadsto t \cdot \color{blue}{\left(-1 \cdot \left(x \cdot y\right)\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(x \cdot y\right)\right)} \]

       5. mul-1-neg N/A

          \[\leadsto t \cdot \color{blue}{\left(\mathsf{neg}\left(x \cdot y\right)\right)} \]

       6. *-commutative N/A

          \[\leadsto t \cdot \left(\mathsf{neg}\left(\color{blue}{y \cdot x}\right)\right) \]

       7. distribute-rgt-neg-in N/A

          \[\leadsto t \cdot \color{blue}{\left(y \cdot \left(\mathsf{neg}\left(x\right)\right)\right)} \]

       8. neg-mul-1 N/A

          \[\leadsto t \cdot \left(y \cdot \color{blue}{\left(-1 \cdot x\right)}\right) \]

       9. lower-*.f64 N/A

          \[\leadsto t \cdot \color{blue}{\left(y \cdot \left(-1 \cdot x\right)\right)} \]

       10. neg-mul-1 N/A

           \[\leadsto t \cdot \left(y \cdot \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right) \]

       11. lower-neg.f64 20.9

           \[\leadsto t \cdot \left(y \cdot \color{blue}{\left(-x\right)}\right) \]

   11. Simplified 20.9%

       \[\leadsto \color{blue}{t \cdot \left(y \cdot \left(-x\right)\right)} \]

   if 0.0 < (exp.f64 (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))))

   1. Initial program 92.9%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in 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. lower-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      5. lower-neg.f64 68.5

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(-t\right)}} \]

   5. Simplified 68.5%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}} \]

   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. lower--.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(1 - t \cdot y\right)} \]

      4. *-commutative N/A

         \[\leadsto x \cdot \left(1 - \color{blue}{y \cdot t}\right) \]

      5. lower-*.f64 51.6

         \[\leadsto x \cdot \left(1 - \color{blue}{y \cdot t}\right) \]

   8. Simplified 51.6%

      \[\leadsto x \cdot \color{blue}{\left(1 - y \cdot t\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 39.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \leq 0:\\ \;\;\;\;t \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - y \cdot t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 25.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \leq 0:\\ \;\;\;\;y \cdot \left(x \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot y, t, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (exp (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b)))) 0.0)
   (* y (* x t))
   (fma (* x y) t x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (exp(((y * (log(z) - t)) + (a * (log((1.0 - z)) - b)))) <= 0.0) {
		tmp = y * (x * t);
	} else {
		tmp = fma((x * y), t, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (exp(Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))) <= 0.0)
		tmp = Float64(y * Float64(x * t));
	else
		tmp = fma(Float64(x * y), t, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[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], 0.0], N[(y * N[(x * t), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] * t + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))) < 0.0

   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. lower-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      5. lower-neg.f64 51.3

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(-t\right)}} \]

   5. Simplified 51.3%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + -1 \cdot \left(t \cdot \left(x \cdot y\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{-1 \cdot \left(t \cdot \left(x \cdot y\right)\right) + x} \]

      2. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(t \cdot \left(x \cdot y\right)\right)\right)} + x \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{t \cdot \left(\mathsf{neg}\left(x \cdot y\right)\right)} + x \]

      4. mul-1-neg N/A

         \[\leadsto t \cdot \color{blue}{\left(-1 \cdot \left(x \cdot y\right)\right)} + x \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, -1 \cdot \left(x \cdot y\right), x\right)} \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(x \cdot y\right)}, x\right) \]

      7. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{x \cdot \left(\mathsf{neg}\left(y\right)\right)}, x\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(t, x \cdot \color{blue}{\left(-1 \cdot y\right)}, x\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{x \cdot \left(-1 \cdot y\right)}, x\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(t, x \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}, x\right) \]

      11. lower-neg.f64 3.0

          \[\leadsto \mathsf{fma}\left(t, x \cdot \color{blue}{\left(-y\right)}, x\right) \]

   8. Simplified 3.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, x \cdot \left(-y\right), x\right)} \]
   9. Step-by-step derivation

      1. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(t, x \cdot \color{blue}{\left(0 - y\right)}, x\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(t, x \cdot \color{blue}{\left(0 + \left(\mathsf{neg}\left(y\right)\right)\right)}, x\right) \]

      3. lift-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(t, x \cdot \left(0 + \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right), x\right) \]

      4. flip3-+ N/A

         \[\leadsto \mathsf{fma}\left(t, x \cdot \color{blue}{\frac{{0}^{3} + {\left(\mathsf{neg}\left(y\right)\right)}^{3}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right) - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}}, x\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(t, x \cdot \color{blue}{\frac{{0}^{3} + {\left(\mathsf{neg}\left(y\right)\right)}^{3}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right) - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}}, x\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(t, x \cdot \frac{\color{blue}{0} + {\left(\mathsf{neg}\left(y\right)\right)}^{3}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right) - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}, x\right) \]

      7. sqr-pow N/A

         \[\leadsto \mathsf{fma}\left(t, x \cdot \frac{0 + \color{blue}{{\left(\mathsf{neg}\left(y\right)\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\mathsf{neg}\left(y\right)\right)}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right) - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}, x\right) \]

      8. pow-prod-down N/A

         \[\leadsto \mathsf{fma}\left(t, x \cdot \frac{0 + \color{blue}{{\left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right) - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}, x\right) \]

      9. lift-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(t, x \cdot \frac{0 + {\left(\color{blue}{\left(\mathsf{neg}\left(y\right)\right)} \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right) - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}, x\right) \]

      10. lift-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, x \cdot \frac{0 + {\left(\left(\mathsf{neg}\left(y\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right) - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}, x\right) \]

      11. sqr-neg N/A

          \[\leadsto \mathsf{fma}\left(t, x \cdot \frac{0 + {\color{blue}{\left(y \cdot y\right)}}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right) - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}, x\right) \]

      12. pow-prod-down N/A

          \[\leadsto \mathsf{fma}\left(t, x \cdot \frac{0 + \color{blue}{{y}^{\left(\frac{3}{2}\right)} \cdot {y}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right) - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}, x\right) \]

      13. sqr-pow N/A

          \[\leadsto \mathsf{fma}\left(t, x \cdot \frac{0 + \color{blue}{{y}^{3}}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right) - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}, x\right) \]

      14. cube-unmult N/A

          \[\leadsto \mathsf{fma}\left(t, x \cdot \frac{0 + \color{blue}{y \cdot \left(y \cdot y\right)}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right) - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}, x\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, x \cdot \frac{0 + y \cdot \color{blue}{\left(y \cdot y\right)}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right) - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}, x\right) \]

      16. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, x \cdot \frac{0 + \color{blue}{y \cdot \left(y \cdot y\right)}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right) - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}, x\right) \]

      17. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, x \cdot \frac{\color{blue}{0 + y \cdot \left(y \cdot y\right)}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right) - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}, x\right) \]

      18. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(t, x \cdot \frac{0 + y \cdot \left(y \cdot y\right)}{\color{blue}{0} + \left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right) - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}, x\right) \]

      19. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, x \cdot \frac{0 + y \cdot \left(y \cdot y\right)}{\color{blue}{0 + \left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right) - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}}, x\right) \]

      20. lift-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, x \cdot \frac{0 + y \cdot \left(y \cdot y\right)}{0 + \left(\color{blue}{\left(\mathsf{neg}\left(y\right)\right)} \cdot \left(\mathsf{neg}\left(y\right)\right) - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}, x\right) \]

      21. lift-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, x \cdot \frac{0 + y \cdot \left(y \cdot y\right)}{0 + \left(\left(\mathsf{neg}\left(y\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)} - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}, x\right) \]

      22. sqr-neg N/A

          \[\leadsto \mathsf{fma}\left(t, x \cdot \frac{0 + y \cdot \left(y \cdot y\right)}{0 + \left(\color{blue}{y \cdot y} - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}, x\right) \]

      23. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, x \cdot \frac{0 + y \cdot \left(y \cdot y\right)}{0 + \left(\color{blue}{y \cdot y} - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}, x\right) \]

   10. Applied egg-rr 1.5%

       \[\leadsto \mathsf{fma}\left(t, x \cdot \color{blue}{\frac{0 + y \cdot \left(y \cdot y\right)}{0 + \left(y \cdot y - 0 \cdot \left(-y\right)\right)}}, x\right) \]

   11. Taylor expanded in t around inf

       \[\leadsto \color{blue}{t \cdot \left(x \cdot y\right)} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot t} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{\left(y \cdot x\right)} \cdot t \]

       3. associate-*r* N/A

          \[\leadsto \color{blue}{y \cdot \left(x \cdot t\right)} \]

       4. *-commutative N/A

          \[\leadsto y \cdot \color{blue}{\left(t \cdot x\right)} \]

       5. lower-*.f64 N/A

          \[\leadsto \color{blue}{y \cdot \left(t \cdot x\right)} \]

       6. *-commutative N/A

          \[\leadsto y \cdot \color{blue}{\left(x \cdot t\right)} \]

       7. lower-*.f64 17.2

          \[\leadsto y \cdot \color{blue}{\left(x \cdot t\right)} \]

   13. Simplified 17.2%

       \[\leadsto \color{blue}{y \cdot \left(x \cdot t\right)} \]

   if 0.0 < (exp.f64 (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))))

   1. Initial program 92.9%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in 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. lower-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      5. lower-neg.f64 68.5

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(-t\right)}} \]

   5. Simplified 68.5%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + -1 \cdot \left(t \cdot \left(x \cdot y\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{-1 \cdot \left(t \cdot \left(x \cdot y\right)\right) + x} \]

      2. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(t \cdot \left(x \cdot y\right)\right)\right)} + x \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{t \cdot \left(\mathsf{neg}\left(x \cdot y\right)\right)} + x \]

      4. mul-1-neg N/A

         \[\leadsto t \cdot \color{blue}{\left(-1 \cdot \left(x \cdot y\right)\right)} + x \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, -1 \cdot \left(x \cdot y\right), x\right)} \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(x \cdot y\right)}, x\right) \]

      7. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{x \cdot \left(\mathsf{neg}\left(y\right)\right)}, x\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(t, x \cdot \color{blue}{\left(-1 \cdot y\right)}, x\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{x \cdot \left(-1 \cdot y\right)}, x\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(t, x \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}, x\right) \]

      11. lower-neg.f64 50.8

          \[\leadsto \mathsf{fma}\left(t, x \cdot \color{blue}{\left(-y\right)}, x\right) \]

   8. Simplified 50.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, x \cdot \left(-y\right), x\right)} \]
   9. Step-by-step derivation

      1. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(t, x \cdot \color{blue}{\left(0 - y\right)}, x\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(t, x \cdot \color{blue}{\left(0 + \left(\mathsf{neg}\left(y\right)\right)\right)}, x\right) \]

      3. lift-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(t, x \cdot \left(0 + \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right), x\right) \]

      4. flip3-+ N/A

         \[\leadsto \mathsf{fma}\left(t, x \cdot \color{blue}{\frac{{0}^{3} + {\left(\mathsf{neg}\left(y\right)\right)}^{3}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right) - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}}, x\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(t, x \cdot \color{blue}{\frac{{0}^{3} + {\left(\mathsf{neg}\left(y\right)\right)}^{3}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right) - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}}, x\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(t, x \cdot \frac{\color{blue}{0} + {\left(\mathsf{neg}\left(y\right)\right)}^{3}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right) - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}, x\right) \]

      7. sqr-pow N/A

         \[\leadsto \mathsf{fma}\left(t, x \cdot \frac{0 + \color{blue}{{\left(\mathsf{neg}\left(y\right)\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\mathsf{neg}\left(y\right)\right)}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right) - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}, x\right) \]

      8. pow-prod-down N/A

         \[\leadsto \mathsf{fma}\left(t, x \cdot \frac{0 + \color{blue}{{\left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right) - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}, x\right) \]

      9. lift-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(t, x \cdot \frac{0 + {\left(\color{blue}{\left(\mathsf{neg}\left(y\right)\right)} \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right) - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}, x\right) \]

      10. lift-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, x \cdot \frac{0 + {\left(\left(\mathsf{neg}\left(y\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right) - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}, x\right) \]

      11. sqr-neg N/A

          \[\leadsto \mathsf{fma}\left(t, x \cdot \frac{0 + {\color{blue}{\left(y \cdot y\right)}}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right) - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}, x\right) \]

      12. pow-prod-down N/A

          \[\leadsto \mathsf{fma}\left(t, x \cdot \frac{0 + \color{blue}{{y}^{\left(\frac{3}{2}\right)} \cdot {y}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right) - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}, x\right) \]

      13. sqr-pow N/A

          \[\leadsto \mathsf{fma}\left(t, x \cdot \frac{0 + \color{blue}{{y}^{3}}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right) - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}, x\right) \]

      14. cube-unmult N/A

          \[\leadsto \mathsf{fma}\left(t, x \cdot \frac{0 + \color{blue}{y \cdot \left(y \cdot y\right)}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right) - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}, x\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, x \cdot \frac{0 + y \cdot \color{blue}{\left(y \cdot y\right)}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right) - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}, x\right) \]

      16. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, x \cdot \frac{0 + \color{blue}{y \cdot \left(y \cdot y\right)}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right) - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}, x\right) \]

      17. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, x \cdot \frac{\color{blue}{0 + y \cdot \left(y \cdot y\right)}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right) - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}, x\right) \]

      18. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(t, x \cdot \frac{0 + y \cdot \left(y \cdot y\right)}{\color{blue}{0} + \left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right) - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}, x\right) \]

      19. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, x \cdot \frac{0 + y \cdot \left(y \cdot y\right)}{\color{blue}{0 + \left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right) - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}}, x\right) \]

      20. lift-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, x \cdot \frac{0 + y \cdot \left(y \cdot y\right)}{0 + \left(\color{blue}{\left(\mathsf{neg}\left(y\right)\right)} \cdot \left(\mathsf{neg}\left(y\right)\right) - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}, x\right) \]

      21. lift-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, x \cdot \frac{0 + y \cdot \left(y \cdot y\right)}{0 + \left(\left(\mathsf{neg}\left(y\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)} - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}, x\right) \]

      22. sqr-neg N/A

          \[\leadsto \mathsf{fma}\left(t, x \cdot \frac{0 + y \cdot \left(y \cdot y\right)}{0 + \left(\color{blue}{y \cdot y} - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}, x\right) \]

      23. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, x \cdot \frac{0 + y \cdot \left(y \cdot y\right)}{0 + \left(\color{blue}{y \cdot y} - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}, x\right) \]

   10. Applied egg-rr 11.8%

       \[\leadsto \mathsf{fma}\left(t, x \cdot \color{blue}{\frac{0 + y \cdot \left(y \cdot y\right)}{0 + \left(y \cdot y - 0 \cdot \left(-y\right)\right)}}, x\right) \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto t \cdot \left(x \cdot \frac{0 + y \cdot \color{blue}{\left(y \cdot y\right)}}{0 + \left(y \cdot y - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}\right) + x \]

       2. lift-*.f64 N/A

          \[\leadsto t \cdot \left(x \cdot \frac{0 + \color{blue}{y \cdot \left(y \cdot y\right)}}{0 + \left(y \cdot y - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}\right) + x \]

       3. lift-+.f64 N/A

          \[\leadsto t \cdot \left(x \cdot \frac{\color{blue}{0 + y \cdot \left(y \cdot y\right)}}{0 + \left(y \cdot y - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}\right) + x \]

       4. lift-*.f64 N/A

          \[\leadsto t \cdot \left(x \cdot \frac{0 + y \cdot \left(y \cdot y\right)}{0 + \left(\color{blue}{y \cdot y} - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}\right) + x \]

       5. lift-neg.f64 N/A

          \[\leadsto t \cdot \left(x \cdot \frac{0 + y \cdot \left(y \cdot y\right)}{0 + \left(y \cdot y - 0 \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)}\right) + x \]

       6. lift-*.f64 N/A

          \[\leadsto t \cdot \left(x \cdot \frac{0 + y \cdot \left(y \cdot y\right)}{0 + \left(y \cdot y - \color{blue}{0 \cdot \left(\mathsf{neg}\left(y\right)\right)}\right)}\right) + x \]

       7. lift--.f64 N/A

          \[\leadsto t \cdot \left(x \cdot \frac{0 + y \cdot \left(y \cdot y\right)}{0 + \color{blue}{\left(y \cdot y - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}}\right) + x \]

       8. lift-+.f64 N/A

          \[\leadsto t \cdot \left(x \cdot \frac{0 + y \cdot \left(y \cdot y\right)}{\color{blue}{0 + \left(y \cdot y - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}}\right) + x \]

       9. lift-/.f64 N/A

          \[\leadsto t \cdot \left(x \cdot \color{blue}{\frac{0 + y \cdot \left(y \cdot y\right)}{0 + \left(y \cdot y - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}}\right) + x \]

       10. lift-*.f64 N/A

           \[\leadsto t \cdot \color{blue}{\left(x \cdot \frac{0 + y \cdot \left(y \cdot y\right)}{0 + \left(y \cdot y - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}\right)} + x \]

   12. Applied egg-rr 34.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot y, t, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 24.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \leq 0:\\ \;\;\;\;y \cdot \left(x \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (exp (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b)))) 0.0)
   (* y (* x t))
   x))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (exp(((y * (log(z) - t)) + (a * (log((1.0 - z)) - b)))) <= 0.0) {
		tmp = y * (x * t);
	} else {
		tmp = 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) :: tmp
    if (exp(((y * (log(z) - t)) + (a * (log((1.0d0 - z)) - b)))) <= 0.0d0) then
        tmp = y * (x * t)
    else
        tmp = 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 tmp;
	if (Math.exp(((y * (Math.log(z) - t)) + (a * (Math.log((1.0 - z)) - b)))) <= 0.0) {
		tmp = y * (x * t);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if math.exp(((y * (math.log(z) - t)) + (a * (math.log((1.0 - z)) - b)))) <= 0.0:
		tmp = y * (x * t)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (exp(Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))) <= 0.0)
		tmp = Float64(y * Float64(x * t));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (exp(((y * (log(z) - t)) + (a * (log((1.0 - z)) - b)))) <= 0.0)
		tmp = y * (x * t);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[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], 0.0], N[(y * N[(x * t), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))) < 0.0

   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. lower-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      5. lower-neg.f64 51.3

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(-t\right)}} \]

   5. Simplified 51.3%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + -1 \cdot \left(t \cdot \left(x \cdot y\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{-1 \cdot \left(t \cdot \left(x \cdot y\right)\right) + x} \]

      2. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(t \cdot \left(x \cdot y\right)\right)\right)} + x \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{t \cdot \left(\mathsf{neg}\left(x \cdot y\right)\right)} + x \]

      4. mul-1-neg N/A

         \[\leadsto t \cdot \color{blue}{\left(-1 \cdot \left(x \cdot y\right)\right)} + x \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, -1 \cdot \left(x \cdot y\right), x\right)} \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(x \cdot y\right)}, x\right) \]

      7. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{x \cdot \left(\mathsf{neg}\left(y\right)\right)}, x\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(t, x \cdot \color{blue}{\left(-1 \cdot y\right)}, x\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{x \cdot \left(-1 \cdot y\right)}, x\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(t, x \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}, x\right) \]

      11. lower-neg.f64 3.0

          \[\leadsto \mathsf{fma}\left(t, x \cdot \color{blue}{\left(-y\right)}, x\right) \]

   8. Simplified 3.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, x \cdot \left(-y\right), x\right)} \]
   9. Step-by-step derivation

      1. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(t, x \cdot \color{blue}{\left(0 - y\right)}, x\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(t, x \cdot \color{blue}{\left(0 + \left(\mathsf{neg}\left(y\right)\right)\right)}, x\right) \]

      3. lift-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(t, x \cdot \left(0 + \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right), x\right) \]

      4. flip3-+ N/A

         \[\leadsto \mathsf{fma}\left(t, x \cdot \color{blue}{\frac{{0}^{3} + {\left(\mathsf{neg}\left(y\right)\right)}^{3}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right) - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}}, x\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(t, x \cdot \color{blue}{\frac{{0}^{3} + {\left(\mathsf{neg}\left(y\right)\right)}^{3}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right) - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}}, x\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(t, x \cdot \frac{\color{blue}{0} + {\left(\mathsf{neg}\left(y\right)\right)}^{3}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right) - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}, x\right) \]

      7. sqr-pow N/A

         \[\leadsto \mathsf{fma}\left(t, x \cdot \frac{0 + \color{blue}{{\left(\mathsf{neg}\left(y\right)\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\mathsf{neg}\left(y\right)\right)}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right) - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}, x\right) \]

      8. pow-prod-down N/A

         \[\leadsto \mathsf{fma}\left(t, x \cdot \frac{0 + \color{blue}{{\left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right) - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}, x\right) \]

      9. lift-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(t, x \cdot \frac{0 + {\left(\color{blue}{\left(\mathsf{neg}\left(y\right)\right)} \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right) - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}, x\right) \]

      10. lift-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, x \cdot \frac{0 + {\left(\left(\mathsf{neg}\left(y\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right) - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}, x\right) \]

      11. sqr-neg N/A

          \[\leadsto \mathsf{fma}\left(t, x \cdot \frac{0 + {\color{blue}{\left(y \cdot y\right)}}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right) - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}, x\right) \]

      12. pow-prod-down N/A

          \[\leadsto \mathsf{fma}\left(t, x \cdot \frac{0 + \color{blue}{{y}^{\left(\frac{3}{2}\right)} \cdot {y}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right) - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}, x\right) \]

      13. sqr-pow N/A

          \[\leadsto \mathsf{fma}\left(t, x \cdot \frac{0 + \color{blue}{{y}^{3}}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right) - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}, x\right) \]

      14. cube-unmult N/A

          \[\leadsto \mathsf{fma}\left(t, x \cdot \frac{0 + \color{blue}{y \cdot \left(y \cdot y\right)}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right) - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}, x\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, x \cdot \frac{0 + y \cdot \color{blue}{\left(y \cdot y\right)}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right) - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}, x\right) \]

      16. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, x \cdot \frac{0 + \color{blue}{y \cdot \left(y \cdot y\right)}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right) - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}, x\right) \]

      17. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, x \cdot \frac{\color{blue}{0 + y \cdot \left(y \cdot y\right)}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right) - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}, x\right) \]

      18. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(t, x \cdot \frac{0 + y \cdot \left(y \cdot y\right)}{\color{blue}{0} + \left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right) - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}, x\right) \]

      19. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, x \cdot \frac{0 + y \cdot \left(y \cdot y\right)}{\color{blue}{0 + \left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right) - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}}, x\right) \]

      20. lift-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, x \cdot \frac{0 + y \cdot \left(y \cdot y\right)}{0 + \left(\color{blue}{\left(\mathsf{neg}\left(y\right)\right)} \cdot \left(\mathsf{neg}\left(y\right)\right) - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}, x\right) \]

      21. lift-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, x \cdot \frac{0 + y \cdot \left(y \cdot y\right)}{0 + \left(\left(\mathsf{neg}\left(y\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)} - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}, x\right) \]

      22. sqr-neg N/A

          \[\leadsto \mathsf{fma}\left(t, x \cdot \frac{0 + y \cdot \left(y \cdot y\right)}{0 + \left(\color{blue}{y \cdot y} - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}, x\right) \]

      23. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, x \cdot \frac{0 + y \cdot \left(y \cdot y\right)}{0 + \left(\color{blue}{y \cdot y} - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}, x\right) \]

   10. Applied egg-rr 1.5%

       \[\leadsto \mathsf{fma}\left(t, x \cdot \color{blue}{\frac{0 + y \cdot \left(y \cdot y\right)}{0 + \left(y \cdot y - 0 \cdot \left(-y\right)\right)}}, x\right) \]

   11. Taylor expanded in t around inf

       \[\leadsto \color{blue}{t \cdot \left(x \cdot y\right)} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot t} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{\left(y \cdot x\right)} \cdot t \]

       3. associate-*r* N/A

          \[\leadsto \color{blue}{y \cdot \left(x \cdot t\right)} \]

       4. *-commutative N/A

          \[\leadsto y \cdot \color{blue}{\left(t \cdot x\right)} \]

       5. lower-*.f64 N/A

          \[\leadsto \color{blue}{y \cdot \left(t \cdot x\right)} \]

       6. *-commutative N/A

          \[\leadsto y \cdot \color{blue}{\left(x \cdot t\right)} \]

       7. lower-*.f64 17.2

          \[\leadsto y \cdot \color{blue}{\left(x \cdot t\right)} \]

   13. Simplified 17.2%

       \[\leadsto \color{blue}{y \cdot \left(x \cdot t\right)} \]

   if 0.0 < (exp.f64 (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))))

   1. Initial program 92.9%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   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. lower-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\log z - t\right)}} \]

      2. lower--.f64 N/A

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(\log z - t\right)}} \]

      3. lower-log.f64 76.8

         \[\leadsto x \cdot e^{y \cdot \left(\color{blue}{\log z} - t\right)} \]

   5. Simplified 76.8%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\log z - t\right)}} \]

   6. Taylor expanded in y around 0

      \[\leadsto x \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Simplified 32.1%

      \[\leadsto x \cdot \color{blue}{1} \]
   9. Step-by-step derivation

      1. *-rgt-identity 32.1

         \[\leadsto \color{blue}{x} \]

   10. Applied egg-rr 32.1%

       \[\leadsto \color{blue}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 58.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 2 \cdot 10^{+75}:\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(t \cdot \mathsf{fma}\left(x, \left(y \cdot y\right) \cdot 0.5, \frac{\mathsf{fma}\left(x, -y, \frac{x}{t}\right)}{t}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b))) 2e+75)
   (* x (pow z y))
   (* t (* t (fma x (* (* y y) 0.5) (/ (fma x (- y) (/ x t)) 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))) <= 2e+75) {
		tmp = x * pow(z, y);
	} else {
		tmp = t * (t * fma(x, ((y * y) * 0.5), (fma(x, -y, (x / t)) / 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))) <= 2e+75)
		tmp = Float64(x * (z ^ y));
	else
		tmp = Float64(t * Float64(t * fma(x, Float64(Float64(y * y) * 0.5), Float64(fma(x, Float64(-y), Float64(x / t)) / t))));
	end
	return 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], 2e+75], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision], N[(t * N[(t * N[(x * N[(N[(y * y), $MachinePrecision] * 0.5), $MachinePrecision] + N[(N[(x * (-y) + N[(x / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $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))) < 1.99999999999999985e75

   1. Initial program 95.4%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   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. lower-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\log z - t\right)}} \]

      2. lower--.f64 N/A

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(\log z - t\right)}} \]

      3. lower-log.f64 75.9

         \[\leadsto x \cdot e^{y \cdot \left(\color{blue}{\log z} - t\right)} \]

   5. Simplified 75.9%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\log z - t\right)}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{x \cdot {z}^{y}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot {z}^{y}} \]

      2. lower-pow.f64 57.9

         \[\leadsto x \cdot \color{blue}{{z}^{y}} \]

   8. Simplified 57.9%

      \[\leadsto \color{blue}{x \cdot {z}^{y}} \]

   if 1.99999999999999985e75 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 96.6%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   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. lower-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      5. lower-neg.f64 62.5

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(-t\right)}} \]

   5. Simplified 62.5%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}} \]

   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. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, -1 \cdot \left(x \cdot y\right) + \frac{1}{2} \cdot \left(t \cdot \left(x \cdot {y}^{2}\right)\right), x\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{\frac{1}{2} \cdot \left(t \cdot \left(x \cdot {y}^{2}\right)\right) + -1 \cdot \left(x \cdot y\right)}, x\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{\left(\frac{1}{2} \cdot t\right) \cdot \left(x \cdot {y}^{2}\right)} + -1 \cdot \left(x \cdot y\right), x\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot t, x \cdot {y}^{2}, -1 \cdot \left(x \cdot y\right)\right)}, x\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(\color{blue}{\frac{1}{2} \cdot t}, x \cdot {y}^{2}, -1 \cdot \left(x \cdot y\right)\right), x\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(\frac{1}{2} \cdot t, \color{blue}{x \cdot {y}^{2}}, -1 \cdot \left(x \cdot y\right)\right), x\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(\frac{1}{2} \cdot t, x \cdot \color{blue}{\left(y \cdot y\right)}, -1 \cdot \left(x \cdot y\right)\right), x\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(\frac{1}{2} \cdot t, x \cdot \color{blue}{\left(y \cdot y\right)}, -1 \cdot \left(x \cdot y\right)\right), x\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(\frac{1}{2} \cdot t, x \cdot \left(y \cdot y\right), \color{blue}{\mathsf{neg}\left(x \cdot y\right)}\right), x\right) \]

      11. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(\frac{1}{2} \cdot t, x \cdot \left(y \cdot y\right), \color{blue}{x \cdot \left(\mathsf{neg}\left(y\right)\right)}\right), x\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(\frac{1}{2} \cdot t, x \cdot \left(y \cdot y\right), x \cdot \color{blue}{\left(-1 \cdot y\right)}\right), x\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(\frac{1}{2} \cdot t, x \cdot \left(y \cdot y\right), \color{blue}{x \cdot \left(-1 \cdot y\right)}\right), x\right) \]

      14. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(\frac{1}{2} \cdot t, x \cdot \left(y \cdot y\right), x \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right), x\right) \]

      15. lower-neg.f64 69.2

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(0.5 \cdot t, x \cdot \left(y \cdot y\right), x \cdot \color{blue}{\left(-y\right)}\right), x\right) \]

   8. Simplified 69.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, \mathsf{fma}\left(0.5 \cdot t, x \cdot \left(y \cdot y\right), x \cdot \left(-y\right)\right), x\right)} \]

   9. Taylor expanded in t around -inf

      \[\leadsto \color{blue}{{t}^{2} \cdot \left(-1 \cdot \frac{-1 \cdot \frac{x}{t} + x \cdot y}{t} + \frac{1}{2} \cdot \left(x \cdot {y}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \color{blue}{\left(t \cdot t\right)} \cdot \left(-1 \cdot \frac{-1 \cdot \frac{x}{t} + x \cdot y}{t} + \frac{1}{2} \cdot \left(x \cdot {y}^{2}\right)\right) \]

       2. associate-*l* N/A

          \[\leadsto \color{blue}{t \cdot \left(t \cdot \left(-1 \cdot \frac{-1 \cdot \frac{x}{t} + x \cdot y}{t} + \frac{1}{2} \cdot \left(x \cdot {y}^{2}\right)\right)\right)} \]

       3. lower-*.f64 N/A

          \[\leadsto \color{blue}{t \cdot \left(t \cdot \left(-1 \cdot \frac{-1 \cdot \frac{x}{t} + x \cdot y}{t} + \frac{1}{2} \cdot \left(x \cdot {y}^{2}\right)\right)\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto t \cdot \color{blue}{\left(t \cdot \left(-1 \cdot \frac{-1 \cdot \frac{x}{t} + x \cdot y}{t} + \frac{1}{2} \cdot \left(x \cdot {y}^{2}\right)\right)\right)} \]

       5. +-commutative N/A

          \[\leadsto t \cdot \left(t \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(x \cdot {y}^{2}\right) + -1 \cdot \frac{-1 \cdot \frac{x}{t} + x \cdot y}{t}\right)}\right) \]

       6. *-commutative N/A

          \[\leadsto t \cdot \left(t \cdot \left(\color{blue}{\left(x \cdot {y}^{2}\right) \cdot \frac{1}{2}} + -1 \cdot \frac{-1 \cdot \frac{x}{t} + x \cdot y}{t}\right)\right) \]

       7. associate-*l* N/A

          \[\leadsto t \cdot \left(t \cdot \left(\color{blue}{x \cdot \left({y}^{2} \cdot \frac{1}{2}\right)} + -1 \cdot \frac{-1 \cdot \frac{x}{t} + x \cdot y}{t}\right)\right) \]

       8. lower-fma.f64 N/A

          \[\leadsto t \cdot \left(t \cdot \color{blue}{\mathsf{fma}\left(x, {y}^{2} \cdot \frac{1}{2}, -1 \cdot \frac{-1 \cdot \frac{x}{t} + x \cdot y}{t}\right)}\right) \]

       9. lower-*.f64 N/A

          \[\leadsto t \cdot \left(t \cdot \mathsf{fma}\left(x, \color{blue}{{y}^{2} \cdot \frac{1}{2}}, -1 \cdot \frac{-1 \cdot \frac{x}{t} + x \cdot y}{t}\right)\right) \]

       10. unpow2 N/A

           \[\leadsto t \cdot \left(t \cdot \mathsf{fma}\left(x, \color{blue}{\left(y \cdot y\right)} \cdot \frac{1}{2}, -1 \cdot \frac{-1 \cdot \frac{x}{t} + x \cdot y}{t}\right)\right) \]

       11. lower-*.f64 N/A

           \[\leadsto t \cdot \left(t \cdot \mathsf{fma}\left(x, \color{blue}{\left(y \cdot y\right)} \cdot \frac{1}{2}, -1 \cdot \frac{-1 \cdot \frac{x}{t} + x \cdot y}{t}\right)\right) \]

       12. associate-*r/ N/A

           \[\leadsto t \cdot \left(t \cdot \mathsf{fma}\left(x, \left(y \cdot y\right) \cdot \frac{1}{2}, \color{blue}{\frac{-1 \cdot \left(-1 \cdot \frac{x}{t} + x \cdot y\right)}{t}}\right)\right) \]

   11. Simplified 79.1%

       \[\leadsto \color{blue}{t \cdot \left(t \cdot \mathsf{fma}\left(x, \left(y \cdot y\right) \cdot 0.5, \frac{\mathsf{fma}\left(x, -y, \frac{x}{t}\right)}{t}\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 86.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot e^{y \cdot \left(\log z - t\right)}\\ \mathbf{if}\;y \leq -6.8 \cdot 10^{-45}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-47}:\\ \;\;\;\;x \cdot e^{-a \cdot \left(z + b\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (exp (* y (- (log z) t))))))
   (if (<= y -6.8e-45)
     t_1
     (if (<= y 2.2e-47) (* 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 <= -6.8e-45) {
		tmp = t_1;
	} else if (y <= 2.2e-47) {
		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 <= (-6.8d-45)) then
        tmp = t_1
    else if (y <= 2.2d-47) 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 <= -6.8e-45) {
		tmp = t_1;
	} else if (y <= 2.2e-47) {
		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 <= -6.8e-45:
		tmp = t_1
	elif y <= 2.2e-47:
		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 <= -6.8e-45)
		tmp = t_1;
	elseif (y <= 2.2e-47)
		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 <= -6.8e-45)
		tmp = t_1;
	elseif (y <= 2.2e-47)
		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, -6.8e-45], t$95$1, If[LessEqual[y, 2.2e-47], 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 < -6.80000000000000008e-45 or 2.20000000000000019e-47 < y

   1. Initial program 98.7%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\log z - t\right)}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\log z - t\right)}} \]

      2. lower--.f64 N/A

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(\log z - t\right)}} \]

      3. lower-log.f64 88.2

         \[\leadsto x \cdot e^{y \cdot \left(\color{blue}{\log z} - t\right)} \]

   5. Simplified 88.2%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\log z - t\right)}} \]

   if -6.80000000000000008e-45 < y < 2.20000000000000019e-47

   1. Initial program 91.8%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in 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. lower-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{a \cdot \left(\log \left(1 - z\right) - b\right)}} \]

      2. lower--.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. lower-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. lower-neg.f64 90.1

         \[\leadsto x \cdot e^{a \cdot \left(\mathsf{log1p}\left(\color{blue}{-z}\right) - b\right)} \]

   5. Simplified 90.1%

      \[\leadsto x \cdot e^{\color{blue}{a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)}} \]

   6. Taylor expanded in z around 0

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right) + -1 \cdot \left(a \cdot z\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot z\right) + -1 \cdot \left(a \cdot b\right)}} \]

      2. associate-*r* N/A

         \[\leadsto x \cdot e^{\color{blue}{\left(-1 \cdot a\right) \cdot z} + -1 \cdot \left(a \cdot b\right)} \]

      3. associate-*r* N/A

         \[\leadsto x \cdot e^{\left(-1 \cdot a\right) \cdot z + \color{blue}{\left(-1 \cdot a\right) \cdot b}} \]

      4. distribute-lft-out N/A

         \[\leadsto x \cdot e^{\color{blue}{\left(-1 \cdot a\right) \cdot \left(z + b\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\left(-1 \cdot a\right) \cdot \left(z + b\right)}} \]

      6. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\left(\mathsf{neg}\left(a\right)\right)} \cdot \left(z + b\right)} \]

      7. lower-neg.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\left(\mathsf{neg}\left(a\right)\right)} \cdot \left(z + b\right)} \]

      8. lower-+.f64 90.1

         \[\leadsto x \cdot e^{\left(-a\right) \cdot \color{blue}{\left(z + b\right)}} \]

   8. Simplified 90.1%

      \[\leadsto x \cdot e^{\color{blue}{\left(-a\right) \cdot \left(z + b\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.8 \cdot 10^{-45}:\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-47}:\\ \;\;\;\;x \cdot e^{-a \cdot \left(z + b\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 74.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.8 \cdot 10^{-45}:\\ \;\;\;\;x \cdot e^{t \cdot \left(-y\right)}\\ \mathbf{elif}\;y \leq 3.7:\\ \;\;\;\;x \cdot e^{-a \cdot \left(z + b\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -6.8e-45)
   (* x (exp (* t (- y))))
   (if (<= y 3.7) (* x (exp (- (* a (+ z b))))) (* x (pow z y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -6.8e-45) {
		tmp = x * exp((t * -y));
	} else if (y <= 3.7) {
		tmp = x * exp(-(a * (z + b)));
	} else {
		tmp = x * pow(z, y);
	}
	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 <= (-6.8d-45)) then
        tmp = x * exp((t * -y))
    else if (y <= 3.7d0) then
        tmp = x * exp(-(a * (z + b)))
    else
        tmp = x * (z ** y)
    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 <= -6.8e-45) {
		tmp = x * Math.exp((t * -y));
	} else if (y <= 3.7) {
		tmp = x * Math.exp(-(a * (z + b)));
	} else {
		tmp = x * Math.pow(z, y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -6.8e-45:
		tmp = x * math.exp((t * -y))
	elif y <= 3.7:
		tmp = x * math.exp(-(a * (z + b)))
	else:
		tmp = x * math.pow(z, y)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -6.8e-45)
		tmp = Float64(x * exp(Float64(t * Float64(-y))));
	elseif (y <= 3.7)
		tmp = Float64(x * exp(Float64(-Float64(a * Float64(z + b)))));
	else
		tmp = Float64(x * (z ^ y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -6.8e-45)
		tmp = x * exp((t * -y));
	elseif (y <= 3.7)
		tmp = x * exp(-(a * (z + b)));
	else
		tmp = x * (z ^ y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -6.8e-45], N[(x * N[Exp[N[(t * (-y)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.7], N[(x * N[Exp[(-N[(a * N[(z + b), $MachinePrecision]), $MachinePrecision])], $MachinePrecision]), $MachinePrecision], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.80000000000000008e-45

   1. Initial program 97.3%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   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. lower-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      5. lower-neg.f64 69.6

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(-t\right)}} \]

   5. Simplified 69.6%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}} \]

   if -6.80000000000000008e-45 < y < 3.7000000000000002

   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 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. lower-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{a \cdot \left(\log \left(1 - z\right) - b\right)}} \]

      2. lower--.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. lower-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. lower-neg.f64 88.8

         \[\leadsto x \cdot e^{a \cdot \left(\mathsf{log1p}\left(\color{blue}{-z}\right) - b\right)} \]

   5. Simplified 88.8%

      \[\leadsto x \cdot e^{\color{blue}{a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)}} \]

   6. Taylor expanded in z around 0

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right) + -1 \cdot \left(a \cdot z\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot z\right) + -1 \cdot \left(a \cdot b\right)}} \]

      2. associate-*r* N/A

         \[\leadsto x \cdot e^{\color{blue}{\left(-1 \cdot a\right) \cdot z} + -1 \cdot \left(a \cdot b\right)} \]

      3. associate-*r* N/A

         \[\leadsto x \cdot e^{\left(-1 \cdot a\right) \cdot z + \color{blue}{\left(-1 \cdot a\right) \cdot b}} \]

      4. distribute-lft-out N/A

         \[\leadsto x \cdot e^{\color{blue}{\left(-1 \cdot a\right) \cdot \left(z + b\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\left(-1 \cdot a\right) \cdot \left(z + b\right)}} \]

      6. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\left(\mathsf{neg}\left(a\right)\right)} \cdot \left(z + b\right)} \]

      7. lower-neg.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\left(\mathsf{neg}\left(a\right)\right)} \cdot \left(z + b\right)} \]

      8. lower-+.f64 88.8

         \[\leadsto x \cdot e^{\left(-a\right) \cdot \color{blue}{\left(z + b\right)}} \]

   8. Simplified 88.8%

      \[\leadsto x \cdot e^{\color{blue}{\left(-a\right) \cdot \left(z + b\right)}} \]

   if 3.7000000000000002 < y

   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 y around inf

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\log z - t\right)}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\log z - t\right)}} \]

      2. lower--.f64 N/A

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(\log z - t\right)}} \]

      3. lower-log.f64 87.5

         \[\leadsto x \cdot e^{y \cdot \left(\color{blue}{\log z} - t\right)} \]

   5. Simplified 87.5%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\log z - t\right)}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{x \cdot {z}^{y}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot {z}^{y}} \]

      2. lower-pow.f64 68.1

         \[\leadsto x \cdot \color{blue}{{z}^{y}} \]

   8. Simplified 68.1%

      \[\leadsto \color{blue}{x \cdot {z}^{y}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.8 \cdot 10^{-45}:\\ \;\;\;\;x \cdot e^{t \cdot \left(-y\right)}\\ \mathbf{elif}\;y \leq 3.7:\\ \;\;\;\;x \cdot e^{-a \cdot \left(z + b\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 72.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot e^{t \cdot \left(-y\right)}\\ \mathbf{if}\;t \leq -270000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-48}:\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (exp (* t (- y))))))
   (if (<= t -270000000.0) t_1 (if (<= t 5e-48) (* x (pow z y)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * exp((t * -y));
	double tmp;
	if (t <= -270000000.0) {
		tmp = t_1;
	} else if (t <= 5e-48) {
		tmp = x * pow(z, y);
	} 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((t * -y))
    if (t <= (-270000000.0d0)) then
        tmp = t_1
    else if (t <= 5d-48) then
        tmp = x * (z ** y)
    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((t * -y));
	double tmp;
	if (t <= -270000000.0) {
		tmp = t_1;
	} else if (t <= 5e-48) {
		tmp = x * Math.pow(z, y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x * math.exp((t * -y))
	tmp = 0
	if t <= -270000000.0:
		tmp = t_1
	elif t <= 5e-48:
		tmp = x * math.pow(z, y)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x * exp(Float64(t * Float64(-y))))
	tmp = 0.0
	if (t <= -270000000.0)
		tmp = t_1;
	elseif (t <= 5e-48)
		tmp = Float64(x * (z ^ y));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x * exp((t * -y));
	tmp = 0.0;
	if (t <= -270000000.0)
		tmp = t_1;
	elseif (t <= 5e-48)
		tmp = x * (z ^ y);
	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[(t * (-y)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -270000000.0], t$95$1, If[LessEqual[t, 5e-48], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -2.7e8 or 4.9999999999999999e-48 < t

   1. Initial program 95.7%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   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. lower-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      5. lower-neg.f64 82.5

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(-t\right)}} \]

   5. Simplified 82.5%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}} \]

   if -2.7e8 < t < 4.9999999999999999e-48

   1. Initial program 95.9%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   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. lower-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\log z - t\right)}} \]

      2. lower--.f64 N/A

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(\log z - t\right)}} \]

      3. lower-log.f64 67.2

         \[\leadsto x \cdot e^{y \cdot \left(\color{blue}{\log z} - t\right)} \]

   5. Simplified 67.2%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\log z - t\right)}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{x \cdot {z}^{y}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot {z}^{y}} \]

      2. lower-pow.f64 67.1

         \[\leadsto x \cdot \color{blue}{{z}^{y}} \]

   8. Simplified 67.1%

      \[\leadsto \color{blue}{x \cdot {z}^{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -270000000:\\ \;\;\;\;x \cdot e^{t \cdot \left(-y\right)}\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-48}:\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{t \cdot \left(-y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 73.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot {z}^{y}\\ \mathbf{if}\;y \leq -1.4 \cdot 10^{+22}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 0.002:\\ \;\;\;\;x \cdot e^{-a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (pow z y))))
   (if (<= y -1.4e+22) t_1 (if (<= y 0.002) (* x (exp (- (* a 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 <= -1.4e+22) {
		tmp = t_1;
	} else if (y <= 0.002) {
		tmp = x * exp(-(a * 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 <= (-1.4d+22)) then
        tmp = t_1
    else if (y <= 0.002d0) then
        tmp = x * exp(-(a * 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 <= -1.4e+22) {
		tmp = t_1;
	} else if (y <= 0.002) {
		tmp = x * Math.exp(-(a * 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 <= -1.4e+22:
		tmp = t_1
	elif y <= 0.002:
		tmp = x * math.exp(-(a * 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 <= -1.4e+22)
		tmp = t_1;
	elseif (y <= 0.002)
		tmp = Float64(x * exp(Float64(-Float64(a * 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 <= -1.4e+22)
		tmp = t_1;
	elseif (y <= 0.002)
		tmp = x * exp(-(a * 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, -1.4e+22], t$95$1, If[LessEqual[y, 0.002], N[(x * N[Exp[(-N[(a * b), $MachinePrecision])], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.4e22 or 2e-3 < y

   1. Initial program 98.5%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\log z - t\right)}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\log z - t\right)}} \]

      2. lower--.f64 N/A

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(\log z - t\right)}} \]

      3. lower-log.f64 89.6

         \[\leadsto x \cdot e^{y \cdot \left(\color{blue}{\log z} - t\right)} \]

   5. Simplified 89.6%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\log z - t\right)}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{x \cdot {z}^{y}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot {z}^{y}} \]

      2. lower-pow.f64 69.4

         \[\leadsto x \cdot \color{blue}{{z}^{y}} \]

   8. Simplified 69.4%

      \[\leadsto \color{blue}{x \cdot {z}^{y}} \]

   if -1.4e22 < y < 2e-3

   1. Initial program 92.9%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   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. lower-neg.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      3. lower-*.f64 77.3

         \[\leadsto x \cdot e^{-\color{blue}{a \cdot b}} \]

   5. Simplified 77.3%

      \[\leadsto x \cdot e^{\color{blue}{-a \cdot b}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 18.4% 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 95.8%

   \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
2. Add Preprocessing

3. Taylor expanded in y around inf

   \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\log z - t\right)}} \]
4. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\log z - t\right)}} \]

   2. lower--.f64 N/A

      \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(\log z - t\right)}} \]

   3. lower-log.f64 75.4

      \[\leadsto x \cdot e^{y \cdot \left(\color{blue}{\log z} - t\right)} \]

5. Simplified 75.4%

   \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\log z - t\right)}} \]

6. Taylor expanded in y around 0

   \[\leadsto x \cdot \color{blue}{1} \]
7. Step-by-step derivation

8. Simplified 20.5%

   \[\leadsto x \cdot \color{blue}{1} \]
9. Step-by-step derivation

   1. *-rgt-identity 20.5

      \[\leadsto \color{blue}{x} \]

10. Applied egg-rr 20.5%

    \[\leadsto \color{blue}{x} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024207 
(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))))))