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

Percentage Accurate: 96.7% → 96.7%

Time: 16.3s

Alternatives: 21

Speedup: 1.0×

• 41.1% 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: 96.7% 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: 96.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.3%

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

Alternative 2: 63.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+161}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(t, \mathsf{fma}\left(t, 0.5 \cdot \left(y \cdot y\right), y\right), 1\right)}\\ \mathbf{elif}\;t\_1 \leq -20000000000000:\\ \;\;\;\;t \cdot \left(\left(x \cdot \left(y \cdot y\right)\right) \cdot \left(t \cdot 0.5\right)\right)\\ \mathbf{elif}\;t\_1 \leq 10^{-23}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot a, \left(-z\right) - b, x\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+124}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, y \cdot \left(-0.16666666666666666 \cdot \left(t \cdot \left(t \cdot t\right)\right)\right), -t\right), 1\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+295}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(t, \mathsf{fma}\left(0.5, t \cdot \left(y \cdot y\right), -y\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(0.5 \cdot \left(a \cdot a\right), b, -a\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b)))))
   (if (<= t_1 -2e+161)
     (/ x (fma t (fma t (* 0.5 (* y y)) y) 1.0))
     (if (<= t_1 -20000000000000.0)
       (* t (* (* x (* y y)) (* t 0.5)))
       (if (<= t_1 1e-23)
         (fma (* x a) (- (- z) b) x)
         (if (<= t_1 1e+124)
           (*
            x
            (fma
             y
             (fma y (* y (* -0.16666666666666666 (* t (* t t)))) (- t))
             1.0))
           (if (<= t_1 5e+295)
             (* x (fma t (fma 0.5 (* t (* y y)) (- y)) 1.0))
             (* x (fma b (fma (* 0.5 (* a a)) b (- a)) 1.0)))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	double tmp;
	if (t_1 <= -2e+161) {
		tmp = x / fma(t, fma(t, (0.5 * (y * y)), y), 1.0);
	} else if (t_1 <= -20000000000000.0) {
		tmp = t * ((x * (y * y)) * (t * 0.5));
	} else if (t_1 <= 1e-23) {
		tmp = fma((x * a), (-z - b), x);
	} else if (t_1 <= 1e+124) {
		tmp = x * fma(y, fma(y, (y * (-0.16666666666666666 * (t * (t * t)))), -t), 1.0);
	} else if (t_1 <= 5e+295) {
		tmp = x * fma(t, fma(0.5, (t * (y * y)), -y), 1.0);
	} else {
		tmp = x * fma(b, fma((0.5 * (a * a)), b, -a), 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))
	tmp = 0.0
	if (t_1 <= -2e+161)
		tmp = Float64(x / fma(t, fma(t, Float64(0.5 * Float64(y * y)), y), 1.0));
	elseif (t_1 <= -20000000000000.0)
		tmp = Float64(t * Float64(Float64(x * Float64(y * y)) * Float64(t * 0.5)));
	elseif (t_1 <= 1e-23)
		tmp = fma(Float64(x * a), Float64(Float64(-z) - b), x);
	elseif (t_1 <= 1e+124)
		tmp = Float64(x * fma(y, fma(y, Float64(y * Float64(-0.16666666666666666 * Float64(t * Float64(t * t)))), Float64(-t)), 1.0));
	elseif (t_1 <= 5e+295)
		tmp = Float64(x * fma(t, fma(0.5, Float64(t * Float64(y * y)), Float64(-y)), 1.0));
	else
		tmp = Float64(x * fma(b, fma(Float64(0.5 * Float64(a * a)), b, Float64(-a)), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+161], N[(x / N[(t * N[(t * N[(0.5 * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -20000000000000.0], N[(t * N[(N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(t * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-23], N[(N[(x * a), $MachinePrecision] * N[((-z) - b), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 1e+124], N[(x * N[(y * N[(y * N[(y * N[(-0.16666666666666666 * N[(t * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + (-t)), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+295], N[(x * N[(t * N[(0.5 * N[(t * N[(y * y), $MachinePrecision]), $MachinePrecision] + (-y)), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(b * N[(N[(0.5 * N[(a * a), $MachinePrecision]), $MachinePrecision] * b + (-a)), $MachinePrecision] + 1.0), $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))) < -2.0000000000000001e161

   1. Initial program 98.6%

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

   3. Taylor expanded in 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 54.4

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

   5. Applied rewrites 54.4%

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

      1. distribute-rgt-neg-out N/A

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

      2. exp-neg N/A

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

      3. pow-exp N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. pow-exp N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-*.f64 54.4

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

   7. Applied rewrites 54.4%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-+l+ N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

      12. associate-+l+ N/A

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

      13. *-commutative N/A

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

      14. distribute-rgt-in N/A

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

   10. Applied rewrites 64.3%

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

   if -2.0000000000000001e161 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -2e13

   1. Initial program 89.1%

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

   3. Taylor expanded in 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 36.5

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

   5. Applied rewrites 36.5%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-neg.f64 3.8

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

   8. Applied rewrites 3.8%

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

   9. Taylor expanded in y 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. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 29.3

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

   11. Applied rewrites 29.3%

       \[\leadsto \color{blue}{\left(0.5 \cdot \left(t \cdot t\right)\right) \cdot \left(x \cdot \left(y \cdot y\right)\right)} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. associate-*r* N/A

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

       9. associate-*r* N/A

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

       10. lower-*.f64 N/A

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

       11. lower-*.f64 N/A

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

       12. *-commutative N/A

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

       13. lower-*.f64 51.7

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

   13. Applied rewrites 51.7%

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

   if -2e13 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 9.9999999999999996e-24

   1. Initial program 92.0%

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

   3. Taylor expanded in y around 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 95.7

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

   5. Applied rewrites 95.7%

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

   6. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 95.7

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

   8. Applied rewrites 95.7%

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

   9. Taylor expanded in a around 0

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

       1. +-commutative N/A

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

       2. mul-1-neg N/A

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

       3. associate-*r* N/A

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

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

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

       5. mul-1-neg N/A

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

       6. lower-fma.f64 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 N/A

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

       9. distribute-lft-in N/A

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

       10. +-commutative N/A

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

       11. mul-1-neg N/A

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

       12. sub-neg N/A

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

       13. lower--.f64 N/A

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

       14. mul-1-neg N/A

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

       15. lower-neg.f64 88.9

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

   11. Applied rewrites 88.9%

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

   if 9.9999999999999996e-24 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 9.99999999999999948e123

   1. Initial program 94.6%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   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 63.1

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

   5. Applied rewrites 63.1%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

   8. Applied rewrites 41.2%

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

   9. Taylor expanded in t around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. cube-mult N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 40.8

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

   11. Applied rewrites 40.8%

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

   if 9.99999999999999948e123 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 4.99999999999999991e295

   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 50.1

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

   5. Applied rewrites 50.1%

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

   6. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 70.7

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

   8. Applied rewrites 70.7%

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

   if 4.99999999999999991e295 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 97.8%

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

   3. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 65.4

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

   5. Applied rewrites 65.4%

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

   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. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 70.0

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

   8. Applied rewrites 70.0%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -2 \cdot 10^{+161}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(t, \mathsf{fma}\left(t, 0.5 \cdot \left(y \cdot y\right), y\right), 1\right)}\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -20000000000000:\\ \;\;\;\;t \cdot \left(\left(x \cdot \left(y \cdot y\right)\right) \cdot \left(t \cdot 0.5\right)\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 10^{-23}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot a, \left(-z\right) - b, x\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 10^{+124}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, y \cdot \left(-0.16666666666666666 \cdot \left(t \cdot \left(t \cdot t\right)\right)\right), -t\right), 1\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 5 \cdot 10^{+295}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(t, \mathsf{fma}\left(0.5, t \cdot \left(y \cdot y\right), -y\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(0.5 \cdot \left(a \cdot a\right), b, -a\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 54.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ t_2 := t \cdot \left(\left(x \cdot \left(y \cdot y\right)\right) \cdot \left(t \cdot 0.5\right)\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(t, y, 1\right)}\\ \mathbf{elif}\;t\_1 \leq -20000000000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5:\\ \;\;\;\;\mathsf{fma}\left(x \cdot a, \left(-z\right) - b, x\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+115}:\\ \;\;\;\;y \cdot \left(y \cdot \left(0.5 \cdot \left(x \cdot \left(t \cdot t\right)\right)\right)\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+295}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(0.5 \cdot \left(a \cdot a\right), b, -a\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b))))
        (t_2 (* t (* (* x (* y y)) (* t 0.5)))))
   (if (<= t_1 (- INFINITY))
     (/ x (fma t y 1.0))
     (if (<= t_1 -20000000000000.0)
       t_2
       (if (<= t_1 5.0)
         (fma (* x a) (- (- z) b) x)
         (if (<= t_1 2e+115)
           (* y (* y (* 0.5 (* x (* t t)))))
           (if (<= t_1 5e+295)
             t_2
             (* x (fma b (fma (* 0.5 (* a a)) b (- a)) 1.0)))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	double t_2 = t * ((x * (y * y)) * (t * 0.5));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = x / fma(t, y, 1.0);
	} else if (t_1 <= -20000000000000.0) {
		tmp = t_2;
	} else if (t_1 <= 5.0) {
		tmp = fma((x * a), (-z - b), x);
	} else if (t_1 <= 2e+115) {
		tmp = y * (y * (0.5 * (x * (t * t))));
	} else if (t_1 <= 5e+295) {
		tmp = t_2;
	} else {
		tmp = x * fma(b, fma((0.5 * (a * a)), b, -a), 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))
	t_2 = Float64(t * Float64(Float64(x * Float64(y * y)) * Float64(t * 0.5)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(x / fma(t, y, 1.0));
	elseif (t_1 <= -20000000000000.0)
		tmp = t_2;
	elseif (t_1 <= 5.0)
		tmp = fma(Float64(x * a), Float64(Float64(-z) - b), x);
	elseif (t_1 <= 2e+115)
		tmp = Float64(y * Float64(y * Float64(0.5 * Float64(x * Float64(t * t)))));
	elseif (t_1 <= 5e+295)
		tmp = t_2;
	else
		tmp = Float64(x * fma(b, fma(Float64(0.5 * Float64(a * a)), b, Float64(-a)), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * N[(N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(t * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(x / N[(t * y + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -20000000000000.0], t$95$2, If[LessEqual[t$95$1, 5.0], N[(N[(x * a), $MachinePrecision] * N[((-z) - b), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 2e+115], N[(y * N[(y * N[(0.5 * N[(x * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+295], t$95$2, N[(x * N[(b * N[(N[(0.5 * N[(a * a), $MachinePrecision]), $MachinePrecision] * b + (-a)), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -inf.0

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 69.1

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

   5. Applied rewrites 69.1%

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

      1. distribute-rgt-neg-out N/A

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

      2. exp-neg N/A

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

      3. pow-exp N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. pow-exp N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-*.f64 69.1

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

   7. Applied rewrites 69.1%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 63.9

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

   10. Applied rewrites 63.9%

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

   if -inf.0 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -2e13 or 2e115 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 4.99999999999999991e295

   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 43.7

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

   5. Applied rewrites 43.7%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-neg.f64 22.3

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

   8. Applied rewrites 22.3%

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

   9. Taylor expanded in y 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. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 37.9

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

   11. Applied rewrites 37.9%

       \[\leadsto \color{blue}{\left(0.5 \cdot \left(t \cdot t\right)\right) \cdot \left(x \cdot \left(y \cdot y\right)\right)} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. associate-*r* N/A

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

       9. associate-*r* N/A

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

       10. lower-*.f64 N/A

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

       11. lower-*.f64 N/A

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

       12. *-commutative N/A

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

       13. lower-*.f64 54.1

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

   13. Applied rewrites 54.1%

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

   if -2e13 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 5

   1. Initial program 92.6%

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

   3. Taylor expanded in 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 91.9

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

   5. Applied rewrites 91.9%

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

   6. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 91.9

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

   8. Applied rewrites 91.9%

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

   9. Taylor expanded in a around 0

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

       1. +-commutative N/A

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

       2. mul-1-neg N/A

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

       3. associate-*r* N/A

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

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

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

       5. mul-1-neg N/A

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

       6. lower-fma.f64 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 N/A

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

       9. distribute-lft-in N/A

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

       10. +-commutative N/A

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

       11. mul-1-neg N/A

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

       12. sub-neg N/A

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

       13. lower--.f64 N/A

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

       14. mul-1-neg N/A

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

       15. lower-neg.f64 85.6

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

   11. Applied rewrites 85.6%

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

   if 5 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 2e115

   1. Initial program 93.4%

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

   3. Taylor expanded in 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 63.2

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

   5. Applied rewrites 63.2%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-neg.f64 29.9

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

   8. Applied rewrites 29.9%

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

   9. Taylor expanded in y 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. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 12.2

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

   11. Applied rewrites 12.2%

       \[\leadsto \color{blue}{\left(0.5 \cdot \left(t \cdot t\right)\right) \cdot \left(x \cdot \left(y \cdot y\right)\right)} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*r* N/A

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

       5. lift-*.f64 N/A

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

       6. associate-*r* N/A

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

       7. lower-*.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. associate-*l* N/A

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

       11. lower-*.f64 N/A

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

       12. lower-*.f64 36.2

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

   13. Applied rewrites 36.2%

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

   if 4.99999999999999991e295 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 97.8%

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

   3. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 65.4

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

   5. Applied rewrites 65.4%

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

   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. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 70.0

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

   8. Applied rewrites 70.0%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(0.5 \cdot \left(a \cdot a\right), b, -a\right), 1\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -\infty:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(t, y, 1\right)}\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -20000000000000:\\ \;\;\;\;t \cdot \left(\left(x \cdot \left(y \cdot y\right)\right) \cdot \left(t \cdot 0.5\right)\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 5:\\ \;\;\;\;\mathsf{fma}\left(x \cdot a, \left(-z\right) - b, x\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 2 \cdot 10^{+115}:\\ \;\;\;\;y \cdot \left(y \cdot \left(0.5 \cdot \left(x \cdot \left(t \cdot t\right)\right)\right)\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 5 \cdot 10^{+295}:\\ \;\;\;\;t \cdot \left(\left(x \cdot \left(y \cdot y\right)\right) \cdot \left(t \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(0.5 \cdot \left(a \cdot a\right), b, -a\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 66.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+161}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(t, \mathsf{fma}\left(t, \left(y \cdot y\right) \cdot \mathsf{fma}\left(t \cdot 0.16666666666666666, y, 0.5\right), y\right), 1\right)}\\ \mathbf{elif}\;t\_1 \leq -20000000000000:\\ \;\;\;\;t \cdot \left(\left(x \cdot \left(y \cdot y\right)\right) \cdot \left(t \cdot 0.5\right)\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+154}:\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+295}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(t, \mathsf{fma}\left(0.5, t \cdot \left(y \cdot y\right), -y\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(0.5 \cdot \left(a \cdot a\right), b, -a\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b)))))
   (if (<= t_1 -2e+161)
     (/
      x
      (fma t (fma t (* (* y y) (fma (* t 0.16666666666666666) y 0.5)) y) 1.0))
     (if (<= t_1 -20000000000000.0)
       (* t (* (* x (* y y)) (* t 0.5)))
       (if (<= t_1 2e+154)
         (* x (pow z y))
         (if (<= t_1 5e+295)
           (* x (fma t (fma 0.5 (* t (* y y)) (- y)) 1.0))
           (* x (fma b (fma (* 0.5 (* a a)) b (- a)) 1.0))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	double tmp;
	if (t_1 <= -2e+161) {
		tmp = x / fma(t, fma(t, ((y * y) * fma((t * 0.16666666666666666), y, 0.5)), y), 1.0);
	} else if (t_1 <= -20000000000000.0) {
		tmp = t * ((x * (y * y)) * (t * 0.5));
	} else if (t_1 <= 2e+154) {
		tmp = x * pow(z, y);
	} else if (t_1 <= 5e+295) {
		tmp = x * fma(t, fma(0.5, (t * (y * y)), -y), 1.0);
	} else {
		tmp = x * fma(b, fma((0.5 * (a * a)), b, -a), 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))
	tmp = 0.0
	if (t_1 <= -2e+161)
		tmp = Float64(x / fma(t, fma(t, Float64(Float64(y * y) * fma(Float64(t * 0.16666666666666666), y, 0.5)), y), 1.0));
	elseif (t_1 <= -20000000000000.0)
		tmp = Float64(t * Float64(Float64(x * Float64(y * y)) * Float64(t * 0.5)));
	elseif (t_1 <= 2e+154)
		tmp = Float64(x * (z ^ y));
	elseif (t_1 <= 5e+295)
		tmp = Float64(x * fma(t, fma(0.5, Float64(t * Float64(y * y)), Float64(-y)), 1.0));
	else
		tmp = Float64(x * fma(b, fma(Float64(0.5 * Float64(a * a)), b, Float64(-a)), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+161], N[(x / N[(t * N[(t * N[(N[(y * y), $MachinePrecision] * N[(N[(t * 0.16666666666666666), $MachinePrecision] * y + 0.5), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -20000000000000.0], N[(t * N[(N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(t * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+154], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+295], N[(x * N[(t * N[(0.5 * N[(t * N[(y * y), $MachinePrecision]), $MachinePrecision] + (-y)), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(b * N[(N[(0.5 * N[(a * a), $MachinePrecision]), $MachinePrecision] * b + (-a)), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -2.0000000000000001e161

   1. Initial program 98.6%

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

   3. Taylor expanded in 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 54.4

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

   5. Applied rewrites 54.4%

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

      1. distribute-rgt-neg-out N/A

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

      2. exp-neg N/A

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

      3. pow-exp N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. pow-exp N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-*.f64 54.4

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

   7. Applied rewrites 54.4%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 65.7%

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

   if -2.0000000000000001e161 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -2e13

   1. Initial program 89.1%

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

   3. Taylor expanded in 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 36.5

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

   5. Applied rewrites 36.5%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-neg.f64 3.8

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

   8. Applied rewrites 3.8%

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

   9. Taylor expanded in y 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. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 29.3

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

   11. Applied rewrites 29.3%

       \[\leadsto \color{blue}{\left(0.5 \cdot \left(t \cdot t\right)\right) \cdot \left(x \cdot \left(y \cdot y\right)\right)} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. associate-*r* N/A

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

       9. associate-*r* N/A

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

       10. lower-*.f64 N/A

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

       11. lower-*.f64 N/A

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

       12. *-commutative N/A

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

       13. lower-*.f64 51.7

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

   13. Applied rewrites 51.7%

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

   if -2e13 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 2.00000000000000007e154

   1. Initial program 93.6%

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

   3. Taylor expanded in y around 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 83.1

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

   5. Applied rewrites 83.1%

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

   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 62.1

         \[\leadsto x \cdot \color{blue}{{z}^{y}} \]

   8. Applied rewrites 62.1%

      \[\leadsto \color{blue}{x \cdot {z}^{y}} \]

   if 2.00000000000000007e154 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 4.99999999999999991e295

   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 51.2

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

   5. Applied rewrites 51.2%

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

   6. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 77.1

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

   8. Applied rewrites 77.1%

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

   if 4.99999999999999991e295 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 97.8%

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

   3. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 65.4

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

   5. Applied rewrites 65.4%

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

   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. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 70.0

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

   8. Applied rewrites 70.0%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(0.5 \cdot \left(a \cdot a\right), b, -a\right), 1\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -2 \cdot 10^{+161}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(t, \mathsf{fma}\left(t, \left(y \cdot y\right) \cdot \mathsf{fma}\left(t \cdot 0.16666666666666666, y, 0.5\right), y\right), 1\right)}\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -20000000000000:\\ \;\;\;\;t \cdot \left(\left(x \cdot \left(y \cdot y\right)\right) \cdot \left(t \cdot 0.5\right)\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 2 \cdot 10^{+154}:\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 5 \cdot 10^{+295}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(t, \mathsf{fma}\left(0.5, t \cdot \left(y \cdot y\right), -y\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(0.5 \cdot \left(a \cdot a\right), b, -a\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 64.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+161}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(t, \mathsf{fma}\left(t, 0.5 \cdot \left(y \cdot y\right), y\right), 1\right)}\\ \mathbf{elif}\;t\_1 \leq -2000:\\ \;\;\;\;t \cdot \left(\left(x \cdot \left(y \cdot y\right)\right) \cdot \left(t \cdot 0.5\right)\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+147}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y, t \cdot \mathsf{fma}\left(t, y \cdot \mathsf{fma}\left(t, y \cdot -0.16666666666666666, 0.5\right), -1\right), 1\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+295}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(t, \mathsf{fma}\left(0.5, t \cdot \left(y \cdot y\right), -y\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(0.5 \cdot \left(a \cdot a\right), b, -a\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b)))))
   (if (<= t_1 -2e+161)
     (/ x (fma t (fma t (* 0.5 (* y y)) y) 1.0))
     (if (<= t_1 -2000.0)
       (* t (* (* x (* y y)) (* t 0.5)))
       (if (<= t_1 1e+147)
         (*
          x
          (fma
           y
           (* t (fma t (* y (fma t (* y -0.16666666666666666) 0.5)) -1.0))
           1.0))
         (if (<= t_1 5e+295)
           (* x (fma t (fma 0.5 (* t (* y y)) (- y)) 1.0))
           (* x (fma b (fma (* 0.5 (* a a)) b (- a)) 1.0))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	double tmp;
	if (t_1 <= -2e+161) {
		tmp = x / fma(t, fma(t, (0.5 * (y * y)), y), 1.0);
	} else if (t_1 <= -2000.0) {
		tmp = t * ((x * (y * y)) * (t * 0.5));
	} else if (t_1 <= 1e+147) {
		tmp = x * fma(y, (t * fma(t, (y * fma(t, (y * -0.16666666666666666), 0.5)), -1.0)), 1.0);
	} else if (t_1 <= 5e+295) {
		tmp = x * fma(t, fma(0.5, (t * (y * y)), -y), 1.0);
	} else {
		tmp = x * fma(b, fma((0.5 * (a * a)), b, -a), 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))
	tmp = 0.0
	if (t_1 <= -2e+161)
		tmp = Float64(x / fma(t, fma(t, Float64(0.5 * Float64(y * y)), y), 1.0));
	elseif (t_1 <= -2000.0)
		tmp = Float64(t * Float64(Float64(x * Float64(y * y)) * Float64(t * 0.5)));
	elseif (t_1 <= 1e+147)
		tmp = Float64(x * fma(y, Float64(t * fma(t, Float64(y * fma(t, Float64(y * -0.16666666666666666), 0.5)), -1.0)), 1.0));
	elseif (t_1 <= 5e+295)
		tmp = Float64(x * fma(t, fma(0.5, Float64(t * Float64(y * y)), Float64(-y)), 1.0));
	else
		tmp = Float64(x * fma(b, fma(Float64(0.5 * Float64(a * a)), b, Float64(-a)), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+161], N[(x / N[(t * N[(t * N[(0.5 * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -2000.0], N[(t * N[(N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(t * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+147], N[(x * N[(y * N[(t * N[(t * N[(y * N[(t * N[(y * -0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+295], N[(x * N[(t * N[(0.5 * N[(t * N[(y * y), $MachinePrecision]), $MachinePrecision] + (-y)), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(b * N[(N[(0.5 * N[(a * a), $MachinePrecision]), $MachinePrecision] * b + (-a)), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -2.0000000000000001e161

   1. Initial program 98.6%

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

   3. Taylor expanded in 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 54.4

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

   5. Applied rewrites 54.4%

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

      1. distribute-rgt-neg-out N/A

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

      2. exp-neg N/A

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

      3. pow-exp N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. pow-exp N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-*.f64 54.4

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

   7. Applied rewrites 54.4%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-+l+ N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

      12. associate-+l+ N/A

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

      13. *-commutative N/A

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

      14. distribute-rgt-in N/A

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

   10. Applied rewrites 64.3%

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

   if -2.0000000000000001e161 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -2e3

   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 39.8

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

   5. Applied rewrites 39.8%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-neg.f64 3.9

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

   8. Applied rewrites 3.9%

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

   9. Taylor expanded in y 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. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 28.0

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

   11. Applied rewrites 28.0%

       \[\leadsto \color{blue}{\left(0.5 \cdot \left(t \cdot t\right)\right) \cdot \left(x \cdot \left(y \cdot y\right)\right)} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. associate-*r* N/A

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

       9. associate-*r* N/A

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

       10. lower-*.f64 N/A

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

       11. lower-*.f64 N/A

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

       12. *-commutative N/A

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

       13. lower-*.f64 49.2

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

   13. Applied rewrites 49.2%

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

   if -2e3 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 9.9999999999999998e146

   1. Initial program 93.4%

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

   3. Taylor expanded in 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 78.0

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

   5. Applied rewrites 78.0%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

   8. Applied rewrites 58.9%

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

   9. Taylor expanded in t around 0

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

       1. lower-*.f64 N/A

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

       2. sub-neg N/A

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

       3. metadata-eval N/A

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

       4. lower-fma.f64 N/A

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

       5. +-commutative N/A

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

       6. associate-*r* N/A

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

       7. unpow2 N/A

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

       8. associate-*r* N/A

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

       9. associate-*r* N/A

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

       10. distribute-rgt-out N/A

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

       11. lower-*.f64 N/A

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

       12. +-commutative N/A

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

       13. *-commutative N/A

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

       14. associate-*l* N/A

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

       15. *-commutative N/A

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

       16. lower-fma.f64 N/A

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

       17. *-commutative N/A

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

       18. lower-*.f64 60.7

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

   11. Applied rewrites 60.7%

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

   if 9.9999999999999998e146 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 4.99999999999999991e295

   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 48.8

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

   5. Applied rewrites 48.8%

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

   6. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 73.4

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

   8. Applied rewrites 73.4%

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

   if 4.99999999999999991e295 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 97.8%

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

   3. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 65.4

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

   5. Applied rewrites 65.4%

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

   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. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 70.0

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

   8. Applied rewrites 70.0%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(0.5 \cdot \left(a \cdot a\right), b, -a\right), 1\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -2 \cdot 10^{+161}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(t, \mathsf{fma}\left(t, 0.5 \cdot \left(y \cdot y\right), y\right), 1\right)}\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -2000:\\ \;\;\;\;t \cdot \left(\left(x \cdot \left(y \cdot y\right)\right) \cdot \left(t \cdot 0.5\right)\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 10^{+147}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y, t \cdot \mathsf{fma}\left(t, y \cdot \mathsf{fma}\left(t, y \cdot -0.16666666666666666, 0.5\right), -1\right), 1\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 5 \cdot 10^{+295}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(t, \mathsf{fma}\left(0.5, t \cdot \left(y \cdot y\right), -y\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(0.5 \cdot \left(a \cdot a\right), b, -a\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 54.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ t_2 := t \cdot \left(\left(x \cdot \left(y \cdot y\right)\right) \cdot \left(t \cdot 0.5\right)\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(t, y, 1\right)}\\ \mathbf{elif}\;t\_1 \leq -20000000000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5:\\ \;\;\;\;\mathsf{fma}\left(x \cdot a, \left(-z\right) - b, x\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+115}:\\ \;\;\;\;y \cdot \left(y \cdot \left(0.5 \cdot \left(x \cdot \left(t \cdot t\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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 (* t (* (* x (* y y)) (* t 0.5)))))
   (if (<= t_1 (- INFINITY))
     (/ x (fma t y 1.0))
     (if (<= t_1 -20000000000000.0)
       t_2
       (if (<= t_1 5.0)
         (fma (* x a) (- (- z) b) x)
         (if (<= t_1 2e+115) (* y (* y (* 0.5 (* x (* t t))))) 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 = t * ((x * (y * y)) * (t * 0.5));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = x / fma(t, y, 1.0);
	} else if (t_1 <= -20000000000000.0) {
		tmp = t_2;
	} else if (t_1 <= 5.0) {
		tmp = fma((x * a), (-z - b), x);
	} else if (t_1 <= 2e+115) {
		tmp = y * (y * (0.5 * (x * (t * t))));
	} 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(t * Float64(Float64(x * Float64(y * y)) * Float64(t * 0.5)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(x / fma(t, y, 1.0));
	elseif (t_1 <= -20000000000000.0)
		tmp = t_2;
	elseif (t_1 <= 5.0)
		tmp = fma(Float64(x * a), Float64(Float64(-z) - b), x);
	elseif (t_1 <= 2e+115)
		tmp = Float64(y * Float64(y * Float64(0.5 * Float64(x * Float64(t * t)))));
	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[(t * N[(N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(t * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(x / N[(t * y + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -20000000000000.0], t$95$2, If[LessEqual[t$95$1, 5.0], N[(N[(x * a), $MachinePrecision] * N[((-z) - b), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 2e+115], N[(y * N[(y * N[(0.5 * N[(x * N[(t * t), $MachinePrecision]), $MachinePrecision]), $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))) < -inf.0

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 69.1

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

   5. Applied rewrites 69.1%

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

      1. distribute-rgt-neg-out N/A

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

      2. exp-neg N/A

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

      3. pow-exp N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. pow-exp N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-*.f64 69.1

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

   7. Applied rewrites 69.1%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 63.9

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

   10. Applied rewrites 63.9%

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

   if -inf.0 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -2e13 or 2e115 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   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 49.1

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

   5. Applied rewrites 49.1%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-neg.f64 35.0

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

   8. Applied rewrites 35.0%

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

   9. Taylor expanded in y 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. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 43.6

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

   11. Applied rewrites 43.6%

       \[\leadsto \color{blue}{\left(0.5 \cdot \left(t \cdot t\right)\right) \cdot \left(x \cdot \left(y \cdot y\right)\right)} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. associate-*r* N/A

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

       9. associate-*r* N/A

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

       10. lower-*.f64 N/A

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

       11. lower-*.f64 N/A

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

       12. *-commutative N/A

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

       13. lower-*.f64 54.8

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

   13. Applied rewrites 54.8%

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

   if -2e13 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 5

   1. Initial program 92.6%

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

   3. Taylor expanded in 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 91.9

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

   5. Applied rewrites 91.9%

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

   6. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 91.9

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

   8. Applied rewrites 91.9%

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

   9. Taylor expanded in a around 0

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

       1. +-commutative N/A

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

       2. mul-1-neg N/A

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

       3. associate-*r* N/A

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

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

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

       5. mul-1-neg N/A

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

       6. lower-fma.f64 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 N/A

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

       9. distribute-lft-in N/A

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

       10. +-commutative N/A

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

       11. mul-1-neg N/A

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

       12. sub-neg N/A

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

       13. lower--.f64 N/A

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

       14. mul-1-neg N/A

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

       15. lower-neg.f64 85.6

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

   11. Applied rewrites 85.6%

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

   if 5 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 2e115

   1. Initial program 93.4%

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

   3. Taylor expanded in 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 63.2

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

   5. Applied rewrites 63.2%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-neg.f64 29.9

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

   8. Applied rewrites 29.9%

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

   9. Taylor expanded in y 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. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 12.2

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

   11. Applied rewrites 12.2%

       \[\leadsto \color{blue}{\left(0.5 \cdot \left(t \cdot t\right)\right) \cdot \left(x \cdot \left(y \cdot y\right)\right)} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*r* N/A

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

       5. lift-*.f64 N/A

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

       6. associate-*r* N/A

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

       7. lower-*.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. associate-*l* N/A

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

       11. lower-*.f64 N/A

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

       12. lower-*.f64 36.2

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

   13. Applied rewrites 36.2%

       \[\leadsto \color{blue}{\left(\left(0.5 \cdot \left(\left(t \cdot t\right) \cdot x\right)\right) \cdot y\right) \cdot y} \]
3. Recombined 4 regimes into one program.

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -\infty:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(t, y, 1\right)}\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -20000000000000:\\ \;\;\;\;t \cdot \left(\left(x \cdot \left(y \cdot y\right)\right) \cdot \left(t \cdot 0.5\right)\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 5:\\ \;\;\;\;\mathsf{fma}\left(x \cdot a, \left(-z\right) - b, x\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 2 \cdot 10^{+115}:\\ \;\;\;\;y \cdot \left(y \cdot \left(0.5 \cdot \left(x \cdot \left(t \cdot t\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\left(x \cdot \left(y \cdot y\right)\right) \cdot \left(t \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t\_1 \leq 10^{-23}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(t, \mathsf{fma}\left(t, \left(y \cdot y\right) \cdot \mathsf{fma}\left(t \cdot 0.16666666666666666, y, 0.5\right), y\right), 1\right)}\\ \mathbf{elif}\;t\_1 \leq 10^{+124}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(-0.16666666666666666 \cdot \left(t \cdot \left(t \cdot t\right)\right), y, 0.5 \cdot \left(t \cdot t\right)\right), -t\right), 1\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+295}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(t, \mathsf{fma}\left(0.5, t \cdot \left(y \cdot y\right), -y\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(0.5 \cdot \left(a \cdot a\right), b, -a\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b)))))
   (if (<= t_1 1e-23)
     (/
      x
      (fma t (fma t (* (* y y) (fma (* t 0.16666666666666666) y 0.5)) y) 1.0))
     (if (<= t_1 1e+124)
       (*
        x
        (fma
         y
         (fma
          y
          (fma (* -0.16666666666666666 (* t (* t t))) y (* 0.5 (* t t)))
          (- t))
         1.0))
       (if (<= t_1 5e+295)
         (* x (fma t (fma 0.5 (* t (* y y)) (- y)) 1.0))
         (* x (fma b (fma (* 0.5 (* a a)) b (- a)) 1.0)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	double tmp;
	if (t_1 <= 1e-23) {
		tmp = x / fma(t, fma(t, ((y * y) * fma((t * 0.16666666666666666), y, 0.5)), y), 1.0);
	} else if (t_1 <= 1e+124) {
		tmp = x * fma(y, fma(y, fma((-0.16666666666666666 * (t * (t * t))), y, (0.5 * (t * t))), -t), 1.0);
	} else if (t_1 <= 5e+295) {
		tmp = x * fma(t, fma(0.5, (t * (y * y)), -y), 1.0);
	} else {
		tmp = x * fma(b, fma((0.5 * (a * a)), b, -a), 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))
	tmp = 0.0
	if (t_1 <= 1e-23)
		tmp = Float64(x / fma(t, fma(t, Float64(Float64(y * y) * fma(Float64(t * 0.16666666666666666), y, 0.5)), y), 1.0));
	elseif (t_1 <= 1e+124)
		tmp = Float64(x * fma(y, fma(y, fma(Float64(-0.16666666666666666 * Float64(t * Float64(t * t))), y, Float64(0.5 * Float64(t * t))), Float64(-t)), 1.0));
	elseif (t_1 <= 5e+295)
		tmp = Float64(x * fma(t, fma(0.5, Float64(t * Float64(y * y)), Float64(-y)), 1.0));
	else
		tmp = Float64(x * fma(b, fma(Float64(0.5 * Float64(a * a)), b, Float64(-a)), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 1e-23], N[(x / N[(t * N[(t * N[(N[(y * y), $MachinePrecision] * N[(N[(t * 0.16666666666666666), $MachinePrecision] * y + 0.5), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+124], N[(x * N[(y * N[(y * N[(N[(-0.16666666666666666 * N[(t * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y + N[(0.5 * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + (-t)), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+295], N[(x * N[(t * N[(0.5 * N[(t * N[(y * y), $MachinePrecision]), $MachinePrecision] + (-y)), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(b * N[(N[(0.5 * N[(a * a), $MachinePrecision]), $MachinePrecision] * b + (-a)), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 9.9999999999999996e-24

   1. Initial program 95.1%

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

   3. Taylor expanded in 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 64.3

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

   5. Applied rewrites 64.3%

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

      1. distribute-rgt-neg-out N/A

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

      2. exp-neg N/A

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

      3. pow-exp N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. pow-exp N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-*.f64 64.2

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

   7. Applied rewrites 64.2%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 65.7%

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

   if 9.9999999999999996e-24 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 9.99999999999999948e123

   1. Initial program 94.6%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   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 63.1

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

   5. Applied rewrites 63.1%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

   8. Applied rewrites 41.2%

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

   if 9.99999999999999948e123 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 4.99999999999999991e295

   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 50.1

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

   5. Applied rewrites 50.1%

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

   6. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 70.7

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

   8. Applied rewrites 70.7%

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

   if 4.99999999999999991e295 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 97.8%

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

   3. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 65.4

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

   5. Applied rewrites 65.4%

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

   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. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 70.0

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

   8. Applied rewrites 70.0%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(0.5 \cdot \left(a \cdot a\right), b, -a\right), 1\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 8: 61.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t\_1 \leq 0.5:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(t, \mathsf{fma}\left(t, \left(y \cdot y\right) \cdot \mathsf{fma}\left(t \cdot 0.16666666666666666, y, 0.5\right), y\right), 1\right)}\\ \mathbf{elif}\;t\_1 \leq 10^{+124}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, y \cdot \left(-0.16666666666666666 \cdot \left(t \cdot \left(t \cdot t\right)\right)\right), -t\right), 1\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+295}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(t, \mathsf{fma}\left(0.5, t \cdot \left(y \cdot y\right), -y\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(0.5 \cdot \left(a \cdot a\right), b, -a\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b)))))
   (if (<= t_1 0.5)
     (/
      x
      (fma t (fma t (* (* y y) (fma (* t 0.16666666666666666) y 0.5)) y) 1.0))
     (if (<= t_1 1e+124)
       (*
        x
        (fma y (fma y (* y (* -0.16666666666666666 (* t (* t t)))) (- t)) 1.0))
       (if (<= t_1 5e+295)
         (* x (fma t (fma 0.5 (* t (* y y)) (- y)) 1.0))
         (* x (fma b (fma (* 0.5 (* a a)) b (- a)) 1.0)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	double tmp;
	if (t_1 <= 0.5) {
		tmp = x / fma(t, fma(t, ((y * y) * fma((t * 0.16666666666666666), y, 0.5)), y), 1.0);
	} else if (t_1 <= 1e+124) {
		tmp = x * fma(y, fma(y, (y * (-0.16666666666666666 * (t * (t * t)))), -t), 1.0);
	} else if (t_1 <= 5e+295) {
		tmp = x * fma(t, fma(0.5, (t * (y * y)), -y), 1.0);
	} else {
		tmp = x * fma(b, fma((0.5 * (a * a)), b, -a), 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))
	tmp = 0.0
	if (t_1 <= 0.5)
		tmp = Float64(x / fma(t, fma(t, Float64(Float64(y * y) * fma(Float64(t * 0.16666666666666666), y, 0.5)), y), 1.0));
	elseif (t_1 <= 1e+124)
		tmp = Float64(x * fma(y, fma(y, Float64(y * Float64(-0.16666666666666666 * Float64(t * Float64(t * t)))), Float64(-t)), 1.0));
	elseif (t_1 <= 5e+295)
		tmp = Float64(x * fma(t, fma(0.5, Float64(t * Float64(y * y)), Float64(-y)), 1.0));
	else
		tmp = Float64(x * fma(b, fma(Float64(0.5 * Float64(a * a)), b, Float64(-a)), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.5], N[(x / N[(t * N[(t * N[(N[(y * y), $MachinePrecision] * N[(N[(t * 0.16666666666666666), $MachinePrecision] * y + 0.5), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+124], N[(x * N[(y * N[(y * N[(y * N[(-0.16666666666666666 * N[(t * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + (-t)), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+295], N[(x * N[(t * N[(0.5 * N[(t * N[(y * y), $MachinePrecision]), $MachinePrecision] + (-y)), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(b * N[(N[(0.5 * N[(a * a), $MachinePrecision]), $MachinePrecision] * b + (-a)), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 0.5

   1. Initial program 95.2%

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

   3. Taylor expanded in 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 64.6

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

   5. Applied rewrites 64.6%

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

      1. distribute-rgt-neg-out N/A

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

      2. exp-neg N/A

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

      3. pow-exp N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. pow-exp N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-*.f64 64.6

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

   7. Applied rewrites 64.6%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 65.7%

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

   if 0.5 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 9.99999999999999948e123

   1. Initial program 94.1%

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

   3. Taylor expanded in 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 61.7

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

   5. Applied rewrites 61.7%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

   8. Applied rewrites 38.9%

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

   9. Taylor expanded in t around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. cube-mult N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 38.8

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

   11. Applied rewrites 38.8%

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

   if 9.99999999999999948e123 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 4.99999999999999991e295

   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 50.1

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

   5. Applied rewrites 50.1%

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

   6. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 70.7

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

   8. Applied rewrites 70.7%

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

   if 4.99999999999999991e295 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 97.8%

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

   3. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 65.4

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

   5. Applied rewrites 65.4%

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

   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. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 70.0

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

   8. Applied rewrites 70.0%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(0.5 \cdot \left(a \cdot a\right), b, -a\right), 1\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 0.5:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(t, \mathsf{fma}\left(t, \left(y \cdot y\right) \cdot \mathsf{fma}\left(t \cdot 0.16666666666666666, y, 0.5\right), y\right), 1\right)}\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 10^{+124}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, y \cdot \left(-0.16666666666666666 \cdot \left(t \cdot \left(t \cdot t\right)\right)\right), -t\right), 1\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 5 \cdot 10^{+295}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(t, \mathsf{fma}\left(0.5, t \cdot \left(y \cdot y\right), -y\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(0.5 \cdot \left(a \cdot a\right), b, -a\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 62.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+161}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(t, \mathsf{fma}\left(t, 0.5 \cdot \left(y \cdot y\right), y\right), 1\right)}\\ \mathbf{elif}\;t\_1 \leq -20000000000000:\\ \;\;\;\;t \cdot \left(\left(x \cdot \left(y \cdot y\right)\right) \cdot \left(t \cdot 0.5\right)\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+295}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(t, \mathsf{fma}\left(0.5, t \cdot \left(y \cdot y\right), -y\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(0.5 \cdot \left(a \cdot a\right), b, -a\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b)))))
   (if (<= t_1 -2e+161)
     (/ x (fma t (fma t (* 0.5 (* y y)) y) 1.0))
     (if (<= t_1 -20000000000000.0)
       (* t (* (* x (* y y)) (* t 0.5)))
       (if (<= t_1 5e+295)
         (* x (fma t (fma 0.5 (* t (* y y)) (- y)) 1.0))
         (* x (fma b (fma (* 0.5 (* a a)) b (- a)) 1.0)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	double tmp;
	if (t_1 <= -2e+161) {
		tmp = x / fma(t, fma(t, (0.5 * (y * y)), y), 1.0);
	} else if (t_1 <= -20000000000000.0) {
		tmp = t * ((x * (y * y)) * (t * 0.5));
	} else if (t_1 <= 5e+295) {
		tmp = x * fma(t, fma(0.5, (t * (y * y)), -y), 1.0);
	} else {
		tmp = x * fma(b, fma((0.5 * (a * a)), b, -a), 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))
	tmp = 0.0
	if (t_1 <= -2e+161)
		tmp = Float64(x / fma(t, fma(t, Float64(0.5 * Float64(y * y)), y), 1.0));
	elseif (t_1 <= -20000000000000.0)
		tmp = Float64(t * Float64(Float64(x * Float64(y * y)) * Float64(t * 0.5)));
	elseif (t_1 <= 5e+295)
		tmp = Float64(x * fma(t, fma(0.5, Float64(t * Float64(y * y)), Float64(-y)), 1.0));
	else
		tmp = Float64(x * fma(b, fma(Float64(0.5 * Float64(a * a)), b, Float64(-a)), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+161], N[(x / N[(t * N[(t * N[(0.5 * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -20000000000000.0], N[(t * N[(N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(t * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+295], N[(x * N[(t * N[(0.5 * N[(t * N[(y * y), $MachinePrecision]), $MachinePrecision] + (-y)), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(b * N[(N[(0.5 * N[(a * a), $MachinePrecision]), $MachinePrecision] * b + (-a)), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -2.0000000000000001e161

   1. Initial program 98.6%

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

   3. Taylor expanded in 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 54.4

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

   5. Applied rewrites 54.4%

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

      1. distribute-rgt-neg-out N/A

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

      2. exp-neg N/A

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

      3. pow-exp N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. pow-exp N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-*.f64 54.4

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

   7. Applied rewrites 54.4%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-+l+ N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

      12. associate-+l+ N/A

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

      13. *-commutative N/A

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

      14. distribute-rgt-in N/A

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

   10. Applied rewrites 64.3%

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

   if -2.0000000000000001e161 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -2e13

   1. Initial program 89.1%

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

   3. Taylor expanded in 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 36.5

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

   5. Applied rewrites 36.5%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-neg.f64 3.8

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

   8. Applied rewrites 3.8%

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

   9. Taylor expanded in y 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. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 29.3

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

   11. Applied rewrites 29.3%

       \[\leadsto \color{blue}{\left(0.5 \cdot \left(t \cdot t\right)\right) \cdot \left(x \cdot \left(y \cdot y\right)\right)} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. associate-*r* N/A

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

       9. associate-*r* N/A

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

       10. lower-*.f64 N/A

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

       11. lower-*.f64 N/A

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

       12. *-commutative N/A

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

       13. lower-*.f64 51.7

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

   13. Applied rewrites 51.7%

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

   if -2e13 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 4.99999999999999991e295

   1. Initial program 95.6%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   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 68.7

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

   5. Applied rewrites 68.7%

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

   6. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 61.1

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

   8. Applied rewrites 61.1%

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

   if 4.99999999999999991e295 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 97.8%

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

   3. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 65.4

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

   5. Applied rewrites 65.4%

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

   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. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 70.0

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

   8. Applied rewrites 70.0%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(0.5 \cdot \left(a \cdot a\right), b, -a\right), 1\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -2 \cdot 10^{+161}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(t, \mathsf{fma}\left(t, 0.5 \cdot \left(y \cdot y\right), y\right), 1\right)}\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -20000000000000:\\ \;\;\;\;t \cdot \left(\left(x \cdot \left(y \cdot y\right)\right) \cdot \left(t \cdot 0.5\right)\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 5 \cdot 10^{+295}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(t, \mathsf{fma}\left(0.5, t \cdot \left(y \cdot y\right), -y\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(0.5 \cdot \left(a \cdot a\right), b, -a\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 53.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(t, y, 1\right)}\\ \mathbf{elif}\;t\_1 \leq -20000000000000:\\ \;\;\;\;t \cdot \left(\left(x \cdot \left(y \cdot y\right)\right) \cdot \left(t \cdot 0.5\right)\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+295}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(t, \mathsf{fma}\left(0.5, t \cdot \left(y \cdot y\right), -y\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(0.5 \cdot \left(a \cdot a\right), b, -a\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b)))))
   (if (<= t_1 (- INFINITY))
     (/ x (fma t y 1.0))
     (if (<= t_1 -20000000000000.0)
       (* t (* (* x (* y y)) (* t 0.5)))
       (if (<= t_1 5e+295)
         (* x (fma t (fma 0.5 (* t (* y y)) (- y)) 1.0))
         (* x (fma b (fma (* 0.5 (* a a)) b (- a)) 1.0)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = x / fma(t, y, 1.0);
	} else if (t_1 <= -20000000000000.0) {
		tmp = t * ((x * (y * y)) * (t * 0.5));
	} else if (t_1 <= 5e+295) {
		tmp = x * fma(t, fma(0.5, (t * (y * y)), -y), 1.0);
	} else {
		tmp = x * fma(b, fma((0.5 * (a * a)), b, -a), 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(x / fma(t, y, 1.0));
	elseif (t_1 <= -20000000000000.0)
		tmp = Float64(t * Float64(Float64(x * Float64(y * y)) * Float64(t * 0.5)));
	elseif (t_1 <= 5e+295)
		tmp = Float64(x * fma(t, fma(0.5, Float64(t * Float64(y * y)), Float64(-y)), 1.0));
	else
		tmp = Float64(x * fma(b, fma(Float64(0.5 * Float64(a * a)), b, Float64(-a)), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(x / N[(t * y + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -20000000000000.0], N[(t * N[(N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(t * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+295], N[(x * N[(t * N[(0.5 * N[(t * N[(y * y), $MachinePrecision]), $MachinePrecision] + (-y)), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(b * N[(N[(0.5 * N[(a * a), $MachinePrecision]), $MachinePrecision] * b + (-a)), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -inf.0

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 69.1

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

   5. Applied rewrites 69.1%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}} \]
   6. Step-by-step derivation

      1. distribute-rgt-neg-out N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(y \cdot t\right)}} \]

      2. exp-neg N/A

         \[\leadsto x \cdot \color{blue}{\frac{1}{e^{y \cdot t}}} \]

      3. pow-exp N/A

         \[\leadsto x \cdot \frac{1}{\color{blue}{{\left(e^{y}\right)}^{t}}} \]

      4. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{x}{{\left(e^{y}\right)}^{t}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{{\left(e^{y}\right)}^{t}}} \]

      6. pow-exp N/A

         \[\leadsto \frac{x}{\color{blue}{e^{y \cdot t}}} \]

      7. lower-exp.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{e^{y \cdot t}}} \]

      8. lower-*.f64 69.1

         \[\leadsto \frac{x}{e^{\color{blue}{y \cdot t}}} \]

   7. Applied rewrites 69.1%

      \[\leadsto \color{blue}{\frac{x}{e^{y \cdot t}}} \]

   8. Taylor expanded in y around 0

      \[\leadsto \frac{x}{\color{blue}{1 + t \cdot y}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{t \cdot y + 1}} \]

      2. lower-fma.f64 63.9

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(t, y, 1\right)}} \]

   10. Applied rewrites 63.9%

       \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(t, y, 1\right)}} \]

   if -inf.0 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -2e13

   1. Initial program 94.5%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   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 39.1

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(-t\right)}} \]

   5. Applied rewrites 39.1%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}} \]

   6. Taylor expanded in y around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + y \cdot \left(-1 \cdot t + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(y \cdot \left(-1 \cdot t + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right)\right) + 1\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y, -1 \cdot t + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right), 1\right)} \]

      3. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{\left(\mathsf{neg}\left(t\right)\right)} + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right), 1\right) \]

      4. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{\frac{1}{2} \cdot \left({t}^{2} \cdot y\right) + \left(\mathsf{neg}\left(t\right)\right)}, 1\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {t}^{2} \cdot y, \mathsf{neg}\left(t\right)\right)}, 1\right) \]

      6. *-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{y \cdot {t}^{2}}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      7. lower-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{y \cdot {t}^{2}}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      8. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, y \cdot \color{blue}{\left(t \cdot t\right)}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      9. lower-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, y \cdot \color{blue}{\left(t \cdot t\right)}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      10. lower-neg.f64 2.8

          \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(0.5, y \cdot \left(t \cdot t\right), \color{blue}{-t}\right), 1\right) \]

   8. Applied rewrites 2.8%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(0.5, y \cdot \left(t \cdot t\right), -t\right), 1\right)} \]

   9. Taylor expanded in y 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. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot {t}^{2}\right) \cdot \left(x \cdot {y}^{2}\right)} \]

       2. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot {t}^{2}\right) \cdot \left(x \cdot {y}^{2}\right)} \]

       3. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot {t}^{2}\right)} \cdot \left(x \cdot {y}^{2}\right) \]

       4. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot \left(x \cdot {y}^{2}\right) \]

       5. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot \left(x \cdot {y}^{2}\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(x \cdot {y}^{2}\right)} \]

       7. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \left(t \cdot t\right)\right) \cdot \left(x \cdot \color{blue}{\left(y \cdot y\right)}\right) \]

       8. lower-*.f64 26.9

          \[\leadsto \left(0.5 \cdot \left(t \cdot t\right)\right) \cdot \left(x \cdot \color{blue}{\left(y \cdot y\right)}\right) \]

   11. Applied rewrites 26.9%

       \[\leadsto \color{blue}{\left(0.5 \cdot \left(t \cdot t\right)\right) \cdot \left(x \cdot \left(y \cdot y\right)\right)} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot \left(x \cdot \left(y \cdot y\right)\right) \]

       2. lift-*.f64 N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \left(t \cdot t\right)\right)} \cdot \left(x \cdot \left(y \cdot y\right)\right) \]

       3. lift-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \left(t \cdot t\right)\right) \cdot \left(x \cdot \color{blue}{\left(y \cdot y\right)}\right) \]

       4. lift-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(x \cdot \left(y \cdot y\right)\right)} \]

       5. *-commutative N/A

          \[\leadsto \color{blue}{\left(x \cdot \left(y \cdot y\right)\right) \cdot \left(\frac{1}{2} \cdot \left(t \cdot t\right)\right)} \]

       6. lift-*.f64 N/A

          \[\leadsto \left(x \cdot \left(y \cdot y\right)\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(t \cdot t\right)\right)} \]

       7. lift-*.f64 N/A

          \[\leadsto \left(x \cdot \left(y \cdot y\right)\right) \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(t \cdot t\right)}\right) \]

       8. associate-*r* N/A

          \[\leadsto \left(x \cdot \left(y \cdot y\right)\right) \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot t\right) \cdot t\right)} \]

       9. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(x \cdot \left(y \cdot y\right)\right) \cdot \left(\frac{1}{2} \cdot t\right)\right) \cdot t} \]

       10. lower-*.f64 N/A

           \[\leadsto \color{blue}{\left(\left(x \cdot \left(y \cdot y\right)\right) \cdot \left(\frac{1}{2} \cdot t\right)\right) \cdot t} \]

       11. lower-*.f64 N/A

           \[\leadsto \color{blue}{\left(\left(x \cdot \left(y \cdot y\right)\right) \cdot \left(\frac{1}{2} \cdot t\right)\right)} \cdot t \]

       12. *-commutative N/A

           \[\leadsto \left(\left(x \cdot \left(y \cdot y\right)\right) \cdot \color{blue}{\left(t \cdot \frac{1}{2}\right)}\right) \cdot t \]

       13. lower-*.f64 43.9

           \[\leadsto \left(\left(x \cdot \left(y \cdot y\right)\right) \cdot \color{blue}{\left(t \cdot 0.5\right)}\right) \cdot t \]

   13. Applied rewrites 43.9%

       \[\leadsto \color{blue}{\left(\left(x \cdot \left(y \cdot y\right)\right) \cdot \left(t \cdot 0.5\right)\right) \cdot t} \]

   if -2e13 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 4.99999999999999991e295

   1. Initial program 95.6%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   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 68.7

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(-t\right)}} \]

   5. Applied rewrites 68.7%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}} \]

   6. Taylor expanded in t around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + t \cdot \left(-1 \cdot y + \frac{1}{2} \cdot \left(t \cdot {y}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(t \cdot \left(-1 \cdot y + \frac{1}{2} \cdot \left(t \cdot {y}^{2}\right)\right) + 1\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(t, -1 \cdot y + \frac{1}{2} \cdot \left(t \cdot {y}^{2}\right), 1\right)} \]

      3. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(t, \color{blue}{\frac{1}{2} \cdot \left(t \cdot {y}^{2}\right) + -1 \cdot y}, 1\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(t, \color{blue}{\mathsf{fma}\left(\frac{1}{2}, t \cdot {y}^{2}, -1 \cdot y\right)}, 1\right) \]

      5. lower-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(t, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{t \cdot {y}^{2}}, -1 \cdot y\right), 1\right) \]

      6. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(t, \mathsf{fma}\left(\frac{1}{2}, t \cdot \color{blue}{\left(y \cdot y\right)}, -1 \cdot y\right), 1\right) \]

      7. lower-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(t, \mathsf{fma}\left(\frac{1}{2}, t \cdot \color{blue}{\left(y \cdot y\right)}, -1 \cdot y\right), 1\right) \]

      8. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(t, \mathsf{fma}\left(\frac{1}{2}, t \cdot \left(y \cdot y\right), \color{blue}{\mathsf{neg}\left(y\right)}\right), 1\right) \]

      9. lower-neg.f64 61.1

         \[\leadsto x \cdot \mathsf{fma}\left(t, \mathsf{fma}\left(0.5, t \cdot \left(y \cdot y\right), \color{blue}{-y}\right), 1\right) \]

   8. Applied rewrites 61.1%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(t, \mathsf{fma}\left(0.5, t \cdot \left(y \cdot y\right), -y\right), 1\right)} \]

   if 4.99999999999999991e295 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 97.8%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      2. lower-neg.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      3. lower-*.f64 65.4

         \[\leadsto x \cdot e^{-\color{blue}{a \cdot b}} \]

   5. Applied rewrites 65.4%

      \[\leadsto x \cdot e^{\color{blue}{-a \cdot b}} \]

   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. associate-*r* N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \color{blue}{\left(\frac{1}{2} \cdot {a}^{2}\right) \cdot b} + -1 \cdot a, 1\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot {a}^{2}, b, -1 \cdot a\right)}, 1\right) \]

      6. lower-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(\color{blue}{\frac{1}{2} \cdot {a}^{2}}, b, -1 \cdot a\right), 1\right) \]

      7. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(\frac{1}{2} \cdot \color{blue}{\left(a \cdot a\right)}, b, -1 \cdot a\right), 1\right) \]

      8. lower-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(\frac{1}{2} \cdot \color{blue}{\left(a \cdot a\right)}, b, -1 \cdot a\right), 1\right) \]

      9. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(\frac{1}{2} \cdot \left(a \cdot a\right), b, \color{blue}{\mathsf{neg}\left(a\right)}\right), 1\right) \]

      10. lower-neg.f64 70.0

          \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(0.5 \cdot \left(a \cdot a\right), b, \color{blue}{-a}\right), 1\right) \]

   8. Applied rewrites 70.0%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(0.5 \cdot \left(a \cdot a\right), b, -a\right), 1\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -\infty:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(t, y, 1\right)}\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -20000000000000:\\ \;\;\;\;t \cdot \left(\left(x \cdot \left(y \cdot y\right)\right) \cdot \left(t \cdot 0.5\right)\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 5 \cdot 10^{+295}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(t, \mathsf{fma}\left(0.5, t \cdot \left(y \cdot y\right), -y\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(0.5 \cdot \left(a \cdot a\right), b, -a\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 53.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ t_2 := x \cdot \left(0.5 \cdot \left(y \cdot \left(y \cdot \left(t \cdot t\right)\right)\right)\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+279}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(t, y, 1\right)}\\ \mathbf{elif}\;t\_1 \leq -20000000000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0.5:\\ \;\;\;\;\mathsf{fma}\left(x \cdot a, \left(-z\right) - b, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b))))
        (t_2 (* x (* 0.5 (* y (* y (* t t)))))))
   (if (<= t_1 -5e+279)
     (/ x (fma t y 1.0))
     (if (<= t_1 -20000000000000.0)
       t_2
       (if (<= t_1 0.5) (fma (* x a) (- (- z) b) x) t_2)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	double t_2 = x * (0.5 * (y * (y * (t * t))));
	double tmp;
	if (t_1 <= -5e+279) {
		tmp = x / fma(t, y, 1.0);
	} else if (t_1 <= -20000000000000.0) {
		tmp = t_2;
	} else if (t_1 <= 0.5) {
		tmp = fma((x * a), (-z - b), x);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))
	t_2 = Float64(x * Float64(0.5 * Float64(y * Float64(y * Float64(t * t)))))
	tmp = 0.0
	if (t_1 <= -5e+279)
		tmp = Float64(x / fma(t, y, 1.0));
	elseif (t_1 <= -20000000000000.0)
		tmp = t_2;
	elseif (t_1 <= 0.5)
		tmp = fma(Float64(x * a), Float64(Float64(-z) - b), x);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x * N[(0.5 * N[(y * N[(y * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+279], N[(x / N[(t * y + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -20000000000000.0], t$95$2, If[LessEqual[t$95$1, 0.5], N[(N[(x * a), $MachinePrecision] * N[((-z) - b), $MachinePrecision] + x), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -5.0000000000000002e279

   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 64.9

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(-t\right)}} \]

   5. Applied rewrites 64.9%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}} \]
   6. Step-by-step derivation

      1. distribute-rgt-neg-out N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(y \cdot t\right)}} \]

      2. exp-neg N/A

         \[\leadsto x \cdot \color{blue}{\frac{1}{e^{y \cdot t}}} \]

      3. pow-exp N/A

         \[\leadsto x \cdot \frac{1}{\color{blue}{{\left(e^{y}\right)}^{t}}} \]

      4. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{x}{{\left(e^{y}\right)}^{t}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{{\left(e^{y}\right)}^{t}}} \]

      6. pow-exp N/A

         \[\leadsto \frac{x}{\color{blue}{e^{y \cdot t}}} \]

      7. lower-exp.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{e^{y \cdot t}}} \]

      8. lower-*.f64 64.9

         \[\leadsto \frac{x}{e^{\color{blue}{y \cdot t}}} \]

   7. Applied rewrites 64.9%

      \[\leadsto \color{blue}{\frac{x}{e^{y \cdot t}}} \]

   8. Taylor expanded in y around 0

      \[\leadsto \frac{x}{\color{blue}{1 + t \cdot y}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{t \cdot y + 1}} \]

      2. lower-fma.f64 56.1

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(t, y, 1\right)}} \]

   10. Applied rewrites 56.1%

       \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(t, y, 1\right)}} \]

   if -5.0000000000000002e279 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -2e13 or 0.5 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 96.5%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   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 52.0

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(-t\right)}} \]

   5. Applied rewrites 52.0%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}} \]

   6. Taylor expanded in y around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + y \cdot \left(-1 \cdot t + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(y \cdot \left(-1 \cdot t + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right)\right) + 1\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y, -1 \cdot t + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right), 1\right)} \]

      3. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{\left(\mathsf{neg}\left(t\right)\right)} + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right), 1\right) \]

      4. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{\frac{1}{2} \cdot \left({t}^{2} \cdot y\right) + \left(\mathsf{neg}\left(t\right)\right)}, 1\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {t}^{2} \cdot y, \mathsf{neg}\left(t\right)\right)}, 1\right) \]

      6. *-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{y \cdot {t}^{2}}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      7. lower-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{y \cdot {t}^{2}}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      8. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, y \cdot \color{blue}{\left(t \cdot t\right)}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      9. lower-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, y \cdot \color{blue}{\left(t \cdot t\right)}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      10. lower-neg.f64 35.4

          \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(0.5, y \cdot \left(t \cdot t\right), \color{blue}{-t}\right), 1\right) \]

   8. Applied rewrites 35.4%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(0.5, y \cdot \left(t \cdot t\right), -t\right), 1\right)} \]

   9. Taylor expanded in y around inf

      \[\leadsto x \cdot \color{blue}{\left(\frac{1}{2} \cdot \left({t}^{2} \cdot {y}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto x \cdot \color{blue}{\left(\frac{1}{2} \cdot \left({t}^{2} \cdot {y}^{2}\right)\right)} \]

       2. unpow2 N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left({t}^{2} \cdot \color{blue}{\left(y \cdot y\right)}\right)\right) \]

       3. associate-*r* N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(\left({t}^{2} \cdot y\right) \cdot y\right)}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(\left({t}^{2} \cdot y\right) \cdot y\right)}\right) \]

       5. lower-*.f64 N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(\color{blue}{\left({t}^{2} \cdot y\right)} \cdot y\right)\right) \]

       6. unpow2 N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(\left(\color{blue}{\left(t \cdot t\right)} \cdot y\right) \cdot y\right)\right) \]

       7. lower-*.f64 46.6

          \[\leadsto x \cdot \left(0.5 \cdot \left(\left(\color{blue}{\left(t \cdot t\right)} \cdot y\right) \cdot y\right)\right) \]

   11. Applied rewrites 46.6%

       \[\leadsto x \cdot \color{blue}{\left(0.5 \cdot \left(\left(\left(t \cdot t\right) \cdot y\right) \cdot y\right)\right)} \]

   if -2e13 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 0.5

   1. Initial program 92.5%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 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 93.5

         \[\leadsto x \cdot e^{a \cdot \left(\mathsf{log1p}\left(\color{blue}{-z}\right) - b\right)} \]

   5. Applied rewrites 93.5%

      \[\leadsto x \cdot e^{\color{blue}{a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)}} \]

   6. Taylor expanded in z around 0

      \[\leadsto x \cdot e^{a \cdot \color{blue}{\left(-1 \cdot z - b\right)}} \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto x \cdot e^{a \cdot \color{blue}{\left(-1 \cdot z - b\right)}} \]

      2. mul-1-neg N/A

         \[\leadsto x \cdot e^{a \cdot \left(\color{blue}{\left(\mathsf{neg}\left(z\right)\right)} - b\right)} \]

      3. lower-neg.f64 93.5

         \[\leadsto x \cdot e^{a \cdot \left(\color{blue}{\left(-z\right)} - b\right)} \]

   8. Applied rewrites 93.5%

      \[\leadsto x \cdot e^{a \cdot \color{blue}{\left(\left(-z\right) - b\right)}} \]

   9. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x + -1 \cdot \left(a \cdot \left(x \cdot \left(b + z\right)\right)\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{-1 \cdot \left(a \cdot \left(x \cdot \left(b + z\right)\right)\right) + x} \]

       2. mul-1-neg N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(a \cdot \left(x \cdot \left(b + z\right)\right)\right)\right)} + x \]

       3. associate-*r* N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(a \cdot x\right) \cdot \left(b + z\right)}\right)\right) + x \]

       4. distribute-rgt-neg-in N/A

          \[\leadsto \color{blue}{\left(a \cdot x\right) \cdot \left(\mathsf{neg}\left(\left(b + z\right)\right)\right)} + x \]

       5. mul-1-neg N/A

          \[\leadsto \left(a \cdot x\right) \cdot \color{blue}{\left(-1 \cdot \left(b + z\right)\right)} + x \]

       6. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot x, -1 \cdot \left(b + z\right), x\right)} \]

       7. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot a}, -1 \cdot \left(b + z\right), x\right) \]

       8. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot a}, -1 \cdot \left(b + z\right), x\right) \]

       9. distribute-lft-in N/A

          \[\leadsto \mathsf{fma}\left(x \cdot a, \color{blue}{-1 \cdot b + -1 \cdot z}, x\right) \]

       10. +-commutative N/A

           \[\leadsto \mathsf{fma}\left(x \cdot a, \color{blue}{-1 \cdot z + -1 \cdot b}, x\right) \]

       11. mul-1-neg N/A

           \[\leadsto \mathsf{fma}\left(x \cdot a, -1 \cdot z + \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}, x\right) \]

       12. sub-neg N/A

           \[\leadsto \mathsf{fma}\left(x \cdot a, \color{blue}{-1 \cdot z - b}, x\right) \]

       13. lower--.f64 N/A

           \[\leadsto \mathsf{fma}\left(x \cdot a, \color{blue}{-1 \cdot z - b}, x\right) \]

       14. mul-1-neg N/A

           \[\leadsto \mathsf{fma}\left(x \cdot a, \color{blue}{\left(\mathsf{neg}\left(z\right)\right)} - b, x\right) \]

       15. lower-neg.f64 87.1

           \[\leadsto \mathsf{fma}\left(x \cdot a, \color{blue}{\left(-z\right)} - b, x\right) \]

   11. Applied rewrites 87.1%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot a, \left(-z\right) - b, x\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 55.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -5 \cdot 10^{+279}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(t, y, 1\right)}\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -20000000000000:\\ \;\;\;\;x \cdot \left(0.5 \cdot \left(y \cdot \left(y \cdot \left(t \cdot t\right)\right)\right)\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 0.5:\\ \;\;\;\;\mathsf{fma}\left(x \cdot a, \left(-z\right) - b, x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 \cdot \left(y \cdot \left(y \cdot \left(t \cdot t\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 39.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+278}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(t, y, 1\right)}\\ \mathbf{elif}\;t\_1 \leq -2000:\\ \;\;\;\;-x \cdot \left(a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - y \cdot t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b)))))
   (if (<= t_1 -4e+278)
     (/ x (fma t y 1.0))
     (if (<= t_1 -2000.0) (- (* x (* a b))) (* x (- 1.0 (* y t)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	double tmp;
	if (t_1 <= -4e+278) {
		tmp = x / fma(t, y, 1.0);
	} else if (t_1 <= -2000.0) {
		tmp = -(x * (a * b));
	} 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 <= -4e+278)
		tmp = Float64(x / fma(t, y, 1.0));
	elseif (t_1 <= -2000.0)
		tmp = Float64(-Float64(x * Float64(a * b)));
	else
		tmp = Float64(x * Float64(1.0 - Float64(y * t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+278], N[(x / N[(t * y + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -2000.0], (-N[(x * N[(a * b), $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))) < -3.99999999999999985e278

   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 63.4

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(-t\right)}} \]

   5. Applied rewrites 63.4%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}} \]
   6. Step-by-step derivation

      1. distribute-rgt-neg-out N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(y \cdot t\right)}} \]

      2. exp-neg N/A

         \[\leadsto x \cdot \color{blue}{\frac{1}{e^{y \cdot t}}} \]

      3. pow-exp N/A

         \[\leadsto x \cdot \frac{1}{\color{blue}{{\left(e^{y}\right)}^{t}}} \]

      4. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{x}{{\left(e^{y}\right)}^{t}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{{\left(e^{y}\right)}^{t}}} \]

      6. pow-exp N/A

         \[\leadsto \frac{x}{\color{blue}{e^{y \cdot t}}} \]

      7. lower-exp.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{e^{y \cdot t}}} \]

      8. lower-*.f64 63.4

         \[\leadsto \frac{x}{e^{\color{blue}{y \cdot t}}} \]

   7. Applied rewrites 63.4%

      \[\leadsto \color{blue}{\frac{x}{e^{y \cdot t}}} \]

   8. Taylor expanded in y around 0

      \[\leadsto \frac{x}{\color{blue}{1 + t \cdot y}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{t \cdot y + 1}} \]

      2. lower-fma.f64 54.8

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(t, y, 1\right)}} \]

   10. Applied rewrites 54.8%

       \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(t, y, 1\right)}} \]

   if -3.99999999999999985e278 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -2e3

   1. Initial program 93.7%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      2. lower-neg.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      3. lower-*.f64 48.2

         \[\leadsto x \cdot e^{-\color{blue}{a \cdot b}} \]

   5. Applied rewrites 48.2%

      \[\leadsto x \cdot e^{\color{blue}{-a \cdot b}} \]

   6. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x + -1 \cdot \left(a \cdot \left(b \cdot x\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{-1 \cdot \left(a \cdot \left(b \cdot x\right)\right) + x} \]

      2. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(a \cdot \left(b \cdot x\right)\right)\right)} + x \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{a \cdot \left(\mathsf{neg}\left(b \cdot x\right)\right)} + x \]

      4. mul-1-neg N/A

         \[\leadsto a \cdot \color{blue}{\left(-1 \cdot \left(b \cdot x\right)\right)} + x \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a, -1 \cdot \left(b \cdot x\right), x\right)} \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\mathsf{neg}\left(b \cdot x\right)}, x\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(a, \mathsf{neg}\left(\color{blue}{x \cdot b}\right), x\right) \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{x \cdot \left(\mathsf{neg}\left(b\right)\right)}, x\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{x \cdot \left(\mathsf{neg}\left(b\right)\right)}, x\right) \]

      10. lower-neg.f64 2.9

          \[\leadsto \mathsf{fma}\left(a, x \cdot \color{blue}{\left(-b\right)}, x\right) \]

   8. Applied rewrites 2.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, x \cdot \left(-b\right), x\right)} \]

   9. Taylor expanded in a around inf

      \[\leadsto \color{blue}{-1 \cdot \left(a \cdot \left(b \cdot x\right)\right)} \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \color{blue}{\mathsf{neg}\left(a \cdot \left(b \cdot x\right)\right)} \]

       2. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(a \cdot \color{blue}{\left(x \cdot b\right)}\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\left(a \cdot x\right) \cdot b}\right) \]

       4. distribute-lft-neg-out N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(a \cdot x\right)\right) \cdot b} \]

       5. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{x \cdot a}\right)\right) \cdot b \]

       6. distribute-rgt-neg-in N/A

          \[\leadsto \color{blue}{\left(x \cdot \left(\mathsf{neg}\left(a\right)\right)\right)} \cdot b \]

       7. mul-1-neg N/A

          \[\leadsto \left(x \cdot \color{blue}{\left(-1 \cdot a\right)}\right) \cdot b \]

       8. associate-*r* N/A

          \[\leadsto \color{blue}{x \cdot \left(\left(-1 \cdot a\right) \cdot b\right)} \]

       9. associate-*r* N/A

          \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \left(a \cdot b\right)\right)} \]

       10. lower-*.f64 N/A

           \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \left(a \cdot b\right)\right)} \]

       11. mul-1-neg N/A

           \[\leadsto x \cdot \color{blue}{\left(\mathsf{neg}\left(a \cdot b\right)\right)} \]

       12. distribute-rgt-neg-in N/A

           \[\leadsto x \cdot \color{blue}{\left(a \cdot \left(\mathsf{neg}\left(b\right)\right)\right)} \]

       13. mul-1-neg N/A

           \[\leadsto x \cdot \left(a \cdot \color{blue}{\left(-1 \cdot b\right)}\right) \]

       14. lower-*.f64 N/A

           \[\leadsto x \cdot \color{blue}{\left(a \cdot \left(-1 \cdot b\right)\right)} \]

       15. mul-1-neg N/A

           \[\leadsto x \cdot \left(a \cdot \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}\right) \]

       16. lower-neg.f64 17.4

           \[\leadsto x \cdot \left(a \cdot \color{blue}{\left(-b\right)}\right) \]

   11. Applied rewrites 17.4%

       \[\leadsto \color{blue}{x \cdot \left(a \cdot \left(-b\right)\right)} \]

   if -2e3 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 96.1%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in 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 66.4

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(-t\right)}} \]

   5. Applied rewrites 66.4%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}} \]

   6. Taylor expanded in y around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + -1 \cdot \left(t \cdot y\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot \left(1 + \color{blue}{\left(\mathsf{neg}\left(t \cdot y\right)\right)}\right) \]

      2. unsub-neg N/A

         \[\leadsto x \cdot \color{blue}{\left(1 - t \cdot y\right)} \]

      3. 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 47.3

         \[\leadsto x \cdot \left(1 - \color{blue}{y \cdot t}\right) \]

   8. Applied rewrites 47.3%

      \[\leadsto x \cdot \color{blue}{\left(1 - y \cdot t\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 43.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -4 \cdot 10^{+278}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(t, y, 1\right)}\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -2000:\\ \;\;\;\;-x \cdot \left(a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - y \cdot t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 33.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -x \cdot \left(a \cdot b\right)\\ t_2 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t\_2 \leq -2000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+74}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- (* x (* a b))))
        (t_2 (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b)))))
   (if (<= t_2 -2000.0) t_1 (if (<= t_2 5e+74) x t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = -(x * (a * b));
	double t_2 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	double tmp;
	if (t_2 <= -2000.0) {
		tmp = t_1;
	} else if (t_2 <= 5e+74) {
		tmp = x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = -(x * (a * b))
    t_2 = (y * (log(z) - t)) + (a * (log((1.0d0 - z)) - b))
    if (t_2 <= (-2000.0d0)) then
        tmp = t_1
    else if (t_2 <= 5d+74) then
        tmp = x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = -(x * (a * b));
	double t_2 = (y * (Math.log(z) - t)) + (a * (Math.log((1.0 - z)) - b));
	double tmp;
	if (t_2 <= -2000.0) {
		tmp = t_1;
	} else if (t_2 <= 5e+74) {
		tmp = x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = -(x * (a * b))
	t_2 = (y * (math.log(z) - t)) + (a * (math.log((1.0 - z)) - b))
	tmp = 0
	if t_2 <= -2000.0:
		tmp = t_1
	elif t_2 <= 5e+74:
		tmp = x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(-Float64(x * Float64(a * b)))
	t_2 = Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))
	tmp = 0.0
	if (t_2 <= -2000.0)
		tmp = t_1;
	elseif (t_2 <= 5e+74)
		tmp = x;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = -(x * (a * b));
	t_2 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	tmp = 0.0;
	if (t_2 <= -2000.0)
		tmp = t_1;
	elseif (t_2 <= 5e+74)
		tmp = x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = (-N[(x * N[(a * b), $MachinePrecision]), $MachinePrecision])}, Block[{t$95$2 = N[(N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2000.0], t$95$1, If[LessEqual[t$95$2, 5e+74], 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))) < -2e3 or 4.99999999999999963e74 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 97.9%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      2. lower-neg.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      3. lower-*.f64 54.7

         \[\leadsto x \cdot e^{-\color{blue}{a \cdot b}} \]

   5. Applied rewrites 54.7%

      \[\leadsto x \cdot e^{\color{blue}{-a \cdot b}} \]

   6. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x + -1 \cdot \left(a \cdot \left(b \cdot x\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{-1 \cdot \left(a \cdot \left(b \cdot x\right)\right) + x} \]

      2. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(a \cdot \left(b \cdot x\right)\right)\right)} + x \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{a \cdot \left(\mathsf{neg}\left(b \cdot x\right)\right)} + x \]

      4. mul-1-neg N/A

         \[\leadsto a \cdot \color{blue}{\left(-1 \cdot \left(b \cdot x\right)\right)} + x \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a, -1 \cdot \left(b \cdot x\right), x\right)} \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\mathsf{neg}\left(b \cdot x\right)}, x\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(a, \mathsf{neg}\left(\color{blue}{x \cdot b}\right), x\right) \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{x \cdot \left(\mathsf{neg}\left(b\right)\right)}, x\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{x \cdot \left(\mathsf{neg}\left(b\right)\right)}, x\right) \]

      10. lower-neg.f64 13.5

          \[\leadsto \mathsf{fma}\left(a, x \cdot \color{blue}{\left(-b\right)}, x\right) \]

   8. Applied rewrites 13.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, x \cdot \left(-b\right), x\right)} \]

   9. Taylor expanded in a around inf

      \[\leadsto \color{blue}{-1 \cdot \left(a \cdot \left(b \cdot x\right)\right)} \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \color{blue}{\mathsf{neg}\left(a \cdot \left(b \cdot x\right)\right)} \]

       2. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(a \cdot \color{blue}{\left(x \cdot b\right)}\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\left(a \cdot x\right) \cdot b}\right) \]

       4. distribute-lft-neg-out N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(a \cdot x\right)\right) \cdot b} \]

       5. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{x \cdot a}\right)\right) \cdot b \]

       6. distribute-rgt-neg-in N/A

          \[\leadsto \color{blue}{\left(x \cdot \left(\mathsf{neg}\left(a\right)\right)\right)} \cdot b \]

       7. mul-1-neg N/A

          \[\leadsto \left(x \cdot \color{blue}{\left(-1 \cdot a\right)}\right) \cdot b \]

       8. associate-*r* N/A

          \[\leadsto \color{blue}{x \cdot \left(\left(-1 \cdot a\right) \cdot b\right)} \]

       9. associate-*r* N/A

          \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \left(a \cdot b\right)\right)} \]

       10. lower-*.f64 N/A

           \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \left(a \cdot b\right)\right)} \]

       11. mul-1-neg N/A

           \[\leadsto x \cdot \color{blue}{\left(\mathsf{neg}\left(a \cdot b\right)\right)} \]

       12. distribute-rgt-neg-in N/A

           \[\leadsto x \cdot \color{blue}{\left(a \cdot \left(\mathsf{neg}\left(b\right)\right)\right)} \]

       13. mul-1-neg N/A

           \[\leadsto x \cdot \left(a \cdot \color{blue}{\left(-1 \cdot b\right)}\right) \]

       14. lower-*.f64 N/A

           \[\leadsto x \cdot \color{blue}{\left(a \cdot \left(-1 \cdot b\right)\right)} \]

       15. mul-1-neg N/A

           \[\leadsto x \cdot \left(a \cdot \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}\right) \]

       16. lower-neg.f64 25.1

           \[\leadsto x \cdot \left(a \cdot \color{blue}{\left(-b\right)}\right) \]

   11. Applied rewrites 25.1%

       \[\leadsto \color{blue}{x \cdot \left(a \cdot \left(-b\right)\right)} \]

   if -2e3 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 4.99999999999999963e74

   1. Initial program 92.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   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. Applied rewrites 87.5%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\log z - t\right)}} \]

   6. Taylor expanded in y around 0

      \[\leadsto x \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 60.4%

      \[\leadsto x \cdot \color{blue}{1} \]
   9. Step-by-step derivation

      1. *-rgt-identity 60.4

         \[\leadsto \color{blue}{x} \]

   10. Applied rewrites 60.4%

       \[\leadsto \color{blue}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -2000:\\ \;\;\;\;-x \cdot \left(a \cdot b\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 5 \cdot 10^{+74}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-x \cdot \left(a \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 34.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -2000:\\ \;\;\;\;-x \cdot \left(a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b))) -2000.0)
   (- (* x (* a b)))
   (* x (- 1.0 (* a b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((y * (log(z) - t)) + (a * (log((1.0 - z)) - b))) <= -2000.0) {
		tmp = -(x * (a * b));
	} else {
		tmp = x * (1.0 - (a * b));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (((y * (log(z) - t)) + (a * (log((1.0d0 - z)) - b))) <= (-2000.0d0)) then
        tmp = -(x * (a * b))
    else
        tmp = x * (1.0d0 - (a * b))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((y * (Math.log(z) - t)) + (a * (Math.log((1.0 - z)) - b))) <= -2000.0) {
		tmp = -(x * (a * b));
	} else {
		tmp = x * (1.0 - (a * b));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if ((y * (math.log(z) - t)) + (a * (math.log((1.0 - z)) - b))) <= -2000.0:
		tmp = -(x * (a * b))
	else:
		tmp = x * (1.0 - (a * b))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b))) <= -2000.0)
		tmp = Float64(-Float64(x * Float64(a * b)));
	else
		tmp = Float64(x * Float64(1.0 - Float64(a * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (((y * (log(z) - t)) + (a * (log((1.0 - z)) - b))) <= -2000.0)
		tmp = -(x * (a * b));
	else
		tmp = x * (1.0 - (a * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2000.0], (-N[(x * N[(a * b), $MachinePrecision]), $MachinePrecision]), N[(x * N[(1.0 - N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -2e3

   1. Initial program 96.7%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in 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 53.1

         \[\leadsto x \cdot e^{-\color{blue}{a \cdot b}} \]

   5. Applied rewrites 53.1%

      \[\leadsto x \cdot e^{\color{blue}{-a \cdot b}} \]

   6. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x + -1 \cdot \left(a \cdot \left(b \cdot x\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{-1 \cdot \left(a \cdot \left(b \cdot x\right)\right) + x} \]

      2. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(a \cdot \left(b \cdot x\right)\right)\right)} + x \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{a \cdot \left(\mathsf{neg}\left(b \cdot x\right)\right)} + x \]

      4. mul-1-neg N/A

         \[\leadsto a \cdot \color{blue}{\left(-1 \cdot \left(b \cdot x\right)\right)} + x \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a, -1 \cdot \left(b \cdot x\right), x\right)} \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\mathsf{neg}\left(b \cdot x\right)}, x\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(a, \mathsf{neg}\left(\color{blue}{x \cdot b}\right), x\right) \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{x \cdot \left(\mathsf{neg}\left(b\right)\right)}, x\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{x \cdot \left(\mathsf{neg}\left(b\right)\right)}, x\right) \]

      10. lower-neg.f64 2.8

          \[\leadsto \mathsf{fma}\left(a, x \cdot \color{blue}{\left(-b\right)}, x\right) \]

   8. Applied rewrites 2.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, x \cdot \left(-b\right), x\right)} \]

   9. Taylor expanded in a around inf

      \[\leadsto \color{blue}{-1 \cdot \left(a \cdot \left(b \cdot x\right)\right)} \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \color{blue}{\mathsf{neg}\left(a \cdot \left(b \cdot x\right)\right)} \]

       2. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(a \cdot \color{blue}{\left(x \cdot b\right)}\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\left(a \cdot x\right) \cdot b}\right) \]

       4. distribute-lft-neg-out N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(a \cdot x\right)\right) \cdot b} \]

       5. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{x \cdot a}\right)\right) \cdot b \]

       6. distribute-rgt-neg-in N/A

          \[\leadsto \color{blue}{\left(x \cdot \left(\mathsf{neg}\left(a\right)\right)\right)} \cdot b \]

       7. mul-1-neg N/A

          \[\leadsto \left(x \cdot \color{blue}{\left(-1 \cdot a\right)}\right) \cdot b \]

       8. associate-*r* N/A

          \[\leadsto \color{blue}{x \cdot \left(\left(-1 \cdot a\right) \cdot b\right)} \]

       9. associate-*r* N/A

          \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \left(a \cdot b\right)\right)} \]

       10. lower-*.f64 N/A

           \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \left(a \cdot b\right)\right)} \]

       11. mul-1-neg N/A

           \[\leadsto x \cdot \color{blue}{\left(\mathsf{neg}\left(a \cdot b\right)\right)} \]

       12. distribute-rgt-neg-in N/A

           \[\leadsto x \cdot \color{blue}{\left(a \cdot \left(\mathsf{neg}\left(b\right)\right)\right)} \]

       13. mul-1-neg N/A

           \[\leadsto x \cdot \left(a \cdot \color{blue}{\left(-1 \cdot b\right)}\right) \]

       14. lower-*.f64 N/A

           \[\leadsto x \cdot \color{blue}{\left(a \cdot \left(-1 \cdot b\right)\right)} \]

       15. mul-1-neg N/A

           \[\leadsto x \cdot \left(a \cdot \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}\right) \]

       16. lower-neg.f64 16.5

           \[\leadsto x \cdot \left(a \cdot \color{blue}{\left(-b\right)}\right) \]

   11. Applied rewrites 16.5%

       \[\leadsto \color{blue}{x \cdot \left(a \cdot \left(-b\right)\right)} \]

   if -2e3 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 96.1%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      2. lower-neg.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      3. lower-*.f64 60.8

         \[\leadsto x \cdot e^{-\color{blue}{a \cdot b}} \]

   5. Applied rewrites 60.8%

      \[\leadsto x \cdot e^{\color{blue}{-a \cdot b}} \]

   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 44.5

         \[\leadsto x \cdot \left(1 - \color{blue}{a \cdot b}\right) \]

   8. Applied rewrites 44.5%

      \[\leadsto x \cdot \color{blue}{\left(1 - a \cdot b\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -2000:\\ \;\;\;\;-x \cdot \left(a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 82.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot e^{a \cdot \left(\left(-z\right) - b\right)}\\ \mathbf{if}\;a \leq -2 \cdot 10^{-38}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 1.06 \cdot 10^{+142}:\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (exp (* a (- (- z) b))))))
   (if (<= a -2e-38)
     t_1
     (if (<= a 1.06e+142) (* x (exp (* y (- (log z) t)))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * exp((a * (-z - b)));
	double tmp;
	if (a <= -2e-38) {
		tmp = t_1;
	} else if (a <= 1.06e+142) {
		tmp = x * exp((y * (log(z) - t)));
	} 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((a * (-z - b)))
    if (a <= (-2d-38)) then
        tmp = t_1
    else if (a <= 1.06d+142) then
        tmp = x * exp((y * (log(z) - t)))
    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((a * (-z - b)));
	double tmp;
	if (a <= -2e-38) {
		tmp = t_1;
	} else if (a <= 1.06e+142) {
		tmp = x * Math.exp((y * (Math.log(z) - t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x * math.exp((a * (-z - b)))
	tmp = 0
	if a <= -2e-38:
		tmp = t_1
	elif a <= 1.06e+142:
		tmp = x * math.exp((y * (math.log(z) - t)))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x * exp(Float64(a * Float64(Float64(-z) - b))))
	tmp = 0.0
	if (a <= -2e-38)
		tmp = t_1;
	elseif (a <= 1.06e+142)
		tmp = Float64(x * exp(Float64(y * Float64(log(z) - t))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x * exp((a * (-z - b)));
	tmp = 0.0;
	if (a <= -2e-38)
		tmp = t_1;
	elseif (a <= 1.06e+142)
		tmp = x * exp((y * (log(z) - t)));
	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[(a * N[((-z) - b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -2e-38], t$95$1, If[LessEqual[a, 1.06e+142], N[(x * N[Exp[N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.9999999999999999e-38 or 1.06e142 < a

   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 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 88.9

         \[\leadsto x \cdot e^{a \cdot \left(\mathsf{log1p}\left(\color{blue}{-z}\right) - b\right)} \]

   5. Applied rewrites 88.9%

      \[\leadsto x \cdot e^{\color{blue}{a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)}} \]

   6. Taylor expanded in z around 0

      \[\leadsto x \cdot e^{a \cdot \color{blue}{\left(-1 \cdot z - b\right)}} \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto x \cdot e^{a \cdot \color{blue}{\left(-1 \cdot z - b\right)}} \]

      2. mul-1-neg N/A

         \[\leadsto x \cdot e^{a \cdot \left(\color{blue}{\left(\mathsf{neg}\left(z\right)\right)} - b\right)} \]

      3. lower-neg.f64 88.9

         \[\leadsto x \cdot e^{a \cdot \left(\color{blue}{\left(-z\right)} - b\right)} \]

   8. Applied rewrites 88.9%

      \[\leadsto x \cdot e^{a \cdot \color{blue}{\left(\left(-z\right) - b\right)}} \]

   if -1.9999999999999999e-38 < a < 1.06e142

   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 89.6

         \[\leadsto x \cdot e^{y \cdot \left(\color{blue}{\log z} - t\right)} \]

   5. Applied rewrites 89.6%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\log z - t\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 74.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot e^{a \cdot \left(\left(-z\right) - b\right)}\\ \mathbf{if}\;a \leq -7.8 \cdot 10^{-47}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 900000:\\ \;\;\;\;\frac{x}{e^{y \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (exp (* a (- (- z) b))))))
   (if (<= a -7.8e-47) t_1 (if (<= a 900000.0) (/ x (exp (* y t))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * exp((a * (-z - b)));
	double tmp;
	if (a <= -7.8e-47) {
		tmp = t_1;
	} else if (a <= 900000.0) {
		tmp = x / exp((y * t));
	} 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((a * (-z - b)))
    if (a <= (-7.8d-47)) then
        tmp = t_1
    else if (a <= 900000.0d0) then
        tmp = x / exp((y * t))
    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((a * (-z - b)));
	double tmp;
	if (a <= -7.8e-47) {
		tmp = t_1;
	} else if (a <= 900000.0) {
		tmp = x / Math.exp((y * t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x * math.exp((a * (-z - b)))
	tmp = 0
	if a <= -7.8e-47:
		tmp = t_1
	elif a <= 900000.0:
		tmp = x / math.exp((y * t))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x * exp(Float64(a * Float64(Float64(-z) - b))))
	tmp = 0.0
	if (a <= -7.8e-47)
		tmp = t_1;
	elseif (a <= 900000.0)
		tmp = Float64(x / exp(Float64(y * t)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x * exp((a * (-z - b)));
	tmp = 0.0;
	if (a <= -7.8e-47)
		tmp = t_1;
	elseif (a <= 900000.0)
		tmp = x / exp((y * t));
	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[(a * N[((-z) - b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -7.8e-47], t$95$1, If[LessEqual[a, 900000.0], N[(x / N[Exp[N[(y * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -7.79999999999999956e-47 or 9e5 < a

   1. Initial program 93.2%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto x \cdot e^{\color{blue}{a \cdot \left(\log \left(1 - z\right) - b\right)}} \]
   4. Step-by-step derivation

      1. 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 81.6

         \[\leadsto x \cdot e^{a \cdot \left(\mathsf{log1p}\left(\color{blue}{-z}\right) - b\right)} \]

   5. Applied rewrites 81.6%

      \[\leadsto x \cdot e^{\color{blue}{a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)}} \]

   6. Taylor expanded in z around 0

      \[\leadsto x \cdot e^{a \cdot \color{blue}{\left(-1 \cdot z - b\right)}} \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto x \cdot e^{a \cdot \color{blue}{\left(-1 \cdot z - b\right)}} \]

      2. mul-1-neg N/A

         \[\leadsto x \cdot e^{a \cdot \left(\color{blue}{\left(\mathsf{neg}\left(z\right)\right)} - b\right)} \]

      3. lower-neg.f64 81.6

         \[\leadsto x \cdot e^{a \cdot \left(\color{blue}{\left(-z\right)} - b\right)} \]

   8. Applied rewrites 81.6%

      \[\leadsto x \cdot e^{a \cdot \color{blue}{\left(\left(-z\right) - b\right)}} \]

   if -7.79999999999999956e-47 < a < 9e5

   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 81.1

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(-t\right)}} \]

   5. Applied rewrites 81.1%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}} \]
   6. Step-by-step derivation

      1. distribute-rgt-neg-out N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(y \cdot t\right)}} \]

      2. exp-neg N/A

         \[\leadsto x \cdot \color{blue}{\frac{1}{e^{y \cdot t}}} \]

      3. pow-exp N/A

         \[\leadsto x \cdot \frac{1}{\color{blue}{{\left(e^{y}\right)}^{t}}} \]

      4. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{x}{{\left(e^{y}\right)}^{t}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{{\left(e^{y}\right)}^{t}}} \]

      6. pow-exp N/A

         \[\leadsto \frac{x}{\color{blue}{e^{y \cdot t}}} \]

      7. lower-exp.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{e^{y \cdot t}}} \]

      8. lower-*.f64 81.1

         \[\leadsto \frac{x}{e^{\color{blue}{y \cdot t}}} \]

   7. Applied rewrites 81.1%

      \[\leadsto \color{blue}{\frac{x}{e^{y \cdot t}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 74.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot e^{a \cdot \left(\left(-z\right) - b\right)}\\ \mathbf{if}\;a \leq -7.8 \cdot 10^{-47}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 900000:\\ \;\;\;\;x \cdot e^{-y \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (exp (* a (- (- z) b))))))
   (if (<= a -7.8e-47) t_1 (if (<= a 900000.0) (* x (exp (- (* y t)))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * exp((a * (-z - b)));
	double tmp;
	if (a <= -7.8e-47) {
		tmp = t_1;
	} else if (a <= 900000.0) {
		tmp = x * exp(-(y * t));
	} 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((a * (-z - b)))
    if (a <= (-7.8d-47)) then
        tmp = t_1
    else if (a <= 900000.0d0) then
        tmp = x * exp(-(y * t))
    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((a * (-z - b)));
	double tmp;
	if (a <= -7.8e-47) {
		tmp = t_1;
	} else if (a <= 900000.0) {
		tmp = x * Math.exp(-(y * t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x * math.exp((a * (-z - b)))
	tmp = 0
	if a <= -7.8e-47:
		tmp = t_1
	elif a <= 900000.0:
		tmp = x * math.exp(-(y * t))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x * exp(Float64(a * Float64(Float64(-z) - b))))
	tmp = 0.0
	if (a <= -7.8e-47)
		tmp = t_1;
	elseif (a <= 900000.0)
		tmp = Float64(x * exp(Float64(-Float64(y * t))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x * exp((a * (-z - b)));
	tmp = 0.0;
	if (a <= -7.8e-47)
		tmp = t_1;
	elseif (a <= 900000.0)
		tmp = x * exp(-(y * t));
	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[(a * N[((-z) - b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -7.8e-47], t$95$1, If[LessEqual[a, 900000.0], N[(x * N[Exp[(-N[(y * t), $MachinePrecision])], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -7.79999999999999956e-47 or 9e5 < a

   1. Initial program 93.2%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto x \cdot e^{\color{blue}{a \cdot \left(\log \left(1 - z\right) - b\right)}} \]
   4. Step-by-step derivation

      1. 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 81.6

         \[\leadsto x \cdot e^{a \cdot \left(\mathsf{log1p}\left(\color{blue}{-z}\right) - b\right)} \]

   5. Applied rewrites 81.6%

      \[\leadsto x \cdot e^{\color{blue}{a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)}} \]

   6. Taylor expanded in z around 0

      \[\leadsto x \cdot e^{a \cdot \color{blue}{\left(-1 \cdot z - b\right)}} \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto x \cdot e^{a \cdot \color{blue}{\left(-1 \cdot z - b\right)}} \]

      2. mul-1-neg N/A

         \[\leadsto x \cdot e^{a \cdot \left(\color{blue}{\left(\mathsf{neg}\left(z\right)\right)} - b\right)} \]

      3. lower-neg.f64 81.6

         \[\leadsto x \cdot e^{a \cdot \left(\color{blue}{\left(-z\right)} - b\right)} \]

   8. Applied rewrites 81.6%

      \[\leadsto x \cdot e^{a \cdot \color{blue}{\left(\left(-z\right) - b\right)}} \]

   if -7.79999999999999956e-47 < a < 9e5

   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 81.1

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(-t\right)}} \]

   5. Applied rewrites 81.1%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -7.8 \cdot 10^{-47}:\\ \;\;\;\;x \cdot e^{a \cdot \left(\left(-z\right) - b\right)}\\ \mathbf{elif}\;a \leq 900000:\\ \;\;\;\;x \cdot e^{-y \cdot t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(\left(-z\right) - b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 72.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot e^{-y \cdot t}\\ \mathbf{if}\;t \leq -1060000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 8 \cdot 10^{+39}:\\ \;\;\;\;x \cdot e^{-a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (exp (- (* y t))))))
   (if (<= t -1060000000.0)
     t_1
     (if (<= t 8e+39) (* 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 * exp(-(y * t));
	double tmp;
	if (t <= -1060000000.0) {
		tmp = t_1;
	} else if (t <= 8e+39) {
		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 * exp(-(y * t))
    if (t <= (-1060000000.0d0)) then
        tmp = t_1
    else if (t <= 8d+39) 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.exp(-(y * t));
	double tmp;
	if (t <= -1060000000.0) {
		tmp = t_1;
	} else if (t <= 8e+39) {
		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.exp(-(y * t))
	tmp = 0
	if t <= -1060000000.0:
		tmp = t_1
	elif t <= 8e+39:
		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 * exp(Float64(-Float64(y * t))))
	tmp = 0.0
	if (t <= -1060000000.0)
		tmp = t_1;
	elseif (t <= 8e+39)
		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 * exp(-(y * t));
	tmp = 0.0;
	if (t <= -1060000000.0)
		tmp = t_1;
	elseif (t <= 8e+39)
		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[Exp[(-N[(y * t), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1060000000.0], t$95$1, If[LessEqual[t, 8e+39], N[(x * N[Exp[(-N[(a * b), $MachinePrecision])], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.06e9 or 7.99999999999999952e39 < t

   1. Initial program 98.4%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in 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 84.6

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(-t\right)}} \]

   5. Applied rewrites 84.6%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}} \]

   if -1.06e9 < t < 7.99999999999999952e39

   1. Initial program 94.2%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      2. lower-neg.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      3. lower-*.f64 69.5

         \[\leadsto x \cdot e^{-\color{blue}{a \cdot b}} \]

   5. Applied rewrites 69.5%

      \[\leadsto x \cdot e^{\color{blue}{-a \cdot b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1060000000:\\ \;\;\;\;x \cdot e^{-y \cdot t}\\ \mathbf{elif}\;t \leq 8 \cdot 10^{+39}:\\ \;\;\;\;x \cdot e^{-a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{-y \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 68.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.35 \cdot 10^{+68}:\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{+25}:\\ \;\;\;\;x \cdot e^{-a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 \cdot \left(y \cdot \left(y \cdot \left(t \cdot t\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -1.35e+68)
   (* x (pow z y))
   (if (<= y 1.4e+25)
     (* x (exp (- (* a b))))
     (* x (* 0.5 (* y (* y (* t t))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -1.35e+68) {
		tmp = x * pow(z, y);
	} else if (y <= 1.4e+25) {
		tmp = x * exp(-(a * b));
	} else {
		tmp = x * (0.5 * (y * (y * (t * t))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (y <= (-1.35d+68)) then
        tmp = x * (z ** y)
    else if (y <= 1.4d+25) then
        tmp = x * exp(-(a * b))
    else
        tmp = x * (0.5d0 * (y * (y * (t * t))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -1.35e+68) {
		tmp = x * Math.pow(z, y);
	} else if (y <= 1.4e+25) {
		tmp = x * Math.exp(-(a * b));
	} else {
		tmp = x * (0.5 * (y * (y * (t * t))));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -1.35e+68:
		tmp = x * math.pow(z, y)
	elif y <= 1.4e+25:
		tmp = x * math.exp(-(a * b))
	else:
		tmp = x * (0.5 * (y * (y * (t * t))))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -1.35e+68)
		tmp = Float64(x * (z ^ y));
	elseif (y <= 1.4e+25)
		tmp = Float64(x * exp(Float64(-Float64(a * b))));
	else
		tmp = Float64(x * Float64(0.5 * Float64(y * Float64(y * Float64(t * t)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -1.35e+68)
		tmp = x * (z ^ y);
	elseif (y <= 1.4e+25)
		tmp = x * exp(-(a * b));
	else
		tmp = x * (0.5 * (y * (y * (t * t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -1.35e+68], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.4e+25], N[(x * N[Exp[(-N[(a * b), $MachinePrecision])], $MachinePrecision]), $MachinePrecision], N[(x * N[(0.5 * N[(y * N[(y * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.34999999999999995e68

   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 92.0

         \[\leadsto x \cdot e^{y \cdot \left(\color{blue}{\log z} - t\right)} \]

   5. Applied rewrites 92.0%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\log z - t\right)}} \]

   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 73.9

         \[\leadsto x \cdot \color{blue}{{z}^{y}} \]

   8. Applied rewrites 73.9%

      \[\leadsto \color{blue}{x \cdot {z}^{y}} \]

   if -1.34999999999999995e68 < y < 1.4000000000000001e25

   1. Initial program 94.9%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   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 75.7

         \[\leadsto x \cdot e^{-\color{blue}{a \cdot b}} \]

   5. Applied rewrites 75.7%

      \[\leadsto x \cdot e^{\color{blue}{-a \cdot b}} \]

   if 1.4000000000000001e25 < y

   1. Initial program 96.7%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(t \cdot y\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(t \cdot y\right)}} \]

      2. *-commutative N/A

         \[\leadsto x \cdot e^{\mathsf{neg}\left(\color{blue}{y \cdot t}\right)} \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      5. lower-neg.f64 71.3

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(-t\right)}} \]

   5. Applied rewrites 71.3%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}} \]

   6. Taylor expanded in y around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + y \cdot \left(-1 \cdot t + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(y \cdot \left(-1 \cdot t + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right)\right) + 1\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y, -1 \cdot t + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right), 1\right)} \]

      3. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{\left(\mathsf{neg}\left(t\right)\right)} + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right), 1\right) \]

      4. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{\frac{1}{2} \cdot \left({t}^{2} \cdot y\right) + \left(\mathsf{neg}\left(t\right)\right)}, 1\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {t}^{2} \cdot y, \mathsf{neg}\left(t\right)\right)}, 1\right) \]

      6. *-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{y \cdot {t}^{2}}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      7. lower-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{y \cdot {t}^{2}}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      8. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, y \cdot \color{blue}{\left(t \cdot t\right)}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      9. lower-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, y \cdot \color{blue}{\left(t \cdot t\right)}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      10. lower-neg.f64 44.1

          \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(0.5, y \cdot \left(t \cdot t\right), \color{blue}{-t}\right), 1\right) \]

   8. Applied rewrites 44.1%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(0.5, y \cdot \left(t \cdot t\right), -t\right), 1\right)} \]

   9. Taylor expanded in y around inf

      \[\leadsto x \cdot \color{blue}{\left(\frac{1}{2} \cdot \left({t}^{2} \cdot {y}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto x \cdot \color{blue}{\left(\frac{1}{2} \cdot \left({t}^{2} \cdot {y}^{2}\right)\right)} \]

       2. unpow2 N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left({t}^{2} \cdot \color{blue}{\left(y \cdot y\right)}\right)\right) \]

       3. associate-*r* N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(\left({t}^{2} \cdot y\right) \cdot y\right)}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(\left({t}^{2} \cdot y\right) \cdot y\right)}\right) \]

       5. lower-*.f64 N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(\color{blue}{\left({t}^{2} \cdot y\right)} \cdot y\right)\right) \]

       6. unpow2 N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(\left(\color{blue}{\left(t \cdot t\right)} \cdot y\right) \cdot y\right)\right) \]

       7. lower-*.f64 61.5

          \[\leadsto x \cdot \left(0.5 \cdot \left(\left(\color{blue}{\left(t \cdot t\right)} \cdot y\right) \cdot y\right)\right) \]

   11. Applied rewrites 61.5%

       \[\leadsto x \cdot \color{blue}{\left(0.5 \cdot \left(\left(\left(t \cdot t\right) \cdot y\right) \cdot y\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.35 \cdot 10^{+68}:\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{+25}:\\ \;\;\;\;x \cdot e^{-a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 \cdot \left(y \cdot \left(y \cdot \left(t \cdot t\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 32.0% accurate, 12.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \left(1 - a \cdot b\right)\\ \mathbf{if}\;b \leq -1.95 \cdot 10^{+99}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{+85}:\\ \;\;\;\;x \cdot \left(1 - y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (- 1.0 (* a b)))))
   (if (<= b -1.95e+99) t_1 (if (<= b 8.5e+85) (* x (- 1.0 (* y t))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * (1.0 - (a * b));
	double tmp;
	if (b <= -1.95e+99) {
		tmp = t_1;
	} else if (b <= 8.5e+85) {
		tmp = x * (1.0 - (y * t));
	} 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 * (1.0d0 - (a * b))
    if (b <= (-1.95d+99)) then
        tmp = t_1
    else if (b <= 8.5d+85) then
        tmp = x * (1.0d0 - (y * t))
    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 * (1.0 - (a * b));
	double tmp;
	if (b <= -1.95e+99) {
		tmp = t_1;
	} else if (b <= 8.5e+85) {
		tmp = x * (1.0 - (y * t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x * (1.0 - (a * b))
	tmp = 0
	if b <= -1.95e+99:
		tmp = t_1
	elif b <= 8.5e+85:
		tmp = x * (1.0 - (y * t))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x * Float64(1.0 - Float64(a * b)))
	tmp = 0.0
	if (b <= -1.95e+99)
		tmp = t_1;
	elseif (b <= 8.5e+85)
		tmp = Float64(x * Float64(1.0 - Float64(y * t)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x * (1.0 - (a * b));
	tmp = 0.0;
	if (b <= -1.95e+99)
		tmp = t_1;
	elseif (b <= 8.5e+85)
		tmp = x * (1.0 - (y * t));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x * N[(1.0 - N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.95e+99], t$95$1, If[LessEqual[b, 8.5e+85], N[(x * N[(1.0 - N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if b < -1.94999999999999997e99 or 8.4999999999999994e85 < b

   1. Initial program 98.8%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      2. lower-neg.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      3. lower-*.f64 81.9

         \[\leadsto x \cdot e^{-\color{blue}{a \cdot b}} \]

   5. Applied rewrites 81.9%

      \[\leadsto x \cdot e^{\color{blue}{-a \cdot b}} \]

   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 38.9

         \[\leadsto x \cdot \left(1 - \color{blue}{a \cdot b}\right) \]

   8. Applied rewrites 38.9%

      \[\leadsto x \cdot \color{blue}{\left(1 - a \cdot b\right)} \]

   if -1.94999999999999997e99 < b < 8.4999999999999994e85

   1. Initial program 95.1%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in 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 71.2

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(-t\right)}} \]

   5. Applied rewrites 71.2%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}} \]

   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 38.5

         \[\leadsto x \cdot \left(1 - \color{blue}{y \cdot t}\right) \]

   8. Applied rewrites 38.5%

      \[\leadsto x \cdot \color{blue}{\left(1 - y \cdot t\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 19.8% accurate, 328.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 x)

C

double code(double x, double y, double z, double t, double a, double b) {
	return x;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = x
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return x;
}

Python

def code(x, y, z, t, a, b):
	return x

Julia

function code(x, y, z, t, a, b)
	return x
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = x;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := x

Derivation

1. Initial program 96.3%

   \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
2. Add Preprocessing

3. Taylor expanded in 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 73.2

      \[\leadsto x \cdot e^{y \cdot \left(\color{blue}{\log z} - t\right)} \]

5. Applied rewrites 73.2%

   \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\log z - t\right)}} \]

6. Taylor expanded in y around 0

   \[\leadsto x \cdot \color{blue}{1} \]
7. Step-by-step derivation

8. Applied rewrites 18.7%

   \[\leadsto x \cdot \color{blue}{1} \]
9. Step-by-step derivation

   1. *-rgt-identity 18.7

      \[\leadsto \color{blue}{x} \]

10. Applied rewrites 18.7%

    \[\leadsto \color{blue}{x} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024219 
(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))))))