Numeric.SpecFunctions:incompleteBetaWorker from math-functions-0.1.5.2, A

Percentage Accurate: 98.5% → 98.5%

Time: 16.2s

Alternatives: 26

Speedup: 1.0×

• 47.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

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 - 1.0d0) * log(a))) - b))) / y
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 - 1.0) * Math.log(a))) - b))) / y;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 - 1.0d0) * log(a))) - b))) / y
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 - 1.0) * Math.log(a))) - b))) / y;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 + (-1.0d0)) * log(a))) - b))) / y
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 + -1.0) * Math.log(a))) - b))) / y;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.1%

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

3. Final simplification 98.1%

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

Alternative 2: 50.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot e^{\left(y \cdot \log z + \left(t + -1\right) \cdot \log a\right) - b}}{y}\\ t_2 := \frac{x}{y \cdot a}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(b, t\_2 \cdot \mathsf{fma}\left(b, 0.5, -1\right), t\_2\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+253}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{a \cdot \left(y \cdot \left(b \cdot b\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (* x (exp (- (+ (* y (log z)) (* (+ t -1.0) (log a))) b))) y))
        (t_2 (/ x (* y a))))
   (if (<= t_1 (- INFINITY))
     (fma b (* t_2 (fma b 0.5 -1.0)) t_2)
     (if (<= t_1 5e+253)
       (/
        x
        (* a (* y (fma b (fma b (fma b 0.16666666666666666 0.5) 1.0) 1.0))))
       (/ (* x 2.0) (* a (* y (* b b))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x * exp((((y * log(z)) + ((t + -1.0) * log(a))) - b))) / y;
	double t_2 = x / (y * a);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(b, (t_2 * fma(b, 0.5, -1.0)), t_2);
	} else if (t_1 <= 5e+253) {
		tmp = x / (a * (y * fma(b, fma(b, fma(b, 0.16666666666666666, 0.5), 1.0), 1.0)));
	} else {
		tmp = (x * 2.0) / (a * (y * (b * b)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x * exp(Float64(Float64(Float64(y * log(z)) + Float64(Float64(t + -1.0) * log(a))) - b))) / y)
	t_2 = Float64(x / Float64(y * a))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = fma(b, Float64(t_2 * fma(b, 0.5, -1.0)), t_2);
	elseif (t_1 <= 5e+253)
		tmp = Float64(x / Float64(a * Float64(y * fma(b, fma(b, fma(b, 0.16666666666666666, 0.5), 1.0), 1.0))));
	else
		tmp = Float64(Float64(x * 2.0) / Float64(a * Float64(y * Float64(b * b))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x * N[Exp[N[(N[(N[(y * N[Log[z], $MachinePrecision]), $MachinePrecision] + N[(N[(t + -1.0), $MachinePrecision] * N[Log[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$2 = N[(x / N[(y * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(b * N[(t$95$2 * N[(b * 0.5 + -1.0), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision], If[LessEqual[t$95$1, 5e+253], N[(x / N[(a * N[(y * N[(b * N[(b * N[(b * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * 2.0), $MachinePrecision] / N[(a * N[(y * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      5. lower-/.f64 N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      6. +-commutative N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z + -1 \cdot \log a}}}{y \cdot e^{b}} \]

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z - \log a}}}{y \cdot e^{b}} \]

      9. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      10. lower-/.f64 N/A

          \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      11. *-commutative N/A

          \[\leadsto x \cdot \frac{\frac{e^{\color{blue}{\log z \cdot y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      12. exp-to-pow N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      13. lower-pow.f64 N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      14. rem-exp-log N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{\color{blue}{a}}}{y \cdot e^{b}} \]

      15. lower-*.f64 N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{\color{blue}{y \cdot e^{b}}} \]

      16. lower-exp.f64 66.7

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 66.7%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot e^{b}}} \]

   6. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-exp.f64 59.1

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

   8. Simplified 59.1%

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

   9. Taylor expanded in b around 0

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

       1. lower-fma.f64 N/A

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

   11. Simplified 44.1%

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

   if -inf.0 < (/.f64 (*.f64 x (exp.f64 (-.f64 (+.f64 (*.f64 y (log.f64 z)) (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))) b))) y) < 4.9999999999999997e253

   1. Initial program 96.7%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      5. lower-/.f64 N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      6. +-commutative N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z + -1 \cdot \log a}}}{y \cdot e^{b}} \]

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z - \log a}}}{y \cdot e^{b}} \]

      9. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      10. lower-/.f64 N/A

          \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      11. *-commutative N/A

          \[\leadsto x \cdot \frac{\frac{e^{\color{blue}{\log z \cdot y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      12. exp-to-pow N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      13. lower-pow.f64 N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      14. rem-exp-log N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{\color{blue}{a}}}{y \cdot e^{b}} \]

      15. lower-*.f64 N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{\color{blue}{y \cdot e^{b}}} \]

      16. lower-exp.f64 71.1

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 71.1%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot e^{b}}} \]

   6. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-exp.f64 59.8

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

   8. Simplified 59.8%

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

   9. Taylor expanded in b around 0

      \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\left(1 + b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)\right)}\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. +-commutative N/A

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

       4. lower-fma.f64 N/A

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

       5. +-commutative N/A

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

       6. *-commutative N/A

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

       7. lower-fma.f64 64.8

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

   11. Simplified 64.8%

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

   if 4.9999999999999997e253 < (/.f64 (*.f64 x (exp.f64 (-.f64 (+.f64 (*.f64 y (log.f64 z)) (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))) b))) y)

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      5. lower-/.f64 N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      6. +-commutative N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z + -1 \cdot \log a}}}{y \cdot e^{b}} \]

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z - \log a}}}{y \cdot e^{b}} \]

      9. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      10. lower-/.f64 N/A

          \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      11. *-commutative N/A

          \[\leadsto x \cdot \frac{\frac{e^{\color{blue}{\log z \cdot y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      12. exp-to-pow N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      13. lower-pow.f64 N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      14. rem-exp-log N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{\color{blue}{a}}}{y \cdot e^{b}} \]

      15. lower-*.f64 N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{\color{blue}{y \cdot e^{b}}} \]

      16. lower-exp.f64 75.5

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 75.5%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot e^{b}}} \]

   6. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-exp.f64 70.8

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

   8. Simplified 70.8%

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

   9. Taylor expanded in b around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. +-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

       8. lower-fma.f64 N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 28.4

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

   11. Simplified 28.4%

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

   12. Taylor expanded in b around inf

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 41.2

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

   14. Simplified 41.2%

       \[\leadsto \color{blue}{\frac{x \cdot 2}{a \cdot \left(y \cdot \left(b \cdot b\right)\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 55.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot e^{\left(y \cdot \log z + \left(t + -1\right) \cdot \log a\right) - b}}{y} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(b, \frac{x}{y \cdot a} \cdot \mathsf{fma}\left(b, 0.5, -1\right), \frac{x}{y \cdot a}\right)\\ \mathbf{elif}\;\frac{x \cdot e^{\left(y \cdot \log z + \left(t + -1\right) \cdot \log a\right) - b}}{y} \leq 5 \cdot 10^{+253}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{a \cdot \left(y \cdot \left(b \cdot b\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 49.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot e^{\left(y \cdot \log z + \left(t + -1\right) \cdot \log a\right) - b}}{y}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+281}:\\ \;\;\;\;\frac{x}{y \cdot a} \cdot \left(1 - b\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+296}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{a \cdot \left(y \cdot \left(b \cdot b\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (* x (exp (- (+ (* y (log z)) (* (+ t -1.0) (log a))) b))) y)))
   (if (<= t_1 -1e+281)
     (* (/ x (* y a)) (- 1.0 b))
     (if (<= t_1 5e+296)
       (/
        x
        (* a (* y (fma b (fma b (fma b 0.16666666666666666 0.5) 1.0) 1.0))))
       (/ (* x 2.0) (* a (* y (* b b))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x * exp((((y * log(z)) + ((t + -1.0) * log(a))) - b))) / y;
	double tmp;
	if (t_1 <= -1e+281) {
		tmp = (x / (y * a)) * (1.0 - b);
	} else if (t_1 <= 5e+296) {
		tmp = x / (a * (y * fma(b, fma(b, fma(b, 0.16666666666666666, 0.5), 1.0), 1.0)));
	} else {
		tmp = (x * 2.0) / (a * (y * (b * b)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x * exp(Float64(Float64(Float64(y * log(z)) + Float64(Float64(t + -1.0) * log(a))) - b))) / y)
	tmp = 0.0
	if (t_1 <= -1e+281)
		tmp = Float64(Float64(x / Float64(y * a)) * Float64(1.0 - b));
	elseif (t_1 <= 5e+296)
		tmp = Float64(x / Float64(a * Float64(y * fma(b, fma(b, fma(b, 0.16666666666666666, 0.5), 1.0), 1.0))));
	else
		tmp = Float64(Float64(x * 2.0) / Float64(a * Float64(y * Float64(b * b))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x * N[Exp[N[(N[(N[(y * N[Log[z], $MachinePrecision]), $MachinePrecision] + N[(N[(t + -1.0), $MachinePrecision] * N[Log[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+281], N[(N[(x / N[(y * a), $MachinePrecision]), $MachinePrecision] * N[(1.0 - b), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+296], N[(x / N[(a * N[(y * N[(b * N[(b * N[(b * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * 2.0), $MachinePrecision] / N[(a * N[(y * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (exp.f64 (-.f64 (+.f64 (*.f64 y (log.f64 z)) (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))) b))) y) < -1e281

   1. Initial program 99.5%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      5. lower-/.f64 N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      6. +-commutative N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z + -1 \cdot \log a}}}{y \cdot e^{b}} \]

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z - \log a}}}{y \cdot e^{b}} \]

      9. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      10. lower-/.f64 N/A

          \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      11. *-commutative N/A

          \[\leadsto x \cdot \frac{\frac{e^{\color{blue}{\log z \cdot y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      12. exp-to-pow N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      13. lower-pow.f64 N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      14. rem-exp-log N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{\color{blue}{a}}}{y \cdot e^{b}} \]

      15. lower-*.f64 N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{\color{blue}{y \cdot e^{b}}} \]

      16. lower-exp.f64 64.6

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 64.6%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot e^{b}}} \]

   6. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-exp.f64 57.2

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

   8. Simplified 57.2%

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

   9. Taylor expanded in b around 0

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

       1. associate-/l* N/A

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

       2. associate-*r* N/A

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

       3. distribute-lft1-in N/A

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

       4. lower-*.f64 N/A

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

       5. lower-+.f64 N/A

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

       6. mul-1-neg N/A

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

       7. lower-neg.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. lower-*.f64 33.3

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

   11. Simplified 33.3%

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

   if -1e281 < (/.f64 (*.f64 x (exp.f64 (-.f64 (+.f64 (*.f64 y (log.f64 z)) (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))) b))) y) < 5.0000000000000001e296

   1. Initial program 96.8%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      5. lower-/.f64 N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      6. +-commutative N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z + -1 \cdot \log a}}}{y \cdot e^{b}} \]

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z - \log a}}}{y \cdot e^{b}} \]

      9. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      10. lower-/.f64 N/A

          \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      11. *-commutative N/A

          \[\leadsto x \cdot \frac{\frac{e^{\color{blue}{\log z \cdot y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      12. exp-to-pow N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      13. lower-pow.f64 N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      14. rem-exp-log N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{\color{blue}{a}}}{y \cdot e^{b}} \]

      15. lower-*.f64 N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{\color{blue}{y \cdot e^{b}}} \]

      16. lower-exp.f64 71.5

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 71.5%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot e^{b}}} \]

   6. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-exp.f64 60.2

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

   8. Simplified 60.2%

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

   9. Taylor expanded in b around 0

      \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\left(1 + b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)\right)}\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. +-commutative N/A

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

       4. lower-fma.f64 N/A

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

       5. +-commutative N/A

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

       6. *-commutative N/A

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

       7. lower-fma.f64 65.1

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

   11. Simplified 65.1%

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

   if 5.0000000000000001e296 < (/.f64 (*.f64 x (exp.f64 (-.f64 (+.f64 (*.f64 y (log.f64 z)) (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))) b))) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      5. lower-/.f64 N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      6. +-commutative N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z + -1 \cdot \log a}}}{y \cdot e^{b}} \]

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z - \log a}}}{y \cdot e^{b}} \]

      9. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      10. lower-/.f64 N/A

          \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      11. *-commutative N/A

          \[\leadsto x \cdot \frac{\frac{e^{\color{blue}{\log z \cdot y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      12. exp-to-pow N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      13. lower-pow.f64 N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      14. rem-exp-log N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{\color{blue}{a}}}{y \cdot e^{b}} \]

      15. lower-*.f64 N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{\color{blue}{y \cdot e^{b}}} \]

      16. lower-exp.f64 76.7

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 76.7%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot e^{b}}} \]

   6. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-exp.f64 71.9

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

   8. Simplified 71.9%

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

   9. Taylor expanded in b around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. +-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

       8. lower-fma.f64 N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 28.8

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

   11. Simplified 28.8%

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

   12. Taylor expanded in b around inf

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 41.8

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

   14. Simplified 41.8%

       \[\leadsto \color{blue}{\frac{x \cdot 2}{a \cdot \left(y \cdot \left(b \cdot b\right)\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 53.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot e^{\left(y \cdot \log z + \left(t + -1\right) \cdot \log a\right) - b}}{y} \leq -1 \cdot 10^{+281}:\\ \;\;\;\;\frac{x}{y \cdot a} \cdot \left(1 - b\right)\\ \mathbf{elif}\;\frac{x \cdot e^{\left(y \cdot \log z + \left(t + -1\right) \cdot \log a\right) - b}}{y} \leq 5 \cdot 10^{+296}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{a \cdot \left(y \cdot \left(b \cdot b\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 46.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot e^{\left(y \cdot \log z + \left(t + -1\right) \cdot \log a\right) - b}}{y}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+281}:\\ \;\;\;\;\frac{x}{y \cdot a} \cdot \left(1 - b\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+253}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5, 1\right), 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{a \cdot \left(y \cdot \left(b \cdot b\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (* x (exp (- (+ (* y (log z)) (* (+ t -1.0) (log a))) b))) y)))
   (if (<= t_1 -1e+281)
     (* (/ x (* y a)) (- 1.0 b))
     (if (<= t_1 5e+253)
       (/ x (* a (* y (fma b (fma b 0.5 1.0) 1.0))))
       (/ (* x 2.0) (* a (* y (* b b))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x * exp((((y * log(z)) + ((t + -1.0) * log(a))) - b))) / y;
	double tmp;
	if (t_1 <= -1e+281) {
		tmp = (x / (y * a)) * (1.0 - b);
	} else if (t_1 <= 5e+253) {
		tmp = x / (a * (y * fma(b, fma(b, 0.5, 1.0), 1.0)));
	} else {
		tmp = (x * 2.0) / (a * (y * (b * b)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x * exp(Float64(Float64(Float64(y * log(z)) + Float64(Float64(t + -1.0) * log(a))) - b))) / y)
	tmp = 0.0
	if (t_1 <= -1e+281)
		tmp = Float64(Float64(x / Float64(y * a)) * Float64(1.0 - b));
	elseif (t_1 <= 5e+253)
		tmp = Float64(x / Float64(a * Float64(y * fma(b, fma(b, 0.5, 1.0), 1.0))));
	else
		tmp = Float64(Float64(x * 2.0) / Float64(a * Float64(y * Float64(b * b))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x * N[Exp[N[(N[(N[(y * N[Log[z], $MachinePrecision]), $MachinePrecision] + N[(N[(t + -1.0), $MachinePrecision] * N[Log[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+281], N[(N[(x / N[(y * a), $MachinePrecision]), $MachinePrecision] * N[(1.0 - b), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+253], N[(x / N[(a * N[(y * N[(b * N[(b * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * 2.0), $MachinePrecision] / N[(a * N[(y * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (exp.f64 (-.f64 (+.f64 (*.f64 y (log.f64 z)) (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))) b))) y) < -1e281

   1. Initial program 99.5%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      5. lower-/.f64 N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      6. +-commutative N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z + -1 \cdot \log a}}}{y \cdot e^{b}} \]

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z - \log a}}}{y \cdot e^{b}} \]

      9. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      10. lower-/.f64 N/A

          \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      11. *-commutative N/A

          \[\leadsto x \cdot \frac{\frac{e^{\color{blue}{\log z \cdot y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      12. exp-to-pow N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      13. lower-pow.f64 N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      14. rem-exp-log N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{\color{blue}{a}}}{y \cdot e^{b}} \]

      15. lower-*.f64 N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{\color{blue}{y \cdot e^{b}}} \]

      16. lower-exp.f64 64.6

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 64.6%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot e^{b}}} \]

   6. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-exp.f64 57.2

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

   8. Simplified 57.2%

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

   9. Taylor expanded in b around 0

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

       1. associate-/l* N/A

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

       2. associate-*r* N/A

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

       3. distribute-lft1-in N/A

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

       4. lower-*.f64 N/A

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

       5. lower-+.f64 N/A

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

       6. mul-1-neg N/A

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

       7. lower-neg.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. lower-*.f64 33.3

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

   11. Simplified 33.3%

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

   if -1e281 < (/.f64 (*.f64 x (exp.f64 (-.f64 (+.f64 (*.f64 y (log.f64 z)) (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))) b))) y) < 4.9999999999999997e253

   1. Initial program 96.9%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      5. lower-/.f64 N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      6. +-commutative N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z + -1 \cdot \log a}}}{y \cdot e^{b}} \]

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z - \log a}}}{y \cdot e^{b}} \]

      9. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      10. lower-/.f64 N/A

          \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      11. *-commutative N/A

          \[\leadsto x \cdot \frac{\frac{e^{\color{blue}{\log z \cdot y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      12. exp-to-pow N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      13. lower-pow.f64 N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      14. rem-exp-log N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{\color{blue}{a}}}{y \cdot e^{b}} \]

      15. lower-*.f64 N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{\color{blue}{y \cdot e^{b}}} \]

      16. lower-exp.f64 71.9

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 71.9%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot e^{b}}} \]

   6. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-exp.f64 60.5

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

   8. Simplified 60.5%

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

   9. Taylor expanded in b around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 60.1

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

   11. Simplified 60.1%

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

   if 4.9999999999999997e253 < (/.f64 (*.f64 x (exp.f64 (-.f64 (+.f64 (*.f64 y (log.f64 z)) (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))) b))) y)

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      5. lower-/.f64 N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      6. +-commutative N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z + -1 \cdot \log a}}}{y \cdot e^{b}} \]

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z - \log a}}}{y \cdot e^{b}} \]

      9. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      10. lower-/.f64 N/A

          \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      11. *-commutative N/A

          \[\leadsto x \cdot \frac{\frac{e^{\color{blue}{\log z \cdot y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      12. exp-to-pow N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      13. lower-pow.f64 N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      14. rem-exp-log N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{\color{blue}{a}}}{y \cdot e^{b}} \]

      15. lower-*.f64 N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{\color{blue}{y \cdot e^{b}}} \]

      16. lower-exp.f64 75.5

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 75.5%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot e^{b}}} \]

   6. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-exp.f64 70.8

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

   8. Simplified 70.8%

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

   9. Taylor expanded in b around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. +-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

       8. lower-fma.f64 N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 28.4

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

   11. Simplified 28.4%

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

   12. Taylor expanded in b around inf

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 41.2

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

   14. Simplified 41.2%

       \[\leadsto \color{blue}{\frac{x \cdot 2}{a \cdot \left(y \cdot \left(b \cdot b\right)\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 50.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot e^{\left(y \cdot \log z + \left(t + -1\right) \cdot \log a\right) - b}}{y} \leq -1 \cdot 10^{+281}:\\ \;\;\;\;\frac{x}{y \cdot a} \cdot \left(1 - b\right)\\ \mathbf{elif}\;\frac{x \cdot e^{\left(y \cdot \log z + \left(t + -1\right) \cdot \log a\right) - b}}{y} \leq 5 \cdot 10^{+253}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5, 1\right), 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{a \cdot \left(y \cdot \left(b \cdot b\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 45.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot e^{\left(y \cdot \log z + \left(t + -1\right) \cdot \log a\right) - b}}{y}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-274}:\\ \;\;\;\;\frac{x}{y \cdot a} \cdot \left(1 - b\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+253}:\\ \;\;\;\;\frac{x}{a \cdot \mathsf{fma}\left(b, y \cdot \left(b \cdot 0.5\right), y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{a \cdot \left(y \cdot \left(b \cdot b\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (* x (exp (- (+ (* y (log z)) (* (+ t -1.0) (log a))) b))) y)))
   (if (<= t_1 -5e-274)
     (* (/ x (* y a)) (- 1.0 b))
     (if (<= t_1 5e+253)
       (/ x (* a (fma b (* y (* b 0.5)) y)))
       (/ (* x 2.0) (* a (* y (* b b))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x * exp((((y * log(z)) + ((t + -1.0) * log(a))) - b))) / y;
	double tmp;
	if (t_1 <= -5e-274) {
		tmp = (x / (y * a)) * (1.0 - b);
	} else if (t_1 <= 5e+253) {
		tmp = x / (a * fma(b, (y * (b * 0.5)), y));
	} else {
		tmp = (x * 2.0) / (a * (y * (b * b)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x * exp(Float64(Float64(Float64(y * log(z)) + Float64(Float64(t + -1.0) * log(a))) - b))) / y)
	tmp = 0.0
	if (t_1 <= -5e-274)
		tmp = Float64(Float64(x / Float64(y * a)) * Float64(1.0 - b));
	elseif (t_1 <= 5e+253)
		tmp = Float64(x / Float64(a * fma(b, Float64(y * Float64(b * 0.5)), y)));
	else
		tmp = Float64(Float64(x * 2.0) / Float64(a * Float64(y * Float64(b * b))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x * N[Exp[N[(N[(N[(y * N[Log[z], $MachinePrecision]), $MachinePrecision] + N[(N[(t + -1.0), $MachinePrecision] * N[Log[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-274], N[(N[(x / N[(y * a), $MachinePrecision]), $MachinePrecision] * N[(1.0 - b), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+253], N[(x / N[(a * N[(b * N[(y * N[(b * 0.5), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * 2.0), $MachinePrecision] / N[(a * N[(y * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (exp.f64 (-.f64 (+.f64 (*.f64 y (log.f64 z)) (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))) b))) y) < -5e-274

   1. Initial program 98.2%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      5. lower-/.f64 N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      6. +-commutative N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z + -1 \cdot \log a}}}{y \cdot e^{b}} \]

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z - \log a}}}{y \cdot e^{b}} \]

      9. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      10. lower-/.f64 N/A

          \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      11. *-commutative N/A

          \[\leadsto x \cdot \frac{\frac{e^{\color{blue}{\log z \cdot y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      12. exp-to-pow N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      13. lower-pow.f64 N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      14. rem-exp-log N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{\color{blue}{a}}}{y \cdot e^{b}} \]

      15. lower-*.f64 N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{\color{blue}{y \cdot e^{b}}} \]

      16. lower-exp.f64 70.2

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 70.2%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot e^{b}}} \]

   6. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-exp.f64 63.1

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

   8. Simplified 63.1%

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

   9. Taylor expanded in b around 0

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

       1. associate-/l* N/A

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

       2. associate-*r* N/A

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

       3. distribute-lft1-in N/A

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

       4. lower-*.f64 N/A

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

       5. lower-+.f64 N/A

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

       6. mul-1-neg N/A

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

       7. lower-neg.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. lower-*.f64 42.6

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

   11. Simplified 42.6%

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

   if -5e-274 < (/.f64 (*.f64 x (exp.f64 (-.f64 (+.f64 (*.f64 y (log.f64 z)) (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))) b))) y) < 4.9999999999999997e253

   1. Initial program 97.3%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      5. lower-/.f64 N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      6. +-commutative N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z + -1 \cdot \log a}}}{y \cdot e^{b}} \]

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z - \log a}}}{y \cdot e^{b}} \]

      9. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      10. lower-/.f64 N/A

          \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      11. *-commutative N/A

          \[\leadsto x \cdot \frac{\frac{e^{\color{blue}{\log z \cdot y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      12. exp-to-pow N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      13. lower-pow.f64 N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      14. rem-exp-log N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{\color{blue}{a}}}{y \cdot e^{b}} \]

      15. lower-*.f64 N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{\color{blue}{y \cdot e^{b}}} \]

      16. lower-exp.f64 69.9

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 69.9%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot e^{b}}} \]

   6. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-exp.f64 58.0

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

   8. Simplified 58.0%

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

   9. Taylor expanded in b around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. +-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

       8. lower-fma.f64 N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 55.5

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

   11. Simplified 55.5%

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

   12. Taylor expanded in b around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 N/A

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

       4. lower-*.f64 55.5

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

   14. Simplified 55.5%

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

   if 4.9999999999999997e253 < (/.f64 (*.f64 x (exp.f64 (-.f64 (+.f64 (*.f64 y (log.f64 z)) (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))) b))) y)

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      5. lower-/.f64 N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      6. +-commutative N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z + -1 \cdot \log a}}}{y \cdot e^{b}} \]

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z - \log a}}}{y \cdot e^{b}} \]

      9. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      10. lower-/.f64 N/A

          \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      11. *-commutative N/A

          \[\leadsto x \cdot \frac{\frac{e^{\color{blue}{\log z \cdot y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      12. exp-to-pow N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      13. lower-pow.f64 N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      14. rem-exp-log N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{\color{blue}{a}}}{y \cdot e^{b}} \]

      15. lower-*.f64 N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{\color{blue}{y \cdot e^{b}}} \]

      16. lower-exp.f64 75.5

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 75.5%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot e^{b}}} \]

   6. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-exp.f64 70.8

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

   8. Simplified 70.8%

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

   9. Taylor expanded in b around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. +-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

       8. lower-fma.f64 N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 28.4

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

   11. Simplified 28.4%

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

   12. Taylor expanded in b around inf

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 41.2

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

   14. Simplified 41.2%

       \[\leadsto \color{blue}{\frac{x \cdot 2}{a \cdot \left(y \cdot \left(b \cdot b\right)\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 49.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot e^{\left(y \cdot \log z + \left(t + -1\right) \cdot \log a\right) - b}}{y} \leq -5 \cdot 10^{-274}:\\ \;\;\;\;\frac{x}{y \cdot a} \cdot \left(1 - b\right)\\ \mathbf{elif}\;\frac{x \cdot e^{\left(y \cdot \log z + \left(t + -1\right) \cdot \log a\right) - b}}{y} \leq 5 \cdot 10^{+253}:\\ \;\;\;\;\frac{x}{a \cdot \mathsf{fma}\left(b, y \cdot \left(b \cdot 0.5\right), y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{a \cdot \left(y \cdot \left(b \cdot b\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 38.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot e^{\left(y \cdot \log z + \left(t + -1\right) \cdot \log a\right) - b}}{y}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-274}:\\ \;\;\;\;\frac{x}{y \cdot a} \cdot \left(1 - b\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+253}:\\ \;\;\;\;\frac{x}{a \cdot \mathsf{fma}\left(y, b, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (* x (exp (- (+ (* y (log z)) (* (+ t -1.0) (log a))) b))) y)))
   (if (<= t_1 -5e-274)
     (* (/ x (* y a)) (- 1.0 b))
     (if (<= t_1 5e+253) (/ x (* a (fma y b y))) (/ (/ x y) a)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x * exp((((y * log(z)) + ((t + -1.0) * log(a))) - b))) / y;
	double tmp;
	if (t_1 <= -5e-274) {
		tmp = (x / (y * a)) * (1.0 - b);
	} else if (t_1 <= 5e+253) {
		tmp = x / (a * fma(y, b, y));
	} else {
		tmp = (x / y) / a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x * exp(Float64(Float64(Float64(y * log(z)) + Float64(Float64(t + -1.0) * log(a))) - b))) / y)
	tmp = 0.0
	if (t_1 <= -5e-274)
		tmp = Float64(Float64(x / Float64(y * a)) * Float64(1.0 - b));
	elseif (t_1 <= 5e+253)
		tmp = Float64(x / Float64(a * fma(y, b, y)));
	else
		tmp = Float64(Float64(x / y) / a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x * N[Exp[N[(N[(N[(y * N[Log[z], $MachinePrecision]), $MachinePrecision] + N[(N[(t + -1.0), $MachinePrecision] * N[Log[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-274], N[(N[(x / N[(y * a), $MachinePrecision]), $MachinePrecision] * N[(1.0 - b), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+253], N[(x / N[(a * N[(y * b + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (exp.f64 (-.f64 (+.f64 (*.f64 y (log.f64 z)) (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))) b))) y) < -5e-274

   1. Initial program 98.2%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      5. lower-/.f64 N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      6. +-commutative N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z + -1 \cdot \log a}}}{y \cdot e^{b}} \]

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z - \log a}}}{y \cdot e^{b}} \]

      9. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      10. lower-/.f64 N/A

          \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      11. *-commutative N/A

          \[\leadsto x \cdot \frac{\frac{e^{\color{blue}{\log z \cdot y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      12. exp-to-pow N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      13. lower-pow.f64 N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      14. rem-exp-log N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{\color{blue}{a}}}{y \cdot e^{b}} \]

      15. lower-*.f64 N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{\color{blue}{y \cdot e^{b}}} \]

      16. lower-exp.f64 70.2

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 70.2%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot e^{b}}} \]

   6. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-exp.f64 63.1

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

   8. Simplified 63.1%

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

   9. Taylor expanded in b around 0

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

       1. associate-/l* N/A

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

       2. associate-*r* N/A

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

       3. distribute-lft1-in N/A

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

       4. lower-*.f64 N/A

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

       5. lower-+.f64 N/A

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

       6. mul-1-neg N/A

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

       7. lower-neg.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. lower-*.f64 42.6

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

   11. Simplified 42.6%

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

   if -5e-274 < (/.f64 (*.f64 x (exp.f64 (-.f64 (+.f64 (*.f64 y (log.f64 z)) (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))) b))) y) < 4.9999999999999997e253

   1. Initial program 97.3%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      5. lower-/.f64 N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      6. +-commutative N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z + -1 \cdot \log a}}}{y \cdot e^{b}} \]

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z - \log a}}}{y \cdot e^{b}} \]

      9. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      10. lower-/.f64 N/A

          \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      11. *-commutative N/A

          \[\leadsto x \cdot \frac{\frac{e^{\color{blue}{\log z \cdot y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      12. exp-to-pow N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      13. lower-pow.f64 N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      14. rem-exp-log N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{\color{blue}{a}}}{y \cdot e^{b}} \]

      15. lower-*.f64 N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{\color{blue}{y \cdot e^{b}}} \]

      16. lower-exp.f64 69.9

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 69.9%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot e^{b}}} \]

   6. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-exp.f64 58.0

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

   8. Simplified 58.0%

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

   9. Taylor expanded in b around 0

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

       1. distribute-lft-out N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 48.5

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

   11. Simplified 48.5%

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

   if 4.9999999999999997e253 < (/.f64 (*.f64 x (exp.f64 (-.f64 (+.f64 (*.f64 y (log.f64 z)) (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))) b))) y)

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. exp-diff N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. exp-prod N/A

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

      8. lower-pow.f64 N/A

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

      9. rem-exp-log N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-exp.f64 66.9

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

   5. Simplified 66.9%

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

   6. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. exp-to-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 54.3

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

   8. Simplified 54.3%

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

   9. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{1}{a}} \cdot \frac{x}{y} \]
   10. Step-by-step derivation

       1. lower-/.f64 33.6

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

   11. Simplified 33.6%

       \[\leadsto \color{blue}{\frac{1}{a}} \cdot \frac{x}{y} \]
   12. Step-by-step derivation

       1. lift-/.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. *-commutative N/A

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

       4. lift-/.f64 N/A

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

       5. un-div-inv N/A

          \[\leadsto \color{blue}{\frac{\frac{x}{y}}{a}} \]

       6. lower-/.f64 33.6

          \[\leadsto \color{blue}{\frac{\frac{x}{y}}{a}} \]

   13. Applied egg-rr 33.6%

       \[\leadsto \color{blue}{\frac{\frac{x}{y}}{a}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 43.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot e^{\left(y \cdot \log z + \left(t + -1\right) \cdot \log a\right) - b}}{y} \leq -5 \cdot 10^{-274}:\\ \;\;\;\;\frac{x}{y \cdot a} \cdot \left(1 - b\right)\\ \mathbf{elif}\;\frac{x \cdot e^{\left(y \cdot \log z + \left(t + -1\right) \cdot \log a\right) - b}}{y} \leq 5 \cdot 10^{+253}:\\ \;\;\;\;\frac{x}{a \cdot \mathsf{fma}\left(y, b, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 37.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot e^{\left(y \cdot \log z + \left(t + -1\right) \cdot \log a\right) - b}}{y}\\ t_2 := \frac{\frac{x}{y}}{a}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+170}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+253}:\\ \;\;\;\;\frac{x}{a \cdot \mathsf{fma}\left(y, b, y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (* x (exp (- (+ (* y (log z)) (* (+ t -1.0) (log a))) b))) y))
        (t_2 (/ (/ x y) a)))
   (if (<= t_1 -4e+170) t_2 (if (<= t_1 5e+253) (/ x (* a (fma y b y))) t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x * exp((((y * log(z)) + ((t + -1.0) * log(a))) - b))) / y;
	double t_2 = (x / y) / a;
	double tmp;
	if (t_1 <= -4e+170) {
		tmp = t_2;
	} else if (t_1 <= 5e+253) {
		tmp = x / (a * fma(y, b, y));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x * exp(Float64(Float64(Float64(y * log(z)) + Float64(Float64(t + -1.0) * log(a))) - b))) / y)
	t_2 = Float64(Float64(x / y) / a)
	tmp = 0.0
	if (t_1 <= -4e+170)
		tmp = t_2;
	elseif (t_1 <= 5e+253)
		tmp = Float64(x / Float64(a * fma(y, b, y)));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x * N[Exp[N[(N[(N[(y * N[Log[z], $MachinePrecision]), $MachinePrecision] + N[(N[(t + -1.0), $MachinePrecision] * N[Log[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x / y), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+170], t$95$2, If[LessEqual[t$95$1, 5e+253], N[(x / N[(a * N[(y * b + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (exp.f64 (-.f64 (+.f64 (*.f64 y (log.f64 z)) (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))) b))) y) < -4.00000000000000014e170 or 4.9999999999999997e253 < (/.f64 (*.f64 x (exp.f64 (-.f64 (+.f64 (*.f64 y (log.f64 z)) (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))) b))) y)

   1. Initial program 99.6%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. exp-diff N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. exp-prod N/A

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

      8. lower-pow.f64 N/A

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

      9. rem-exp-log N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-exp.f64 71.0

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

   5. Simplified 71.0%

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

   6. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. exp-to-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 55.0

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

   8. Simplified 55.0%

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

   9. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{1}{a}} \cdot \frac{x}{y} \]
   10. Step-by-step derivation

       1. lower-/.f64 34.7

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

   11. Simplified 34.7%

       \[\leadsto \color{blue}{\frac{1}{a}} \cdot \frac{x}{y} \]
   12. Step-by-step derivation

       1. lift-/.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. *-commutative N/A

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

       4. lift-/.f64 N/A

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

       5. un-div-inv N/A

          \[\leadsto \color{blue}{\frac{\frac{x}{y}}{a}} \]

       6. lower-/.f64 34.7

          \[\leadsto \color{blue}{\frac{\frac{x}{y}}{a}} \]

   13. Applied egg-rr 34.7%

       \[\leadsto \color{blue}{\frac{\frac{x}{y}}{a}} \]

   if -4.00000000000000014e170 < (/.f64 (*.f64 x (exp.f64 (-.f64 (+.f64 (*.f64 y (log.f64 z)) (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))) b))) y) < 4.9999999999999997e253

   1. Initial program 96.9%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      5. lower-/.f64 N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      6. +-commutative N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z + -1 \cdot \log a}}}{y \cdot e^{b}} \]

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z - \log a}}}{y \cdot e^{b}} \]

      9. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      10. lower-/.f64 N/A

          \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      11. *-commutative N/A

          \[\leadsto x \cdot \frac{\frac{e^{\color{blue}{\log z \cdot y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      12. exp-to-pow N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      13. lower-pow.f64 N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      14. rem-exp-log N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{\color{blue}{a}}}{y \cdot e^{b}} \]

      15. lower-*.f64 N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{\color{blue}{y \cdot e^{b}}} \]

      16. lower-exp.f64 71.7

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 71.7%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot e^{b}}} \]

   6. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-exp.f64 60.3

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

   8. Simplified 60.3%

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

   9. Taylor expanded in b around 0

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

       1. distribute-lft-out N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 51.2

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

   11. Simplified 51.2%

       \[\leadsto \frac{x}{\color{blue}{a \cdot \mathsf{fma}\left(y, b, y\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot e^{\left(y \cdot \log z + \left(t + -1\right) \cdot \log a\right) - b}}{y} \leq -4 \cdot 10^{+170}:\\ \;\;\;\;\frac{\frac{x}{y}}{a}\\ \mathbf{elif}\;\frac{x \cdot e^{\left(y \cdot \log z + \left(t + -1\right) \cdot \log a\right) - b}}{y} \leq 5 \cdot 10^{+253}:\\ \;\;\;\;\frac{x}{a \cdot \mathsf{fma}\left(y, b, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 38.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot e^{\left(y \cdot \log z + \left(t + -1\right) \cdot \log a\right) - b}}{y}\\ t_2 := \frac{\frac{x}{a}}{y}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+281}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{x}{a \cdot \mathsf{fma}\left(y, b, y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (* x (exp (- (+ (* y (log z)) (* (+ t -1.0) (log a))) b))) y))
        (t_2 (/ (/ x a) y)))
   (if (<= t_1 -1e+281) t_2 (if (<= t_1 0.0) (/ x (* a (fma y b y))) t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x * exp((((y * log(z)) + ((t + -1.0) * log(a))) - b))) / y;
	double t_2 = (x / a) / y;
	double tmp;
	if (t_1 <= -1e+281) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = x / (a * fma(y, b, y));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x * exp(Float64(Float64(Float64(y * log(z)) + Float64(Float64(t + -1.0) * log(a))) - b))) / y)
	t_2 = Float64(Float64(x / a) / y)
	tmp = 0.0
	if (t_1 <= -1e+281)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = Float64(x / Float64(a * fma(y, b, y)));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x * N[Exp[N[(N[(N[(y * N[Log[z], $MachinePrecision]), $MachinePrecision] + N[(N[(t + -1.0), $MachinePrecision] * N[Log[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x / a), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+281], t$95$2, If[LessEqual[t$95$1, 0.0], N[(x / N[(a * N[(y * b + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (exp.f64 (-.f64 (+.f64 (*.f64 y (log.f64 z)) (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))) b))) y) < -1e281 or -0.0 < (/.f64 (*.f64 x (exp.f64 (-.f64 (+.f64 (*.f64 y (log.f64 z)) (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))) b))) y)

   1. Initial program 99.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. exp-diff N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. exp-prod N/A

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

      8. lower-pow.f64 N/A

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

      9. rem-exp-log N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-exp.f64 72.7

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

   5. Simplified 72.7%

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

   6. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. exp-to-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 56.8

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

   8. Simplified 56.8%

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

   9. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{1}{a}} \cdot \frac{x}{y} \]
   10. Step-by-step derivation

       1. lower-/.f64 37.7

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

   11. Simplified 37.7%

       \[\leadsto \color{blue}{\frac{1}{a}} \cdot \frac{x}{y} \]
   12. Step-by-step derivation

       1. lift-/.f64 N/A

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

       2. associate-*r/ N/A

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

       3. lower-/.f64 N/A

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

       4. *-commutative N/A

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

       5. lift-/.f64 N/A

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

       6. div-inv N/A

          \[\leadsto \frac{\color{blue}{\frac{x}{a}}}{y} \]

       7. lower-/.f64 37.7

          \[\leadsto \frac{\color{blue}{\frac{x}{a}}}{y} \]

   13. Applied egg-rr 37.7%

       \[\leadsto \color{blue}{\frac{\frac{x}{a}}{y}} \]

   if -1e281 < (/.f64 (*.f64 x (exp.f64 (-.f64 (+.f64 (*.f64 y (log.f64 z)) (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))) b))) y) < -0.0

   1. Initial program 97.3%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      5. lower-/.f64 N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      6. +-commutative N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z + -1 \cdot \log a}}}{y \cdot e^{b}} \]

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z - \log a}}}{y \cdot e^{b}} \]

      9. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      10. lower-/.f64 N/A

          \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      11. *-commutative N/A

          \[\leadsto x \cdot \frac{\frac{e^{\color{blue}{\log z \cdot y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      12. exp-to-pow N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      13. lower-pow.f64 N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      14. rem-exp-log N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{\color{blue}{a}}}{y \cdot e^{b}} \]

      15. lower-*.f64 N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{\color{blue}{y \cdot e^{b}}} \]

      16. lower-exp.f64 70.3

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 70.3%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot e^{b}}} \]

   6. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-exp.f64 58.2

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

   8. Simplified 58.2%

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

   9. Taylor expanded in b around 0

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

       1. distribute-lft-out N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 48.6

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

   11. Simplified 48.6%

       \[\leadsto \frac{x}{\color{blue}{a \cdot \mathsf{fma}\left(y, b, y\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot e^{\left(y \cdot \log z + \left(t + -1\right) \cdot \log a\right) - b}}{y} \leq -1 \cdot 10^{+281}:\\ \;\;\;\;\frac{\frac{x}{a}}{y}\\ \mathbf{elif}\;\frac{x \cdot e^{\left(y \cdot \log z + \left(t + -1\right) \cdot \log a\right) - b}}{y} \leq 0:\\ \;\;\;\;\frac{x}{a \cdot \mathsf{fma}\left(y, b, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{a}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 37.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot e^{\left(y \cdot \log z + \left(t + -1\right) \cdot \log a\right) - b}}{y}\\ t_2 := \frac{x}{y \cdot a}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{+139}:\\ \;\;\;\;\frac{x}{a \cdot \mathsf{fma}\left(y, b, y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (* x (exp (- (+ (* y (log z)) (* (+ t -1.0) (log a))) b))) y))
        (t_2 (/ x (* y a))))
   (if (<= t_1 (- INFINITY))
     t_2
     (if (<= t_1 1e+139) (/ x (* a (fma y b y))) t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x * exp((((y * log(z)) + ((t + -1.0) * log(a))) - b))) / y;
	double t_2 = x / (y * a);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_1 <= 1e+139) {
		tmp = x / (a * fma(y, b, y));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x * exp(Float64(Float64(Float64(y * log(z)) + Float64(Float64(t + -1.0) * log(a))) - b))) / y)
	t_2 = Float64(x / Float64(y * a))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_1 <= 1e+139)
		tmp = Float64(x / Float64(a * fma(y, b, y)));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x * N[Exp[N[(N[(N[(y * N[Log[z], $MachinePrecision]), $MachinePrecision] + N[(N[(t + -1.0), $MachinePrecision] * N[Log[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$2 = N[(x / N[(y * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$2, If[LessEqual[t$95$1, 1e+139], N[(x / N[(a * N[(y * b + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (exp.f64 (-.f64 (+.f64 (*.f64 y (log.f64 z)) (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))) b))) y) < -inf.0 or 1.00000000000000003e139 < (/.f64 (*.f64 x (exp.f64 (-.f64 (+.f64 (*.f64 y (log.f64 z)) (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))) b))) y)

   1. Initial program 99.6%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      5. lower-/.f64 N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      6. +-commutative N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z + -1 \cdot \log a}}}{y \cdot e^{b}} \]

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z - \log a}}}{y \cdot e^{b}} \]

      9. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      10. lower-/.f64 N/A

          \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      11. *-commutative N/A

          \[\leadsto x \cdot \frac{\frac{e^{\color{blue}{\log z \cdot y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      12. exp-to-pow N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      13. lower-pow.f64 N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      14. rem-exp-log N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{\color{blue}{a}}}{y \cdot e^{b}} \]

      15. lower-*.f64 N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{\color{blue}{y \cdot e^{b}}} \]

      16. lower-exp.f64 72.3

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 72.3%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot e^{b}}} \]

   6. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-exp.f64 66.4

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

   8. Simplified 66.4%

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

   9. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot y}} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{a \cdot y}} \]

       2. lower-*.f64 33.9

          \[\leadsto \frac{x}{\color{blue}{a \cdot y}} \]

   11. Simplified 33.9%

       \[\leadsto \color{blue}{\frac{x}{a \cdot y}} \]

   if -inf.0 < (/.f64 (*.f64 x (exp.f64 (-.f64 (+.f64 (*.f64 y (log.f64 z)) (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))) b))) y) < 1.00000000000000003e139

   1. Initial program 96.9%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      5. lower-/.f64 N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      6. +-commutative N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z + -1 \cdot \log a}}}{y \cdot e^{b}} \]

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z - \log a}}}{y \cdot e^{b}} \]

      9. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      10. lower-/.f64 N/A

          \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      11. *-commutative N/A

          \[\leadsto x \cdot \frac{\frac{e^{\color{blue}{\log z \cdot y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      12. exp-to-pow N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      13. lower-pow.f64 N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      14. rem-exp-log N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{\color{blue}{a}}}{y \cdot e^{b}} \]

      15. lower-*.f64 N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{\color{blue}{y \cdot e^{b}}} \]

      16. lower-exp.f64 70.5

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 70.5%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot e^{b}}} \]

   6. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-exp.f64 59.0

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

   8. Simplified 59.0%

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

   9. Taylor expanded in b around 0

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

       1. distribute-lft-out N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 49.9

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

   11. Simplified 49.9%

       \[\leadsto \frac{x}{\color{blue}{a \cdot \mathsf{fma}\left(y, b, y\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot e^{\left(y \cdot \log z + \left(t + -1\right) \cdot \log a\right) - b}}{y} \leq -\infty:\\ \;\;\;\;\frac{x}{y \cdot a}\\ \mathbf{elif}\;\frac{x \cdot e^{\left(y \cdot \log z + \left(t + -1\right) \cdot \log a\right) - b}}{y} \leq 10^{+139}:\\ \;\;\;\;\frac{x}{a \cdot \mathsf{fma}\left(y, b, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 83.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t + -1\right) \cdot \log a\\ t_2 := \frac{x \cdot e^{t \cdot \log a - b}}{y}\\ \mathbf{if}\;t\_1 \leq -50000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -20:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot e^{b}\right)}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+157}:\\ \;\;\;\;\frac{x \cdot e^{y \cdot \log z - b}}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (+ t -1.0) (log a)))
        (t_2 (/ (* x (exp (- (* t (log a)) b))) y)))
   (if (<= t_1 -50000.0)
     t_2
     (if (<= t_1 -20.0)
       (/ x (* a (* y (exp b))))
       (if (<= t_1 5e+157) (/ (* x (exp (- (* y (log z)) b))) y) t_2)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t + -1.0) * log(a);
	double t_2 = (x * exp(((t * log(a)) - b))) / y;
	double tmp;
	if (t_1 <= -50000.0) {
		tmp = t_2;
	} else if (t_1 <= -20.0) {
		tmp = x / (a * (y * exp(b)));
	} else if (t_1 <= 5e+157) {
		tmp = (x * exp(((y * log(z)) - b))) / y;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (t + (-1.0d0)) * log(a)
    t_2 = (x * exp(((t * log(a)) - b))) / y
    if (t_1 <= (-50000.0d0)) then
        tmp = t_2
    else if (t_1 <= (-20.0d0)) then
        tmp = x / (a * (y * exp(b)))
    else if (t_1 <= 5d+157) then
        tmp = (x * exp(((y * log(z)) - b))) / y
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t + -1.0) * Math.log(a);
	double t_2 = (x * Math.exp(((t * Math.log(a)) - b))) / y;
	double tmp;
	if (t_1 <= -50000.0) {
		tmp = t_2;
	} else if (t_1 <= -20.0) {
		tmp = x / (a * (y * Math.exp(b)));
	} else if (t_1 <= 5e+157) {
		tmp = (x * Math.exp(((y * Math.log(z)) - b))) / y;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (t + -1.0) * math.log(a)
	t_2 = (x * math.exp(((t * math.log(a)) - b))) / y
	tmp = 0
	if t_1 <= -50000.0:
		tmp = t_2
	elif t_1 <= -20.0:
		tmp = x / (a * (y * math.exp(b)))
	elif t_1 <= 5e+157:
		tmp = (x * math.exp(((y * math.log(z)) - b))) / y
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t + -1.0) * log(a))
	t_2 = Float64(Float64(x * exp(Float64(Float64(t * log(a)) - b))) / y)
	tmp = 0.0
	if (t_1 <= -50000.0)
		tmp = t_2;
	elseif (t_1 <= -20.0)
		tmp = Float64(x / Float64(a * Float64(y * exp(b))));
	elseif (t_1 <= 5e+157)
		tmp = Float64(Float64(x * exp(Float64(Float64(y * log(z)) - b))) / y);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (t + -1.0) * log(a);
	t_2 = (x * exp(((t * log(a)) - b))) / y;
	tmp = 0.0;
	if (t_1 <= -50000.0)
		tmp = t_2;
	elseif (t_1 <= -20.0)
		tmp = x / (a * (y * exp(b)));
	elseif (t_1 <= 5e+157)
		tmp = (x * exp(((y * log(z)) - b))) / y;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t + -1.0), $MachinePrecision] * N[Log[a], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * N[Exp[N[(N[(t * N[Log[a], $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$1, -50000.0], t$95$2, If[LessEqual[t$95$1, -20.0], N[(x / N[(a * N[(y * N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+157], N[(N[(x * N[Exp[N[(N[(y * N[Log[z], $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < -5e4 or 4.99999999999999976e157 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{t \cdot \log a} - b}}{y} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

      3. rem-exp-log N/A

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

      4. lower-log.f64 N/A

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

      5. rem-exp-log 94.8

         \[\leadsto \frac{x \cdot e^{\log \color{blue}{a} \cdot t - b}}{y} \]

   5. Simplified 94.8%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

   if -5e4 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < -20

   1. Initial program 93.8%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      5. lower-/.f64 N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      6. +-commutative N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z + -1 \cdot \log a}}}{y \cdot e^{b}} \]

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z - \log a}}}{y \cdot e^{b}} \]

      9. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      10. lower-/.f64 N/A

          \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      11. *-commutative N/A

          \[\leadsto x \cdot \frac{\frac{e^{\color{blue}{\log z \cdot y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      12. exp-to-pow N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      13. lower-pow.f64 N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      14. rem-exp-log N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{\color{blue}{a}}}{y \cdot e^{b}} \]

      15. lower-*.f64 N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{\color{blue}{y \cdot e^{b}}} \]

      16. lower-exp.f64 95.6

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 95.6%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot e^{b}}} \]

   6. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-exp.f64 86.5

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

   8. Simplified 86.5%

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

   if -20 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < 4.99999999999999976e157

   1. Initial program 99.4%

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

   3. Taylor expanded in y around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]

      2. lower-log.f64 80.1

         \[\leadsto \frac{x \cdot e^{y \cdot \color{blue}{\log z} - b}}{y} \]

   5. Simplified 80.1%

      \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(t + -1\right) \cdot \log a \leq -50000:\\ \;\;\;\;\frac{x \cdot e^{t \cdot \log a - b}}{y}\\ \mathbf{elif}\;\left(t + -1\right) \cdot \log a \leq -20:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot e^{b}\right)}\\ \mathbf{elif}\;\left(t + -1\right) \cdot \log a \leq 5 \cdot 10^{+157}:\\ \;\;\;\;\frac{x \cdot e^{y \cdot \log z - b}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot e^{t \cdot \log a - b}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 79.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t + -1\right) \cdot \log a\\ t_2 := x \cdot \frac{{a}^{\left(t + -1\right)}}{y}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+40}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -20:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot e^{b}\right)}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+157}:\\ \;\;\;\;\frac{x \cdot e^{y \cdot \log z - b}}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (+ t -1.0) (log a))) (t_2 (* x (/ (pow a (+ t -1.0)) y))))
   (if (<= t_1 -4e+40)
     t_2
     (if (<= t_1 -20.0)
       (/ x (* a (* y (exp b))))
       (if (<= t_1 5e+157) (/ (* x (exp (- (* y (log z)) b))) y) t_2)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t + -1.0) * log(a);
	double t_2 = x * (pow(a, (t + -1.0)) / y);
	double tmp;
	if (t_1 <= -4e+40) {
		tmp = t_2;
	} else if (t_1 <= -20.0) {
		tmp = x / (a * (y * exp(b)));
	} else if (t_1 <= 5e+157) {
		tmp = (x * exp(((y * log(z)) - b))) / y;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (t + (-1.0d0)) * log(a)
    t_2 = x * ((a ** (t + (-1.0d0))) / y)
    if (t_1 <= (-4d+40)) then
        tmp = t_2
    else if (t_1 <= (-20.0d0)) then
        tmp = x / (a * (y * exp(b)))
    else if (t_1 <= 5d+157) then
        tmp = (x * exp(((y * log(z)) - b))) / y
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t + -1.0) * Math.log(a);
	double t_2 = x * (Math.pow(a, (t + -1.0)) / y);
	double tmp;
	if (t_1 <= -4e+40) {
		tmp = t_2;
	} else if (t_1 <= -20.0) {
		tmp = x / (a * (y * Math.exp(b)));
	} else if (t_1 <= 5e+157) {
		tmp = (x * Math.exp(((y * Math.log(z)) - b))) / y;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (t + -1.0) * math.log(a)
	t_2 = x * (math.pow(a, (t + -1.0)) / y)
	tmp = 0
	if t_1 <= -4e+40:
		tmp = t_2
	elif t_1 <= -20.0:
		tmp = x / (a * (y * math.exp(b)))
	elif t_1 <= 5e+157:
		tmp = (x * math.exp(((y * math.log(z)) - b))) / y
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t + -1.0) * log(a))
	t_2 = Float64(x * Float64((a ^ Float64(t + -1.0)) / y))
	tmp = 0.0
	if (t_1 <= -4e+40)
		tmp = t_2;
	elseif (t_1 <= -20.0)
		tmp = Float64(x / Float64(a * Float64(y * exp(b))));
	elseif (t_1 <= 5e+157)
		tmp = Float64(Float64(x * exp(Float64(Float64(y * log(z)) - b))) / y);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (t + -1.0) * log(a);
	t_2 = x * ((a ^ (t + -1.0)) / y);
	tmp = 0.0;
	if (t_1 <= -4e+40)
		tmp = t_2;
	elseif (t_1 <= -20.0)
		tmp = x / (a * (y * exp(b)));
	elseif (t_1 <= 5e+157)
		tmp = (x * exp(((y * log(z)) - b))) / y;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t + -1.0), $MachinePrecision] * N[Log[a], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x * N[(N[Power[a, N[(t + -1.0), $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+40], t$95$2, If[LessEqual[t$95$1, -20.0], N[(x / N[(a * N[(y * N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+157], N[(N[(x * N[Exp[N[(N[(y * N[Log[z], $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < -4.00000000000000012e40 or 4.99999999999999976e157 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. exp-diff N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. exp-prod N/A

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

      8. lower-pow.f64 N/A

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

      9. rem-exp-log N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-exp.f64 64.4

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

   5. Simplified 64.4%

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

   6. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. exp-to-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 77.2

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

   8. Simplified 77.2%

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

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. div-inv N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. associate-*l/ N/A

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

      10. metadata-eval N/A

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

      11. *-lft-identity N/A

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

      12. lower-/.f64 87.6

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

   10. Applied egg-rr 87.6%

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

   if -4.00000000000000012e40 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < -20

   1. Initial program 94.5%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      5. lower-/.f64 N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      6. +-commutative N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z + -1 \cdot \log a}}}{y \cdot e^{b}} \]

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z - \log a}}}{y \cdot e^{b}} \]

      9. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      10. lower-/.f64 N/A

          \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      11. *-commutative N/A

          \[\leadsto x \cdot \frac{\frac{e^{\color{blue}{\log z \cdot y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      12. exp-to-pow N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      13. lower-pow.f64 N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      14. rem-exp-log N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{\color{blue}{a}}}{y \cdot e^{b}} \]

      15. lower-*.f64 N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{\color{blue}{y \cdot e^{b}}} \]

      16. lower-exp.f64 94.8

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 94.8%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot e^{b}}} \]

   6. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-exp.f64 84.2

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

   8. Simplified 84.2%

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

   if -20 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < 4.99999999999999976e157

   1. Initial program 99.4%

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

   3. Taylor expanded in y around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]

      2. lower-log.f64 80.1

         \[\leadsto \frac{x \cdot e^{y \cdot \color{blue}{\log z} - b}}{y} \]

   5. Simplified 80.1%

      \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(t + -1\right) \cdot \log a \leq -4 \cdot 10^{+40}:\\ \;\;\;\;x \cdot \frac{{a}^{\left(t + -1\right)}}{y}\\ \mathbf{elif}\;\left(t + -1\right) \cdot \log a \leq -20:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot e^{b}\right)}\\ \mathbf{elif}\;\left(t + -1\right) \cdot \log a \leq 5 \cdot 10^{+157}:\\ \;\;\;\;\frac{x \cdot e^{y \cdot \log z - b}}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{{a}^{\left(t + -1\right)}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 74.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t + -1\right) \cdot \log a\\ t_2 := x \cdot \frac{{a}^{\left(t + -1\right)}}{y}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+40}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 110:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot e^{b}\right)}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+157}:\\ \;\;\;\;x \cdot \frac{{z}^{y}}{y \cdot a}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (+ t -1.0) (log a))) (t_2 (* x (/ (pow a (+ t -1.0)) y))))
   (if (<= t_1 -4e+40)
     t_2
     (if (<= t_1 110.0)
       (/ x (* a (* y (exp b))))
       (if (<= t_1 5e+157) (* x (/ (pow z y) (* y a))) t_2)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t + -1.0) * log(a);
	double t_2 = x * (pow(a, (t + -1.0)) / y);
	double tmp;
	if (t_1 <= -4e+40) {
		tmp = t_2;
	} else if (t_1 <= 110.0) {
		tmp = x / (a * (y * exp(b)));
	} else if (t_1 <= 5e+157) {
		tmp = x * (pow(z, y) / (y * a));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (t + (-1.0d0)) * log(a)
    t_2 = x * ((a ** (t + (-1.0d0))) / y)
    if (t_1 <= (-4d+40)) then
        tmp = t_2
    else if (t_1 <= 110.0d0) then
        tmp = x / (a * (y * exp(b)))
    else if (t_1 <= 5d+157) then
        tmp = x * ((z ** y) / (y * a))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t + -1.0) * Math.log(a);
	double t_2 = x * (Math.pow(a, (t + -1.0)) / y);
	double tmp;
	if (t_1 <= -4e+40) {
		tmp = t_2;
	} else if (t_1 <= 110.0) {
		tmp = x / (a * (y * Math.exp(b)));
	} else if (t_1 <= 5e+157) {
		tmp = x * (Math.pow(z, y) / (y * a));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (t + -1.0) * math.log(a)
	t_2 = x * (math.pow(a, (t + -1.0)) / y)
	tmp = 0
	if t_1 <= -4e+40:
		tmp = t_2
	elif t_1 <= 110.0:
		tmp = x / (a * (y * math.exp(b)))
	elif t_1 <= 5e+157:
		tmp = x * (math.pow(z, y) / (y * a))
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t + -1.0) * log(a))
	t_2 = Float64(x * Float64((a ^ Float64(t + -1.0)) / y))
	tmp = 0.0
	if (t_1 <= -4e+40)
		tmp = t_2;
	elseif (t_1 <= 110.0)
		tmp = Float64(x / Float64(a * Float64(y * exp(b))));
	elseif (t_1 <= 5e+157)
		tmp = Float64(x * Float64((z ^ y) / Float64(y * a)));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (t + -1.0) * log(a);
	t_2 = x * ((a ^ (t + -1.0)) / y);
	tmp = 0.0;
	if (t_1 <= -4e+40)
		tmp = t_2;
	elseif (t_1 <= 110.0)
		tmp = x / (a * (y * exp(b)));
	elseif (t_1 <= 5e+157)
		tmp = x * ((z ^ y) / (y * a));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t + -1.0), $MachinePrecision] * N[Log[a], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x * N[(N[Power[a, N[(t + -1.0), $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+40], t$95$2, If[LessEqual[t$95$1, 110.0], N[(x / N[(a * N[(y * N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+157], N[(x * N[(N[Power[z, y], $MachinePrecision] / N[(y * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < -4.00000000000000012e40 or 4.99999999999999976e157 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. exp-diff N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. exp-prod N/A

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

      8. lower-pow.f64 N/A

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

      9. rem-exp-log N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-exp.f64 64.4

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

   5. Simplified 64.4%

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

   6. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. exp-to-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 77.2

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

   8. Simplified 77.2%

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

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. div-inv N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. associate-*l/ N/A

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

      10. metadata-eval N/A

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

      11. *-lft-identity N/A

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

      12. lower-/.f64 87.6

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

   10. Applied egg-rr 87.6%

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

   if -4.00000000000000012e40 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < 110

   1. Initial program 95.0%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      5. lower-/.f64 N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      6. +-commutative N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z + -1 \cdot \log a}}}{y \cdot e^{b}} \]

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z - \log a}}}{y \cdot e^{b}} \]

      9. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      10. lower-/.f64 N/A

          \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      11. *-commutative N/A

          \[\leadsto x \cdot \frac{\frac{e^{\color{blue}{\log z \cdot y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      12. exp-to-pow N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      13. lower-pow.f64 N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      14. rem-exp-log N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{\color{blue}{a}}}{y \cdot e^{b}} \]

      15. lower-*.f64 N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{\color{blue}{y \cdot e^{b}}} \]

      16. lower-exp.f64 94.2

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 94.2%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot e^{b}}} \]

   6. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-exp.f64 84.7

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

   8. Simplified 84.7%

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

   if 110 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < 4.99999999999999976e157

   1. Initial program 99.3%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      5. lower-/.f64 N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      6. +-commutative N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z + -1 \cdot \log a}}}{y \cdot e^{b}} \]

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z - \log a}}}{y \cdot e^{b}} \]

      9. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      10. lower-/.f64 N/A

          \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      11. *-commutative N/A

          \[\leadsto x \cdot \frac{\frac{e^{\color{blue}{\log z \cdot y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      12. exp-to-pow N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      13. lower-pow.f64 N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      14. rem-exp-log N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{\color{blue}{a}}}{y \cdot e^{b}} \]

      15. lower-*.f64 N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{\color{blue}{y \cdot e^{b}}} \]

      16. lower-exp.f64 77.5

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 77.5%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot e^{b}}} \]

   6. Taylor expanded in b around 0

      \[\leadsto x \cdot \color{blue}{\frac{{z}^{y}}{a \cdot y}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 71.8

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

   8. Simplified 71.8%

      \[\leadsto x \cdot \color{blue}{\frac{{z}^{y}}{y \cdot a}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(t + -1\right) \cdot \log a \leq -4 \cdot 10^{+40}:\\ \;\;\;\;x \cdot \frac{{a}^{\left(t + -1\right)}}{y}\\ \mathbf{elif}\;\left(t + -1\right) \cdot \log a \leq 110:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot e^{b}\right)}\\ \mathbf{elif}\;\left(t + -1\right) \cdot \log a \leq 5 \cdot 10^{+157}:\\ \;\;\;\;x \cdot \frac{{z}^{y}}{y \cdot a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{{a}^{\left(t + -1\right)}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 86.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t + -1\right) \cdot \log a\\ t_2 := \frac{x \cdot e^{t \cdot \log a - b}}{y}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+40}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+43}:\\ \;\;\;\;x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot e^{b}}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (+ t -1.0) (log a)))
        (t_2 (/ (* x (exp (- (* t (log a)) b))) y)))
   (if (<= t_1 -4e+40)
     t_2
     (if (<= t_1 5e+43) (* x (/ (/ (pow z y) a) (* y (exp b)))) t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t + -1.0) * log(a);
	double t_2 = (x * exp(((t * log(a)) - b))) / y;
	double tmp;
	if (t_1 <= -4e+40) {
		tmp = t_2;
	} else if (t_1 <= 5e+43) {
		tmp = x * ((pow(z, y) / a) / (y * exp(b)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (t + (-1.0d0)) * log(a)
    t_2 = (x * exp(((t * log(a)) - b))) / y
    if (t_1 <= (-4d+40)) then
        tmp = t_2
    else if (t_1 <= 5d+43) then
        tmp = x * (((z ** y) / a) / (y * exp(b)))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t + -1.0) * Math.log(a);
	double t_2 = (x * Math.exp(((t * Math.log(a)) - b))) / y;
	double tmp;
	if (t_1 <= -4e+40) {
		tmp = t_2;
	} else if (t_1 <= 5e+43) {
		tmp = x * ((Math.pow(z, y) / a) / (y * Math.exp(b)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (t + -1.0) * math.log(a)
	t_2 = (x * math.exp(((t * math.log(a)) - b))) / y
	tmp = 0
	if t_1 <= -4e+40:
		tmp = t_2
	elif t_1 <= 5e+43:
		tmp = x * ((math.pow(z, y) / a) / (y * math.exp(b)))
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t + -1.0) * log(a))
	t_2 = Float64(Float64(x * exp(Float64(Float64(t * log(a)) - b))) / y)
	tmp = 0.0
	if (t_1 <= -4e+40)
		tmp = t_2;
	elseif (t_1 <= 5e+43)
		tmp = Float64(x * Float64(Float64((z ^ y) / a) / Float64(y * exp(b))));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (t + -1.0) * log(a);
	t_2 = (x * exp(((t * log(a)) - b))) / y;
	tmp = 0.0;
	if (t_1 <= -4e+40)
		tmp = t_2;
	elseif (t_1 <= 5e+43)
		tmp = x * (((z ^ y) / a) / (y * exp(b)));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t + -1.0), $MachinePrecision] * N[Log[a], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * N[Exp[N[(N[(t * N[Log[a], $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+40], t$95$2, If[LessEqual[t$95$1, 5e+43], N[(x * N[(N[(N[Power[z, y], $MachinePrecision] / a), $MachinePrecision] / N[(y * N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < -4.00000000000000012e40 or 5.0000000000000004e43 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{t \cdot \log a} - b}}{y} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

      3. rem-exp-log N/A

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

      4. lower-log.f64 N/A

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

      5. rem-exp-log 91.6

         \[\leadsto \frac{x \cdot e^{\log \color{blue}{a} \cdot t - b}}{y} \]

   5. Simplified 91.6%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

   if -4.00000000000000012e40 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < 5.0000000000000004e43

   1. Initial program 96.7%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      5. lower-/.f64 N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      6. +-commutative N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z + -1 \cdot \log a}}}{y \cdot e^{b}} \]

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z - \log a}}}{y \cdot e^{b}} \]

      9. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      10. lower-/.f64 N/A

          \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      11. *-commutative N/A

          \[\leadsto x \cdot \frac{\frac{e^{\color{blue}{\log z \cdot y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      12. exp-to-pow N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      13. lower-pow.f64 N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      14. rem-exp-log N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{\color{blue}{a}}}{y \cdot e^{b}} \]

      15. lower-*.f64 N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{\color{blue}{y \cdot e^{b}}} \]

      16. lower-exp.f64 90.2

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 90.2%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot e^{b}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(t + -1\right) \cdot \log a \leq -4 \cdot 10^{+40}:\\ \;\;\;\;\frac{x \cdot e^{t \cdot \log a - b}}{y}\\ \mathbf{elif}\;\left(t + -1\right) \cdot \log a \leq 5 \cdot 10^{+43}:\\ \;\;\;\;x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot e^{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot e^{t \cdot \log a - b}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 73.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t + -1\right) \cdot \log a\\ t_2 := x \cdot \frac{{a}^{\left(t + -1\right)}}{y}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+40}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+151}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot e^{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (+ t -1.0) (log a))) (t_2 (* x (/ (pow a (+ t -1.0)) y))))
   (if (<= t_1 -4e+40)
     t_2
     (if (<= t_1 5e+151) (/ x (* a (* y (exp b)))) t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t + -1.0) * log(a);
	double t_2 = x * (pow(a, (t + -1.0)) / y);
	double tmp;
	if (t_1 <= -4e+40) {
		tmp = t_2;
	} else if (t_1 <= 5e+151) {
		tmp = x / (a * (y * exp(b)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (t + (-1.0d0)) * log(a)
    t_2 = x * ((a ** (t + (-1.0d0))) / y)
    if (t_1 <= (-4d+40)) then
        tmp = t_2
    else if (t_1 <= 5d+151) then
        tmp = x / (a * (y * exp(b)))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t + -1.0) * Math.log(a);
	double t_2 = x * (Math.pow(a, (t + -1.0)) / y);
	double tmp;
	if (t_1 <= -4e+40) {
		tmp = t_2;
	} else if (t_1 <= 5e+151) {
		tmp = x / (a * (y * Math.exp(b)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (t + -1.0) * math.log(a)
	t_2 = x * (math.pow(a, (t + -1.0)) / y)
	tmp = 0
	if t_1 <= -4e+40:
		tmp = t_2
	elif t_1 <= 5e+151:
		tmp = x / (a * (y * math.exp(b)))
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t + -1.0) * log(a))
	t_2 = Float64(x * Float64((a ^ Float64(t + -1.0)) / y))
	tmp = 0.0
	if (t_1 <= -4e+40)
		tmp = t_2;
	elseif (t_1 <= 5e+151)
		tmp = Float64(x / Float64(a * Float64(y * exp(b))));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (t + -1.0) * log(a);
	t_2 = x * ((a ^ (t + -1.0)) / y);
	tmp = 0.0;
	if (t_1 <= -4e+40)
		tmp = t_2;
	elseif (t_1 <= 5e+151)
		tmp = x / (a * (y * exp(b)));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t + -1.0), $MachinePrecision] * N[Log[a], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x * N[(N[Power[a, N[(t + -1.0), $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+40], t$95$2, If[LessEqual[t$95$1, 5e+151], N[(x / N[(a * N[(y * N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < -4.00000000000000012e40 or 5.0000000000000002e151 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. exp-diff N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. exp-prod N/A

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

      8. lower-pow.f64 N/A

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

      9. rem-exp-log N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-exp.f64 64.5

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

   5. Simplified 64.5%

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

   6. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. exp-to-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 76.8

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

   8. Simplified 76.8%

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

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. div-inv N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. associate-*l/ N/A

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

      10. metadata-eval N/A

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

      11. *-lft-identity N/A

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

      12. lower-/.f64 86.9

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

   10. Applied egg-rr 86.9%

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

   if -4.00000000000000012e40 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < 5.0000000000000002e151

   1. Initial program 97.1%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      5. lower-/.f64 N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      6. +-commutative N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z + -1 \cdot \log a}}}{y \cdot e^{b}} \]

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z - \log a}}}{y \cdot e^{b}} \]

      9. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      10. lower-/.f64 N/A

          \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      11. *-commutative N/A

          \[\leadsto x \cdot \frac{\frac{e^{\color{blue}{\log z \cdot y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      12. exp-to-pow N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      13. lower-pow.f64 N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      14. rem-exp-log N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{\color{blue}{a}}}{y \cdot e^{b}} \]

      15. lower-*.f64 N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{\color{blue}{y \cdot e^{b}}} \]

      16. lower-exp.f64 87.0

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 87.0%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot e^{b}}} \]

   6. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-exp.f64 74.6

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

   8. Simplified 74.6%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(t + -1\right) \cdot \log a \leq -4 \cdot 10^{+40}:\\ \;\;\;\;x \cdot \frac{{a}^{\left(t + -1\right)}}{y}\\ \mathbf{elif}\;\left(t + -1\right) \cdot \log a \leq 5 \cdot 10^{+151}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot e^{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{{a}^{\left(t + -1\right)}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 65.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \frac{{z}^{y}}{y}\\ \mathbf{if}\;y \leq -1.6:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -2.8 \cdot 10^{-124}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y \cdot 0.5, a \cdot \left(b \cdot b\right), a \cdot \mathsf{fma}\left(y, b, y\right)\right)}\\ \mathbf{elif}\;y \leq 3.85 \cdot 10^{+15}:\\ \;\;\;\;\frac{x \cdot e^{-b}}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (/ (pow z y) y))))
   (if (<= y -1.6)
     t_1
     (if (<= y -2.8e-124)
       (/ x (fma (* y 0.5) (* a (* b b)) (* a (fma y b y))))
       (if (<= y 3.85e+15) (/ (* x (exp (- b))) y) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * (pow(z, y) / y);
	double tmp;
	if (y <= -1.6) {
		tmp = t_1;
	} else if (y <= -2.8e-124) {
		tmp = x / fma((y * 0.5), (a * (b * b)), (a * fma(y, b, y)));
	} else if (y <= 3.85e+15) {
		tmp = (x * exp(-b)) / y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x * Float64((z ^ y) / y))
	tmp = 0.0
	if (y <= -1.6)
		tmp = t_1;
	elseif (y <= -2.8e-124)
		tmp = Float64(x / fma(Float64(y * 0.5), Float64(a * Float64(b * b)), Float64(a * fma(y, b, y))));
	elseif (y <= 3.85e+15)
		tmp = Float64(Float64(x * exp(Float64(-b))) / y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x * N[(N[Power[z, y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.6], t$95$1, If[LessEqual[y, -2.8e-124], N[(x / N[(N[(y * 0.5), $MachinePrecision] * N[(a * N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(a * N[(y * b + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.85e+15], N[(N[(x * N[Exp[(-b)], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.6000000000000001 or 3.85e15 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]

      2. lower-log.f64 83.5

         \[\leadsto \frac{x \cdot e^{y \cdot \color{blue}{\log z} - b}}{y} \]

   5. Simplified 83.5%

      \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]

   6. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{x \cdot {z}^{y}}{y}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 80.1

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

   8. Simplified 80.1%

      \[\leadsto \color{blue}{\frac{{z}^{y} \cdot x}{y}} \]
   9. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 80.1

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

   10. Applied egg-rr 80.1%

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

   if -1.6000000000000001 < y < -2.79999999999999998e-124

   1. Initial program 93.9%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      5. lower-/.f64 N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      6. +-commutative N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z + -1 \cdot \log a}}}{y \cdot e^{b}} \]

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z - \log a}}}{y \cdot e^{b}} \]

      9. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      10. lower-/.f64 N/A

          \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      11. *-commutative N/A

          \[\leadsto x \cdot \frac{\frac{e^{\color{blue}{\log z \cdot y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      12. exp-to-pow N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      13. lower-pow.f64 N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      14. rem-exp-log N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{\color{blue}{a}}}{y \cdot e^{b}} \]

      15. lower-*.f64 N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{\color{blue}{y \cdot e^{b}}} \]

      16. lower-exp.f64 83.5

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 83.5%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot e^{b}}} \]

   6. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-exp.f64 82.9

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

   8. Simplified 82.9%

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

   9. Taylor expanded in b around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. +-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

       8. lower-fma.f64 N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 60.1

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

   11. Simplified 60.1%

       \[\leadsto \frac{x}{a \cdot \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(0.5, y \cdot b, y\right), y\right)}} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-fma.f64 N/A

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

       3. lift-fma.f64 N/A

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

       4. distribute-lft-in N/A

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

       5. *-commutative N/A

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

       6. lift-*.f64 N/A

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

       7. associate-+l+ N/A

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

       8. distribute-rgt-in N/A

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

       9. *-commutative N/A

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

       10. lift-*.f64 N/A

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

       11. associate-*r* N/A

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

       12. associate-*l* N/A

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

       13. associate-*l* N/A

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

       14. lower-fma.f64 N/A

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

   13. Applied egg-rr 63.2%

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

   if -2.79999999999999998e-124 < y < 3.85e15

   1. Initial program 97.1%

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

   3. Taylor expanded in y around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]

      2. lower-log.f64 56.9

         \[\leadsto \frac{x \cdot e^{y \cdot \color{blue}{\log z} - b}}{y} \]

   5. Simplified 56.9%

      \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\mathsf{neg}\left(b\right)}}{y}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 56.9

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

   8. Simplified 56.9%

      \[\leadsto \color{blue}{\frac{x \cdot e^{-b}}{y}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.6:\\ \;\;\;\;x \cdot \frac{{z}^{y}}{y}\\ \mathbf{elif}\;y \leq -2.8 \cdot 10^{-124}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y \cdot 0.5, a \cdot \left(b \cdot b\right), a \cdot \mathsf{fma}\left(y, b, y\right)\right)}\\ \mathbf{elif}\;y \leq 3.85 \cdot 10^{+15}:\\ \;\;\;\;\frac{x \cdot e^{-b}}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{{z}^{y}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 73.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.8 \cdot 10^{+111}:\\ \;\;\;\;\frac{x \cdot e^{-b}}{y}\\ \mathbf{elif}\;b \leq 1.85 \cdot 10^{+100}:\\ \;\;\;\;x \cdot \frac{{a}^{\left(t + -1\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -5.8e+111)
   (/ (* x (exp (- b))) y)
   (if (<= b 1.85e+100)
     (* x (/ (pow a (+ t -1.0)) y))
     (/
      x
      (* a (* y (fma b (fma b (fma b 0.16666666666666666 0.5) 1.0) 1.0)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -5.8e+111) {
		tmp = (x * exp(-b)) / y;
	} else if (b <= 1.85e+100) {
		tmp = x * (pow(a, (t + -1.0)) / y);
	} else {
		tmp = x / (a * (y * fma(b, fma(b, fma(b, 0.16666666666666666, 0.5), 1.0), 1.0)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -5.8e+111)
		tmp = Float64(Float64(x * exp(Float64(-b))) / y);
	elseif (b <= 1.85e+100)
		tmp = Float64(x * Float64((a ^ Float64(t + -1.0)) / y));
	else
		tmp = Float64(x / Float64(a * Float64(y * fma(b, fma(b, fma(b, 0.16666666666666666, 0.5), 1.0), 1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -5.8e+111], N[(N[(x * N[Exp[(-b)], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[b, 1.85e+100], N[(x * N[(N[Power[a, N[(t + -1.0), $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x / N[(a * N[(y * N[(b * N[(b * N[(b * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.7999999999999999e111

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]

      2. lower-log.f64 91.1

         \[\leadsto \frac{x \cdot e^{y \cdot \color{blue}{\log z} - b}}{y} \]

   5. Simplified 91.1%

      \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\mathsf{neg}\left(b\right)}}{y}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 84.3

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

   8. Simplified 84.3%

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

   if -5.7999999999999999e111 < b < 1.8500000000000001e100

   1. Initial program 97.1%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. exp-diff N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. exp-prod N/A

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

      8. lower-pow.f64 N/A

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

      9. rem-exp-log N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-exp.f64 67.3

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

   5. Simplified 67.3%

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

   6. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. exp-to-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. sub-neg N/A

         \[\leadsto {a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot \frac{x}{y} \]

      7. metadata-eval N/A

         \[\leadsto {a}^{\left(t + \color{blue}{-1}\right)} \cdot \frac{x}{y} \]

      8. +-commutative N/A

         \[\leadsto {a}^{\color{blue}{\left(-1 + t\right)}} \cdot \frac{x}{y} \]

      9. lower-+.f64 N/A

         \[\leadsto {a}^{\color{blue}{\left(-1 + t\right)}} \cdot \frac{x}{y} \]

      10. lower-/.f64 63.6

          \[\leadsto {a}^{\left(-1 + t\right)} \cdot \color{blue}{\frac{x}{y}} \]

   8. Simplified 63.6%

      \[\leadsto \color{blue}{{a}^{\left(-1 + t\right)} \cdot \frac{x}{y}} \]
   9. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto {a}^{\color{blue}{\left(-1 + t\right)}} \cdot \frac{x}{y} \]

      2. lift-pow.f64 N/A

         \[\leadsto \color{blue}{{a}^{\left(-1 + t\right)}} \cdot \frac{x}{y} \]

      3. div-inv N/A

         \[\leadsto {a}^{\left(-1 + t\right)} \cdot \color{blue}{\left(x \cdot \frac{1}{y}\right)} \]

      4. *-commutative N/A

         \[\leadsto {a}^{\left(-1 + t\right)} \cdot \color{blue}{\left(\frac{1}{y} \cdot x\right)} \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left({a}^{\left(-1 + t\right)} \cdot \frac{1}{y}\right) \cdot x} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{y} \cdot {a}^{\left(-1 + t\right)}\right)} \cdot x \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{y} \cdot {a}^{\left(-1 + t\right)}\right) \cdot x} \]

      8. metadata-eval N/A

         \[\leadsto \left(\frac{\color{blue}{-1 \cdot -1}}{y} \cdot {a}^{\left(-1 + t\right)}\right) \cdot x \]

      9. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\left(-1 \cdot -1\right) \cdot {a}^{\left(-1 + t\right)}}{y}} \cdot x \]

      10. metadata-eval N/A

          \[\leadsto \frac{\color{blue}{1} \cdot {a}^{\left(-1 + t\right)}}{y} \cdot x \]

      11. *-lft-identity N/A

          \[\leadsto \frac{\color{blue}{{a}^{\left(-1 + t\right)}}}{y} \cdot x \]

      12. lower-/.f64 69.5

          \[\leadsto \color{blue}{\frac{{a}^{\left(-1 + t\right)}}{y}} \cdot x \]

   10. Applied egg-rr 69.5%

       \[\leadsto \color{blue}{\frac{{a}^{\left(-1 + t\right)}}{y} \cdot x} \]

   if 1.8500000000000001e100 < b

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\left(-1 \cdot \log a + y \cdot \log z\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{e^{\left(-1 \cdot \log a + y \cdot \log z\right) - b}}{y}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \frac{e^{\left(-1 \cdot \log a + y \cdot \log z\right) - b}}{y}} \]

      3. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      5. lower-/.f64 N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      6. +-commutative N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z + -1 \cdot \log a}}}{y \cdot e^{b}} \]

      7. mul-1-neg N/A

         \[\leadsto x \cdot \frac{e^{y \cdot \log z + \color{blue}{\left(\mathsf{neg}\left(\log a\right)\right)}}}{y \cdot e^{b}} \]

      8. unsub-neg N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z - \log a}}}{y \cdot e^{b}} \]

      9. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      10. lower-/.f64 N/A

          \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      11. *-commutative N/A

          \[\leadsto x \cdot \frac{\frac{e^{\color{blue}{\log z \cdot y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      12. exp-to-pow N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      13. lower-pow.f64 N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      14. rem-exp-log N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{\color{blue}{a}}}{y \cdot e^{b}} \]

      15. lower-*.f64 N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{\color{blue}{y \cdot e^{b}}} \]

      16. lower-exp.f64 80.5

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 80.5%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot e^{b}}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. lower-exp.f64 89.3

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 89.3%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\left(1 + b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)\right)}\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\left(b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right) + 1\right)}\right)} \]

       2. lower-fma.f64 N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left(b, 1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right), 1\right)}\right)} \]

       3. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \color{blue}{b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right) + 1}, 1\right)\right)} \]

       4. lower-fma.f64 N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, \frac{1}{2} + \frac{1}{6} \cdot b, 1\right)}, 1\right)\right)} \]

       5. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\frac{1}{6} \cdot b + \frac{1}{2}}, 1\right), 1\right)\right)} \]

       6. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{b \cdot \frac{1}{6}} + \frac{1}{2}, 1\right), 1\right)\right)} \]

       7. lower-fma.f64 89.3

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, 0.16666666666666666, 0.5\right)}, 1\right), 1\right)\right)} \]

   11. Simplified 89.3%

       \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), 1\right)}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.8 \cdot 10^{+111}:\\ \;\;\;\;\frac{x \cdot e^{-b}}{y}\\ \mathbf{elif}\;b \leq 1.85 \cdot 10^{+100}:\\ \;\;\;\;x \cdot \frac{{a}^{\left(t + -1\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 62.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \frac{{z}^{y}}{y}\\ \mathbf{if}\;y \leq -1.6:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 3.85 \cdot 10^{+15}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y \cdot 0.5, a \cdot \left(b \cdot b\right), a \cdot \mathsf{fma}\left(y, b, y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (/ (pow z y) y))))
   (if (<= y -1.6)
     t_1
     (if (<= y 3.85e+15)
       (/ x (fma (* y 0.5) (* a (* b b)) (* a (fma y b y))))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * (pow(z, y) / y);
	double tmp;
	if (y <= -1.6) {
		tmp = t_1;
	} else if (y <= 3.85e+15) {
		tmp = x / fma((y * 0.5), (a * (b * b)), (a * fma(y, b, y)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x * Float64((z ^ y) / y))
	tmp = 0.0
	if (y <= -1.6)
		tmp = t_1;
	elseif (y <= 3.85e+15)
		tmp = Float64(x / fma(Float64(y * 0.5), Float64(a * Float64(b * b)), Float64(a * fma(y, b, y))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x * N[(N[Power[z, y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.6], t$95$1, If[LessEqual[y, 3.85e+15], N[(x / N[(N[(y * 0.5), $MachinePrecision] * N[(a * N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(a * N[(y * b + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.6000000000000001 or 3.85e15 < y

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]

      2. lower-log.f64 83.5

         \[\leadsto \frac{x \cdot e^{y \cdot \color{blue}{\log z} - b}}{y} \]

   5. Simplified 83.5%

      \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]

   6. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{x \cdot {z}^{y}}{y}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot {z}^{y}}{y}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{{z}^{y} \cdot x}}{y} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{{z}^{y} \cdot x}}{y} \]

      4. lower-pow.f64 80.1

         \[\leadsto \frac{\color{blue}{{z}^{y}} \cdot x}{y} \]

   8. Simplified 80.1%

      \[\leadsto \color{blue}{\frac{{z}^{y} \cdot x}{y}} \]
   9. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{z}^{y}} \cdot x}{y} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{{z}^{y} \cdot x}}{y} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{{z}^{y} \cdot x}}{y} \]

      4. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{{z}^{y}}{y} \cdot x} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{{z}^{y}}{y} \cdot x} \]

      6. lower-/.f64 80.1

         \[\leadsto \color{blue}{\frac{{z}^{y}}{y}} \cdot x \]

   10. Applied egg-rr 80.1%

       \[\leadsto \color{blue}{\frac{{z}^{y}}{y} \cdot x} \]

   if -1.6000000000000001 < y < 3.85e15

   1. Initial program 96.4%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\left(-1 \cdot \log a + y \cdot \log z\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{e^{\left(-1 \cdot \log a + y \cdot \log z\right) - b}}{y}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \frac{e^{\left(-1 \cdot \log a + y \cdot \log z\right) - b}}{y}} \]

      3. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      5. lower-/.f64 N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      6. +-commutative N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z + -1 \cdot \log a}}}{y \cdot e^{b}} \]

      7. mul-1-neg N/A

         \[\leadsto x \cdot \frac{e^{y \cdot \log z + \color{blue}{\left(\mathsf{neg}\left(\log a\right)\right)}}}{y \cdot e^{b}} \]

      8. unsub-neg N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z - \log a}}}{y \cdot e^{b}} \]

      9. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      10. lower-/.f64 N/A

          \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      11. *-commutative N/A

          \[\leadsto x \cdot \frac{\frac{e^{\color{blue}{\log z \cdot y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      12. exp-to-pow N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      13. lower-pow.f64 N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      14. rem-exp-log N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{\color{blue}{a}}}{y \cdot e^{b}} \]

      15. lower-*.f64 N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{\color{blue}{y \cdot e^{b}}} \]

      16. lower-exp.f64 74.6

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 74.6%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot e^{b}}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. lower-exp.f64 74.9

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 74.9%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y + b \cdot \left(y + \frac{1}{2} \cdot \left(b \cdot y\right)\right)\right)}} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\left(b \cdot \left(y + \frac{1}{2} \cdot \left(b \cdot y\right)\right) + y\right)}} \]

       2. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(b \cdot \left(y + \frac{1}{2} \cdot \color{blue}{\left(y \cdot b\right)}\right) + y\right)} \]

       3. associate-*r* N/A

          \[\leadsto \frac{x}{a \cdot \left(b \cdot \left(y + \color{blue}{\left(\frac{1}{2} \cdot y\right) \cdot b}\right) + y\right)} \]

       4. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(b \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot y\right) \cdot b + y\right)} + y\right)} \]

       5. lower-fma.f64 N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\mathsf{fma}\left(b, \left(\frac{1}{2} \cdot y\right) \cdot b + y, y\right)}} \]

       6. associate-*r* N/A

          \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \color{blue}{\frac{1}{2} \cdot \left(y \cdot b\right)} + y, y\right)} \]

       7. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \frac{1}{2} \cdot \color{blue}{\left(b \cdot y\right)} + y, y\right)} \]

       8. lower-fma.f64 N/A

          \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(\frac{1}{2}, b \cdot y, y\right)}, y\right)} \]

       9. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{y \cdot b}, y\right), y\right)} \]

       10. lower-*.f64 45.3

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(0.5, \color{blue}{y \cdot b}, y\right), y\right)} \]

   11. Simplified 45.3%

       \[\leadsto \frac{x}{a \cdot \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(0.5, y \cdot b, y\right), y\right)}} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{x}{a \cdot \left(b \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(y \cdot b\right)} + y\right) + y\right)} \]

       2. lift-fma.f64 N/A

          \[\leadsto \frac{x}{a \cdot \left(b \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, y \cdot b, y\right)} + y\right)} \]

       3. lift-fma.f64 N/A

          \[\leadsto \frac{x}{a \cdot \left(b \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(y \cdot b\right) + y\right)} + y\right)} \]

       4. distribute-lft-in N/A

          \[\leadsto \frac{x}{a \cdot \left(\color{blue}{\left(b \cdot \left(\frac{1}{2} \cdot \left(y \cdot b\right)\right) + b \cdot y\right)} + y\right)} \]

       5. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(\left(b \cdot \left(\frac{1}{2} \cdot \left(y \cdot b\right)\right) + \color{blue}{y \cdot b}\right) + y\right)} \]

       6. lift-*.f64 N/A

          \[\leadsto \frac{x}{a \cdot \left(\left(b \cdot \left(\frac{1}{2} \cdot \left(y \cdot b\right)\right) + \color{blue}{y \cdot b}\right) + y\right)} \]

       7. associate-+l+ N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\left(b \cdot \left(\frac{1}{2} \cdot \left(y \cdot b\right)\right) + \left(y \cdot b + y\right)\right)}} \]

       8. distribute-rgt-in N/A

          \[\leadsto \frac{x}{\color{blue}{\left(b \cdot \left(\frac{1}{2} \cdot \left(y \cdot b\right)\right)\right) \cdot a + \left(y \cdot b + y\right) \cdot a}} \]

       9. *-commutative N/A

          \[\leadsto \frac{x}{\color{blue}{\left(\left(\frac{1}{2} \cdot \left(y \cdot b\right)\right) \cdot b\right)} \cdot a + \left(y \cdot b + y\right) \cdot a} \]

       10. lift-*.f64 N/A

           \[\leadsto \frac{x}{\left(\left(\frac{1}{2} \cdot \color{blue}{\left(y \cdot b\right)}\right) \cdot b\right) \cdot a + \left(y \cdot b + y\right) \cdot a} \]

       11. associate-*r* N/A

           \[\leadsto \frac{x}{\left(\color{blue}{\left(\left(\frac{1}{2} \cdot y\right) \cdot b\right)} \cdot b\right) \cdot a + \left(y \cdot b + y\right) \cdot a} \]

       12. associate-*l* N/A

           \[\leadsto \frac{x}{\color{blue}{\left(\left(\frac{1}{2} \cdot y\right) \cdot \left(b \cdot b\right)\right)} \cdot a + \left(y \cdot b + y\right) \cdot a} \]

       13. associate-*l* N/A

           \[\leadsto \frac{x}{\color{blue}{\left(\frac{1}{2} \cdot y\right) \cdot \left(\left(b \cdot b\right) \cdot a\right)} + \left(y \cdot b + y\right) \cdot a} \]

       14. lower-fma.f64 N/A

           \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot y, \left(b \cdot b\right) \cdot a, \left(y \cdot b + y\right) \cdot a\right)}} \]

   13. Applied egg-rr 50.7%

       \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(y \cdot 0.5, \left(b \cdot b\right) \cdot a, \mathsf{fma}\left(y, b, y\right) \cdot a\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.6:\\ \;\;\;\;x \cdot \frac{{z}^{y}}{y}\\ \mathbf{elif}\;y \leq 3.85 \cdot 10^{+15}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y \cdot 0.5, a \cdot \left(b \cdot b\right), a \cdot \mathsf{fma}\left(y, b, y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{{z}^{y}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 52.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, b \cdot 0.5, y\right)\\ t_2 := \mathsf{fma}\left(t\_1, -b, y\right)\\ t_3 := a \cdot \mathsf{fma}\left(b, t\_1, -y\right)\\ t_4 := \frac{x}{y \cdot a}\\ t_5 := \mathsf{fma}\left(\mathsf{fma}\left(b, 0.5, 1\right), y \cdot b, y\right)\\ \mathbf{if}\;b \leq -2.55 \cdot 10^{+26}:\\ \;\;\;\;\mathsf{fma}\left(b, \mathsf{fma}\left(b, \left(b \cdot t\_4\right) \cdot -0.16666666666666666, t\_4 \cdot \mathsf{fma}\left(b, 0.5, -1\right)\right), t\_4\right)\\ \mathbf{elif}\;b \leq -1.05 \cdot 10^{-262}:\\ \;\;\;\;\frac{x}{\frac{a \cdot \left(t\_5 \cdot t\_2\right)}{t\_2}}\\ \mathbf{elif}\;b \leq 2.6 \cdot 10^{-90}:\\ \;\;\;\;\frac{x}{b \cdot \left(b \cdot \mathsf{fma}\left(\frac{y}{b}, a + \frac{a}{b}, a \cdot \left(y \cdot 0.5\right)\right)\right)}\\ \mathbf{elif}\;b \leq 1450000000:\\ \;\;\;\;\frac{x}{\frac{\left(a \cdot t\_5\right) \cdot t\_3}{t\_3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma y (* b 0.5) y))
        (t_2 (fma t_1 (- b) y))
        (t_3 (* a (fma b t_1 (- y))))
        (t_4 (/ x (* y a)))
        (t_5 (fma (fma b 0.5 1.0) (* y b) y)))
   (if (<= b -2.55e+26)
     (fma
      b
      (fma b (* (* b t_4) -0.16666666666666666) (* t_4 (fma b 0.5 -1.0)))
      t_4)
     (if (<= b -1.05e-262)
       (/ x (/ (* a (* t_5 t_2)) t_2))
       (if (<= b 2.6e-90)
         (/ x (* b (* b (fma (/ y b) (+ a (/ a b)) (* a (* y 0.5))))))
         (if (<= b 1450000000.0)
           (/ x (/ (* (* a t_5) t_3) t_3))
           (/
            x
            (*
             a
             (*
              y
              (fma b (fma b (fma b 0.16666666666666666 0.5) 1.0) 1.0))))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(y, (b * 0.5), y);
	double t_2 = fma(t_1, -b, y);
	double t_3 = a * fma(b, t_1, -y);
	double t_4 = x / (y * a);
	double t_5 = fma(fma(b, 0.5, 1.0), (y * b), y);
	double tmp;
	if (b <= -2.55e+26) {
		tmp = fma(b, fma(b, ((b * t_4) * -0.16666666666666666), (t_4 * fma(b, 0.5, -1.0))), t_4);
	} else if (b <= -1.05e-262) {
		tmp = x / ((a * (t_5 * t_2)) / t_2);
	} else if (b <= 2.6e-90) {
		tmp = x / (b * (b * fma((y / b), (a + (a / b)), (a * (y * 0.5)))));
	} else if (b <= 1450000000.0) {
		tmp = x / (((a * t_5) * t_3) / t_3);
	} else {
		tmp = x / (a * (y * fma(b, fma(b, fma(b, 0.16666666666666666, 0.5), 1.0), 1.0)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(y, Float64(b * 0.5), y)
	t_2 = fma(t_1, Float64(-b), y)
	t_3 = Float64(a * fma(b, t_1, Float64(-y)))
	t_4 = Float64(x / Float64(y * a))
	t_5 = fma(fma(b, 0.5, 1.0), Float64(y * b), y)
	tmp = 0.0
	if (b <= -2.55e+26)
		tmp = fma(b, fma(b, Float64(Float64(b * t_4) * -0.16666666666666666), Float64(t_4 * fma(b, 0.5, -1.0))), t_4);
	elseif (b <= -1.05e-262)
		tmp = Float64(x / Float64(Float64(a * Float64(t_5 * t_2)) / t_2));
	elseif (b <= 2.6e-90)
		tmp = Float64(x / Float64(b * Float64(b * fma(Float64(y / b), Float64(a + Float64(a / b)), Float64(a * Float64(y * 0.5))))));
	elseif (b <= 1450000000.0)
		tmp = Float64(x / Float64(Float64(Float64(a * t_5) * t_3) / t_3));
	else
		tmp = Float64(x / Float64(a * Float64(y * fma(b, fma(b, fma(b, 0.16666666666666666, 0.5), 1.0), 1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(y * N[(b * 0.5), $MachinePrecision] + y), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * (-b) + y), $MachinePrecision]}, Block[{t$95$3 = N[(a * N[(b * t$95$1 + (-y)), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(x / N[(y * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(b * 0.5 + 1.0), $MachinePrecision] * N[(y * b), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[b, -2.55e+26], N[(b * N[(b * N[(N[(b * t$95$4), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + N[(t$95$4 * N[(b * 0.5 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision], If[LessEqual[b, -1.05e-262], N[(x / N[(N[(a * N[(t$95$5 * t$95$2), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.6e-90], N[(x / N[(b * N[(b * N[(N[(y / b), $MachinePrecision] * N[(a + N[(a / b), $MachinePrecision]), $MachinePrecision] + N[(a * N[(y * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1450000000.0], N[(x / N[(N[(N[(a * t$95$5), $MachinePrecision] * t$95$3), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision], N[(x / N[(a * N[(y * N[(b * N[(b * N[(b * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if b < -2.5499999999999999e26

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\left(-1 \cdot \log a + y \cdot \log z\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{e^{\left(-1 \cdot \log a + y \cdot \log z\right) - b}}{y}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \frac{e^{\left(-1 \cdot \log a + y \cdot \log z\right) - b}}{y}} \]

      3. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      5. lower-/.f64 N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      6. +-commutative N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z + -1 \cdot \log a}}}{y \cdot e^{b}} \]

      7. mul-1-neg N/A

         \[\leadsto x \cdot \frac{e^{y \cdot \log z + \color{blue}{\left(\mathsf{neg}\left(\log a\right)\right)}}}{y \cdot e^{b}} \]

      8. unsub-neg N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z - \log a}}}{y \cdot e^{b}} \]

      9. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      10. lower-/.f64 N/A

          \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      11. *-commutative N/A

          \[\leadsto x \cdot \frac{\frac{e^{\color{blue}{\log z \cdot y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      12. exp-to-pow N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      13. lower-pow.f64 N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      14. rem-exp-log N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{\color{blue}{a}}}{y \cdot e^{b}} \]

      15. lower-*.f64 N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{\color{blue}{y \cdot e^{b}}} \]

      16. lower-exp.f64 69.0

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 69.0%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot e^{b}}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. lower-exp.f64 75.4

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 75.4%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \color{blue}{b \cdot \left(b \cdot \left(-1 \cdot \left(b \cdot \left(-1 \cdot \left(-1 \cdot \frac{x}{a \cdot y} + \frac{1}{2} \cdot \frac{x}{a \cdot y}\right) + \left(\frac{-1}{2} \cdot \frac{x}{a \cdot y} + \frac{1}{6} \cdot \frac{x}{a \cdot y}\right)\right)\right) - \left(-1 \cdot \frac{x}{a \cdot y} + \frac{1}{2} \cdot \frac{x}{a \cdot y}\right)\right) - \frac{x}{a \cdot y}\right) + \frac{x}{a \cdot y}} \]

   11. Simplified 63.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \left(b \cdot \frac{x}{a \cdot y}\right) \cdot -0.16666666666666666, \frac{x}{a \cdot y} \cdot \mathsf{fma}\left(b, 0.5, -1\right)\right), \frac{x}{a \cdot y}\right)} \]

   if -2.5499999999999999e26 < b < -1.05e-262

   1. Initial program 95.5%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\left(-1 \cdot \log a + y \cdot \log z\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{e^{\left(-1 \cdot \log a + y \cdot \log z\right) - b}}{y}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \frac{e^{\left(-1 \cdot \log a + y \cdot \log z\right) - b}}{y}} \]

      3. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      5. lower-/.f64 N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      6. +-commutative N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z + -1 \cdot \log a}}}{y \cdot e^{b}} \]

      7. mul-1-neg N/A

         \[\leadsto x \cdot \frac{e^{y \cdot \log z + \color{blue}{\left(\mathsf{neg}\left(\log a\right)\right)}}}{y \cdot e^{b}} \]

      8. unsub-neg N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z - \log a}}}{y \cdot e^{b}} \]

      9. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      10. lower-/.f64 N/A

          \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      11. *-commutative N/A

          \[\leadsto x \cdot \frac{\frac{e^{\color{blue}{\log z \cdot y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      12. exp-to-pow N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      13. lower-pow.f64 N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      14. rem-exp-log N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{\color{blue}{a}}}{y \cdot e^{b}} \]

      15. lower-*.f64 N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{\color{blue}{y \cdot e^{b}}} \]

      16. lower-exp.f64 67.3

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 67.3%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot e^{b}}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. lower-exp.f64 49.5

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 49.5%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y + b \cdot \left(y + \frac{1}{2} \cdot \left(b \cdot y\right)\right)\right)}} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\left(b \cdot \left(y + \frac{1}{2} \cdot \left(b \cdot y\right)\right) + y\right)}} \]

       2. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(b \cdot \left(y + \frac{1}{2} \cdot \color{blue}{\left(y \cdot b\right)}\right) + y\right)} \]

       3. associate-*r* N/A

          \[\leadsto \frac{x}{a \cdot \left(b \cdot \left(y + \color{blue}{\left(\frac{1}{2} \cdot y\right) \cdot b}\right) + y\right)} \]

       4. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(b \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot y\right) \cdot b + y\right)} + y\right)} \]

       5. lower-fma.f64 N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\mathsf{fma}\left(b, \left(\frac{1}{2} \cdot y\right) \cdot b + y, y\right)}} \]

       6. associate-*r* N/A

          \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \color{blue}{\frac{1}{2} \cdot \left(y \cdot b\right)} + y, y\right)} \]

       7. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \frac{1}{2} \cdot \color{blue}{\left(b \cdot y\right)} + y, y\right)} \]

       8. lower-fma.f64 N/A

          \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(\frac{1}{2}, b \cdot y, y\right)}, y\right)} \]

       9. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{y \cdot b}, y\right), y\right)} \]

       10. lower-*.f64 46.2

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(0.5, \color{blue}{y \cdot b}, y\right), y\right)} \]

   11. Simplified 46.2%

       \[\leadsto \frac{x}{a \cdot \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(0.5, y \cdot b, y\right), y\right)}} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{x}{a \cdot \left(b \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(y \cdot b\right)} + y\right) + y\right)} \]

       2. lift-fma.f64 N/A

          \[\leadsto \frac{x}{a \cdot \left(b \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, y \cdot b, y\right)} + y\right)} \]

       3. flip-+ N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\frac{\left(b \cdot \mathsf{fma}\left(\frac{1}{2}, y \cdot b, y\right)\right) \cdot \left(b \cdot \mathsf{fma}\left(\frac{1}{2}, y \cdot b, y\right)\right) - y \cdot y}{b \cdot \mathsf{fma}\left(\frac{1}{2}, y \cdot b, y\right) - y}}} \]

       4. frac-2neg N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\frac{\mathsf{neg}\left(\left(\left(b \cdot \mathsf{fma}\left(\frac{1}{2}, y \cdot b, y\right)\right) \cdot \left(b \cdot \mathsf{fma}\left(\frac{1}{2}, y \cdot b, y\right)\right) - y \cdot y\right)\right)}{\mathsf{neg}\left(\left(b \cdot \mathsf{fma}\left(\frac{1}{2}, y \cdot b, y\right) - y\right)\right)}}} \]

       5. associate-*r/ N/A

          \[\leadsto \frac{x}{\color{blue}{\frac{a \cdot \left(\mathsf{neg}\left(\left(\left(b \cdot \mathsf{fma}\left(\frac{1}{2}, y \cdot b, y\right)\right) \cdot \left(b \cdot \mathsf{fma}\left(\frac{1}{2}, y \cdot b, y\right)\right) - y \cdot y\right)\right)\right)}{\mathsf{neg}\left(\left(b \cdot \mathsf{fma}\left(\frac{1}{2}, y \cdot b, y\right) - y\right)\right)}}} \]

       6. lower-/.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{\frac{a \cdot \left(\mathsf{neg}\left(\left(\left(b \cdot \mathsf{fma}\left(\frac{1}{2}, y \cdot b, y\right)\right) \cdot \left(b \cdot \mathsf{fma}\left(\frac{1}{2}, y \cdot b, y\right)\right) - y \cdot y\right)\right)\right)}{\mathsf{neg}\left(\left(b \cdot \mathsf{fma}\left(\frac{1}{2}, y \cdot b, y\right) - y\right)\right)}}} \]

   13. Applied egg-rr 61.4%

       \[\leadsto \frac{x}{\color{blue}{\frac{a \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b, 0.5, 1\right), y \cdot b, y\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, b \cdot 0.5, y\right), -b, y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(y, b \cdot 0.5, y\right), -b, y\right)}}} \]

   if -1.05e-262 < b < 2.6e-90

   1. Initial program 96.3%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\left(-1 \cdot \log a + y \cdot \log z\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{e^{\left(-1 \cdot \log a + y \cdot \log z\right) - b}}{y}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \frac{e^{\left(-1 \cdot \log a + y \cdot \log z\right) - b}}{y}} \]

      3. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      5. lower-/.f64 N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      6. +-commutative N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z + -1 \cdot \log a}}}{y \cdot e^{b}} \]

      7. mul-1-neg N/A

         \[\leadsto x \cdot \frac{e^{y \cdot \log z + \color{blue}{\left(\mathsf{neg}\left(\log a\right)\right)}}}{y \cdot e^{b}} \]

      8. unsub-neg N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z - \log a}}}{y \cdot e^{b}} \]

      9. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      10. lower-/.f64 N/A

          \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      11. *-commutative N/A

          \[\leadsto x \cdot \frac{\frac{e^{\color{blue}{\log z \cdot y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      12. exp-to-pow N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      13. lower-pow.f64 N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      14. rem-exp-log N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{\color{blue}{a}}}{y \cdot e^{b}} \]

      15. lower-*.f64 N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{\color{blue}{y \cdot e^{b}}} \]

      16. lower-exp.f64 73.0

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 73.0%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot e^{b}}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. lower-exp.f64 42.0

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 42.0%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y + b \cdot \left(y + \frac{1}{2} \cdot \left(b \cdot y\right)\right)\right)}} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\left(b \cdot \left(y + \frac{1}{2} \cdot \left(b \cdot y\right)\right) + y\right)}} \]

       2. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(b \cdot \left(y + \frac{1}{2} \cdot \color{blue}{\left(y \cdot b\right)}\right) + y\right)} \]

       3. associate-*r* N/A

          \[\leadsto \frac{x}{a \cdot \left(b \cdot \left(y + \color{blue}{\left(\frac{1}{2} \cdot y\right) \cdot b}\right) + y\right)} \]

       4. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(b \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot y\right) \cdot b + y\right)} + y\right)} \]

       5. lower-fma.f64 N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\mathsf{fma}\left(b, \left(\frac{1}{2} \cdot y\right) \cdot b + y, y\right)}} \]

       6. associate-*r* N/A

          \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \color{blue}{\frac{1}{2} \cdot \left(y \cdot b\right)} + y, y\right)} \]

       7. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \frac{1}{2} \cdot \color{blue}{\left(b \cdot y\right)} + y, y\right)} \]

       8. lower-fma.f64 N/A

          \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(\frac{1}{2}, b \cdot y, y\right)}, y\right)} \]

       9. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{y \cdot b}, y\right), y\right)} \]

       10. lower-*.f64 42.0

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(0.5, \color{blue}{y \cdot b}, y\right), y\right)} \]

   11. Simplified 42.0%

       \[\leadsto \frac{x}{a \cdot \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(0.5, y \cdot b, y\right), y\right)}} \]

   12. Taylor expanded in b around inf

       \[\leadsto \frac{x}{\color{blue}{{b}^{2} \cdot \left(\frac{1}{2} \cdot \left(a \cdot y\right) + \left(\frac{a \cdot y}{b} + \frac{a \cdot y}{{b}^{2}}\right)\right)}} \]
   13. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \frac{x}{\color{blue}{\left(b \cdot b\right)} \cdot \left(\frac{1}{2} \cdot \left(a \cdot y\right) + \left(\frac{a \cdot y}{b} + \frac{a \cdot y}{{b}^{2}}\right)\right)} \]

       2. associate-*l* N/A

          \[\leadsto \frac{x}{\color{blue}{b \cdot \left(b \cdot \left(\frac{1}{2} \cdot \left(a \cdot y\right) + \left(\frac{a \cdot y}{b} + \frac{a \cdot y}{{b}^{2}}\right)\right)\right)}} \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{b \cdot \left(b \cdot \left(\frac{1}{2} \cdot \left(a \cdot y\right) + \left(\frac{a \cdot y}{b} + \frac{a \cdot y}{{b}^{2}}\right)\right)\right)}} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{x}{b \cdot \color{blue}{\left(b \cdot \left(\frac{1}{2} \cdot \left(a \cdot y\right) + \left(\frac{a \cdot y}{b} + \frac{a \cdot y}{{b}^{2}}\right)\right)\right)}} \]

       5. +-commutative N/A

          \[\leadsto \frac{x}{b \cdot \left(b \cdot \color{blue}{\left(\left(\frac{a \cdot y}{b} + \frac{a \cdot y}{{b}^{2}}\right) + \frac{1}{2} \cdot \left(a \cdot y\right)\right)}\right)} \]

       6. associate-/l* N/A

          \[\leadsto \frac{x}{b \cdot \left(b \cdot \left(\left(\color{blue}{a \cdot \frac{y}{b}} + \frac{a \cdot y}{{b}^{2}}\right) + \frac{1}{2} \cdot \left(a \cdot y\right)\right)\right)} \]

       7. unpow2 N/A

          \[\leadsto \frac{x}{b \cdot \left(b \cdot \left(\left(a \cdot \frac{y}{b} + \frac{a \cdot y}{\color{blue}{b \cdot b}}\right) + \frac{1}{2} \cdot \left(a \cdot y\right)\right)\right)} \]

       8. times-frac N/A

          \[\leadsto \frac{x}{b \cdot \left(b \cdot \left(\left(a \cdot \frac{y}{b} + \color{blue}{\frac{a}{b} \cdot \frac{y}{b}}\right) + \frac{1}{2} \cdot \left(a \cdot y\right)\right)\right)} \]

       9. distribute-rgt-out N/A

          \[\leadsto \frac{x}{b \cdot \left(b \cdot \left(\color{blue}{\frac{y}{b} \cdot \left(a + \frac{a}{b}\right)} + \frac{1}{2} \cdot \left(a \cdot y\right)\right)\right)} \]

       10. lower-fma.f64 N/A

           \[\leadsto \frac{x}{b \cdot \left(b \cdot \color{blue}{\mathsf{fma}\left(\frac{y}{b}, a + \frac{a}{b}, \frac{1}{2} \cdot \left(a \cdot y\right)\right)}\right)} \]

       11. lower-/.f64 N/A

           \[\leadsto \frac{x}{b \cdot \left(b \cdot \mathsf{fma}\left(\color{blue}{\frac{y}{b}}, a + \frac{a}{b}, \frac{1}{2} \cdot \left(a \cdot y\right)\right)\right)} \]

       12. lower-+.f64 N/A

           \[\leadsto \frac{x}{b \cdot \left(b \cdot \mathsf{fma}\left(\frac{y}{b}, \color{blue}{a + \frac{a}{b}}, \frac{1}{2} \cdot \left(a \cdot y\right)\right)\right)} \]

       13. lower-/.f64 N/A

           \[\leadsto \frac{x}{b \cdot \left(b \cdot \mathsf{fma}\left(\frac{y}{b}, a + \color{blue}{\frac{a}{b}}, \frac{1}{2} \cdot \left(a \cdot y\right)\right)\right)} \]

       14. *-commutative N/A

           \[\leadsto \frac{x}{b \cdot \left(b \cdot \mathsf{fma}\left(\frac{y}{b}, a + \frac{a}{b}, \color{blue}{\left(a \cdot y\right) \cdot \frac{1}{2}}\right)\right)} \]

       15. associate-*l* N/A

           \[\leadsto \frac{x}{b \cdot \left(b \cdot \mathsf{fma}\left(\frac{y}{b}, a + \frac{a}{b}, \color{blue}{a \cdot \left(y \cdot \frac{1}{2}\right)}\right)\right)} \]

       16. *-commutative N/A

           \[\leadsto \frac{x}{b \cdot \left(b \cdot \mathsf{fma}\left(\frac{y}{b}, a + \frac{a}{b}, a \cdot \color{blue}{\left(\frac{1}{2} \cdot y\right)}\right)\right)} \]

       17. lower-*.f64 N/A

           \[\leadsto \frac{x}{b \cdot \left(b \cdot \mathsf{fma}\left(\frac{y}{b}, a + \frac{a}{b}, \color{blue}{a \cdot \left(\frac{1}{2} \cdot y\right)}\right)\right)} \]

       18. *-commutative N/A

           \[\leadsto \frac{x}{b \cdot \left(b \cdot \mathsf{fma}\left(\frac{y}{b}, a + \frac{a}{b}, a \cdot \color{blue}{\left(y \cdot \frac{1}{2}\right)}\right)\right)} \]

       19. lower-*.f64 55.8

           \[\leadsto \frac{x}{b \cdot \left(b \cdot \mathsf{fma}\left(\frac{y}{b}, a + \frac{a}{b}, a \cdot \color{blue}{\left(y \cdot 0.5\right)}\right)\right)} \]

   14. Simplified 55.8%

       \[\leadsto \frac{x}{\color{blue}{b \cdot \left(b \cdot \mathsf{fma}\left(\frac{y}{b}, a + \frac{a}{b}, a \cdot \left(y \cdot 0.5\right)\right)\right)}} \]

   if 2.6e-90 < b < 1.45e9

   1. Initial program 97.8%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\left(-1 \cdot \log a + y \cdot \log z\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{e^{\left(-1 \cdot \log a + y \cdot \log z\right) - b}}{y}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \frac{e^{\left(-1 \cdot \log a + y \cdot \log z\right) - b}}{y}} \]

      3. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      5. lower-/.f64 N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      6. +-commutative N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z + -1 \cdot \log a}}}{y \cdot e^{b}} \]

      7. mul-1-neg N/A

         \[\leadsto x \cdot \frac{e^{y \cdot \log z + \color{blue}{\left(\mathsf{neg}\left(\log a\right)\right)}}}{y \cdot e^{b}} \]

      8. unsub-neg N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z - \log a}}}{y \cdot e^{b}} \]

      9. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      10. lower-/.f64 N/A

          \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      11. *-commutative N/A

          \[\leadsto x \cdot \frac{\frac{e^{\color{blue}{\log z \cdot y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      12. exp-to-pow N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      13. lower-pow.f64 N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      14. rem-exp-log N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{\color{blue}{a}}}{y \cdot e^{b}} \]

      15. lower-*.f64 N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{\color{blue}{y \cdot e^{b}}} \]

      16. lower-exp.f64 79.4

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 79.4%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot e^{b}}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. lower-exp.f64 51.8

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 51.8%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y + b \cdot \left(y + \frac{1}{2} \cdot \left(b \cdot y\right)\right)\right)}} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\left(b \cdot \left(y + \frac{1}{2} \cdot \left(b \cdot y\right)\right) + y\right)}} \]

       2. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(b \cdot \left(y + \frac{1}{2} \cdot \color{blue}{\left(y \cdot b\right)}\right) + y\right)} \]

       3. associate-*r* N/A

          \[\leadsto \frac{x}{a \cdot \left(b \cdot \left(y + \color{blue}{\left(\frac{1}{2} \cdot y\right) \cdot b}\right) + y\right)} \]

       4. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(b \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot y\right) \cdot b + y\right)} + y\right)} \]

       5. lower-fma.f64 N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\mathsf{fma}\left(b, \left(\frac{1}{2} \cdot y\right) \cdot b + y, y\right)}} \]

       6. associate-*r* N/A

          \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \color{blue}{\frac{1}{2} \cdot \left(y \cdot b\right)} + y, y\right)} \]

       7. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \frac{1}{2} \cdot \color{blue}{\left(b \cdot y\right)} + y, y\right)} \]

       8. lower-fma.f64 N/A

          \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(\frac{1}{2}, b \cdot y, y\right)}, y\right)} \]

       9. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{y \cdot b}, y\right), y\right)} \]

       10. lower-*.f64 41.4

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(0.5, \color{blue}{y \cdot b}, y\right), y\right)} \]

   11. Simplified 41.4%

       \[\leadsto \frac{x}{a \cdot \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(0.5, y \cdot b, y\right), y\right)}} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{x}{a \cdot \left(b \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(y \cdot b\right)} + y\right) + y\right)} \]

       2. lift-fma.f64 N/A

          \[\leadsto \frac{x}{a \cdot \left(b \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, y \cdot b, y\right)} + y\right)} \]

       3. distribute-lft-in N/A

          \[\leadsto \frac{x}{\color{blue}{a \cdot \left(b \cdot \mathsf{fma}\left(\frac{1}{2}, y \cdot b, y\right)\right) + a \cdot y}} \]

       4. flip-+ N/A

          \[\leadsto \frac{x}{\color{blue}{\frac{\left(a \cdot \left(b \cdot \mathsf{fma}\left(\frac{1}{2}, y \cdot b, y\right)\right)\right) \cdot \left(a \cdot \left(b \cdot \mathsf{fma}\left(\frac{1}{2}, y \cdot b, y\right)\right)\right) - \left(a \cdot y\right) \cdot \left(a \cdot y\right)}{a \cdot \left(b \cdot \mathsf{fma}\left(\frac{1}{2}, y \cdot b, y\right)\right) - a \cdot y}}} \]

       5. lower-/.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{\frac{\left(a \cdot \left(b \cdot \mathsf{fma}\left(\frac{1}{2}, y \cdot b, y\right)\right)\right) \cdot \left(a \cdot \left(b \cdot \mathsf{fma}\left(\frac{1}{2}, y \cdot b, y\right)\right)\right) - \left(a \cdot y\right) \cdot \left(a \cdot y\right)}{a \cdot \left(b \cdot \mathsf{fma}\left(\frac{1}{2}, y \cdot b, y\right)\right) - a \cdot y}}} \]

   13. Applied egg-rr 61.3%

       \[\leadsto \frac{x}{\color{blue}{\frac{\left(a \cdot \mathsf{fma}\left(\mathsf{fma}\left(b, 0.5, 1\right), y \cdot b, y\right)\right) \cdot \left(a \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(y, b \cdot 0.5, y\right), -y\right)\right)}{a \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(y, b \cdot 0.5, y\right), -y\right)}}} \]

   if 1.45e9 < b

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\left(-1 \cdot \log a + y \cdot \log z\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{e^{\left(-1 \cdot \log a + y \cdot \log z\right) - b}}{y}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \frac{e^{\left(-1 \cdot \log a + y \cdot \log z\right) - b}}{y}} \]

      3. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      5. lower-/.f64 N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      6. +-commutative N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z + -1 \cdot \log a}}}{y \cdot e^{b}} \]

      7. mul-1-neg N/A

         \[\leadsto x \cdot \frac{e^{y \cdot \log z + \color{blue}{\left(\mathsf{neg}\left(\log a\right)\right)}}}{y \cdot e^{b}} \]

      8. unsub-neg N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z - \log a}}}{y \cdot e^{b}} \]

      9. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      10. lower-/.f64 N/A

          \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      11. *-commutative N/A

          \[\leadsto x \cdot \frac{\frac{e^{\color{blue}{\log z \cdot y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      12. exp-to-pow N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      13. lower-pow.f64 N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      14. rem-exp-log N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{\color{blue}{a}}}{y \cdot e^{b}} \]

      15. lower-*.f64 N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{\color{blue}{y \cdot e^{b}}} \]

      16. lower-exp.f64 72.3

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 72.3%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot e^{b}}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. lower-exp.f64 80.6

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 80.6%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\left(1 + b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)\right)}\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\left(b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right) + 1\right)}\right)} \]

       2. lower-fma.f64 N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left(b, 1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right), 1\right)}\right)} \]

       3. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \color{blue}{b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right) + 1}, 1\right)\right)} \]

       4. lower-fma.f64 N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, \frac{1}{2} + \frac{1}{6} \cdot b, 1\right)}, 1\right)\right)} \]

       5. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\frac{1}{6} \cdot b + \frac{1}{2}}, 1\right), 1\right)\right)} \]

       6. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{b \cdot \frac{1}{6}} + \frac{1}{2}, 1\right), 1\right)\right)} \]

       7. lower-fma.f64 74.3

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, 0.16666666666666666, 0.5\right)}, 1\right), 1\right)\right)} \]

   11. Simplified 74.3%

       \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), 1\right)}\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.55 \cdot 10^{+26}:\\ \;\;\;\;\mathsf{fma}\left(b, \mathsf{fma}\left(b, \left(b \cdot \frac{x}{y \cdot a}\right) \cdot -0.16666666666666666, \frac{x}{y \cdot a} \cdot \mathsf{fma}\left(b, 0.5, -1\right)\right), \frac{x}{y \cdot a}\right)\\ \mathbf{elif}\;b \leq -1.05 \cdot 10^{-262}:\\ \;\;\;\;\frac{x}{\frac{a \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b, 0.5, 1\right), y \cdot b, y\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, b \cdot 0.5, y\right), -b, y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(y, b \cdot 0.5, y\right), -b, y\right)}}\\ \mathbf{elif}\;b \leq 2.6 \cdot 10^{-90}:\\ \;\;\;\;\frac{x}{b \cdot \left(b \cdot \mathsf{fma}\left(\frac{y}{b}, a + \frac{a}{b}, a \cdot \left(y \cdot 0.5\right)\right)\right)}\\ \mathbf{elif}\;b \leq 1450000000:\\ \;\;\;\;\frac{x}{\frac{\left(a \cdot \mathsf{fma}\left(\mathsf{fma}\left(b, 0.5, 1\right), y \cdot b, y\right)\right) \cdot \left(a \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(y, b \cdot 0.5, y\right), -y\right)\right)}{a \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(y, b \cdot 0.5, y\right), -y\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 52.8% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\mathsf{fma}\left(y, b \cdot 0.5, y\right), -b, y\right)\\ t_2 := \frac{x}{y \cdot a}\\ \mathbf{if}\;b \leq -2.55 \cdot 10^{+26}:\\ \;\;\;\;\mathsf{fma}\left(b, \mathsf{fma}\left(b, \left(b \cdot t\_2\right) \cdot -0.16666666666666666, t\_2 \cdot \mathsf{fma}\left(b, 0.5, -1\right)\right), t\_2\right)\\ \mathbf{elif}\;b \leq 800000000:\\ \;\;\;\;\frac{x}{\frac{a \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b, 0.5, 1\right), y \cdot b, y\right) \cdot t\_1\right)}{t\_1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (fma y (* b 0.5) y) (- b) y)) (t_2 (/ x (* y a))))
   (if (<= b -2.55e+26)
     (fma
      b
      (fma b (* (* b t_2) -0.16666666666666666) (* t_2 (fma b 0.5 -1.0)))
      t_2)
     (if (<= b 800000000.0)
       (/ x (/ (* a (* (fma (fma b 0.5 1.0) (* y b) y) t_1)) t_1))
       (/
        x
        (*
         a
         (* y (fma b (fma b (fma b 0.16666666666666666 0.5) 1.0) 1.0))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(fma(y, (b * 0.5), y), -b, y);
	double t_2 = x / (y * a);
	double tmp;
	if (b <= -2.55e+26) {
		tmp = fma(b, fma(b, ((b * t_2) * -0.16666666666666666), (t_2 * fma(b, 0.5, -1.0))), t_2);
	} else if (b <= 800000000.0) {
		tmp = x / ((a * (fma(fma(b, 0.5, 1.0), (y * b), y) * t_1)) / t_1);
	} else {
		tmp = x / (a * (y * fma(b, fma(b, fma(b, 0.16666666666666666, 0.5), 1.0), 1.0)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(fma(y, Float64(b * 0.5), y), Float64(-b), y)
	t_2 = Float64(x / Float64(y * a))
	tmp = 0.0
	if (b <= -2.55e+26)
		tmp = fma(b, fma(b, Float64(Float64(b * t_2) * -0.16666666666666666), Float64(t_2 * fma(b, 0.5, -1.0))), t_2);
	elseif (b <= 800000000.0)
		tmp = Float64(x / Float64(Float64(a * Float64(fma(fma(b, 0.5, 1.0), Float64(y * b), y) * t_1)) / t_1));
	else
		tmp = Float64(x / Float64(a * Float64(y * fma(b, fma(b, fma(b, 0.16666666666666666, 0.5), 1.0), 1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(y * N[(b * 0.5), $MachinePrecision] + y), $MachinePrecision] * (-b) + y), $MachinePrecision]}, Block[{t$95$2 = N[(x / N[(y * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -2.55e+26], N[(b * N[(b * N[(N[(b * t$95$2), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + N[(t$95$2 * N[(b * 0.5 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision], If[LessEqual[b, 800000000.0], N[(x / N[(N[(a * N[(N[(N[(b * 0.5 + 1.0), $MachinePrecision] * N[(y * b), $MachinePrecision] + y), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], N[(x / N[(a * N[(y * N[(b * N[(b * N[(b * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.5499999999999999e26

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\left(-1 \cdot \log a + y \cdot \log z\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{e^{\left(-1 \cdot \log a + y \cdot \log z\right) - b}}{y}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \frac{e^{\left(-1 \cdot \log a + y \cdot \log z\right) - b}}{y}} \]

      3. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      5. lower-/.f64 N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      6. +-commutative N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z + -1 \cdot \log a}}}{y \cdot e^{b}} \]

      7. mul-1-neg N/A

         \[\leadsto x \cdot \frac{e^{y \cdot \log z + \color{blue}{\left(\mathsf{neg}\left(\log a\right)\right)}}}{y \cdot e^{b}} \]

      8. unsub-neg N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z - \log a}}}{y \cdot e^{b}} \]

      9. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      10. lower-/.f64 N/A

          \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      11. *-commutative N/A

          \[\leadsto x \cdot \frac{\frac{e^{\color{blue}{\log z \cdot y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      12. exp-to-pow N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      13. lower-pow.f64 N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      14. rem-exp-log N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{\color{blue}{a}}}{y \cdot e^{b}} \]

      15. lower-*.f64 N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{\color{blue}{y \cdot e^{b}}} \]

      16. lower-exp.f64 69.0

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 69.0%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot e^{b}}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. lower-exp.f64 75.4

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 75.4%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \color{blue}{b \cdot \left(b \cdot \left(-1 \cdot \left(b \cdot \left(-1 \cdot \left(-1 \cdot \frac{x}{a \cdot y} + \frac{1}{2} \cdot \frac{x}{a \cdot y}\right) + \left(\frac{-1}{2} \cdot \frac{x}{a \cdot y} + \frac{1}{6} \cdot \frac{x}{a \cdot y}\right)\right)\right) - \left(-1 \cdot \frac{x}{a \cdot y} + \frac{1}{2} \cdot \frac{x}{a \cdot y}\right)\right) - \frac{x}{a \cdot y}\right) + \frac{x}{a \cdot y}} \]

   11. Simplified 63.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \left(b \cdot \frac{x}{a \cdot y}\right) \cdot -0.16666666666666666, \frac{x}{a \cdot y} \cdot \mathsf{fma}\left(b, 0.5, -1\right)\right), \frac{x}{a \cdot y}\right)} \]

   if -2.5499999999999999e26 < b < 8e8

   1. Initial program 96.2%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\left(-1 \cdot \log a + y \cdot \log z\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{e^{\left(-1 \cdot \log a + y \cdot \log z\right) - b}}{y}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \frac{e^{\left(-1 \cdot \log a + y \cdot \log z\right) - b}}{y}} \]

      3. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      5. lower-/.f64 N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      6. +-commutative N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z + -1 \cdot \log a}}}{y \cdot e^{b}} \]

      7. mul-1-neg N/A

         \[\leadsto x \cdot \frac{e^{y \cdot \log z + \color{blue}{\left(\mathsf{neg}\left(\log a\right)\right)}}}{y \cdot e^{b}} \]

      8. unsub-neg N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z - \log a}}}{y \cdot e^{b}} \]

      9. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      10. lower-/.f64 N/A

          \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      11. *-commutative N/A

          \[\leadsto x \cdot \frac{\frac{e^{\color{blue}{\log z \cdot y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      12. exp-to-pow N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      13. lower-pow.f64 N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      14. rem-exp-log N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{\color{blue}{a}}}{y \cdot e^{b}} \]

      15. lower-*.f64 N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{\color{blue}{y \cdot e^{b}}} \]

      16. lower-exp.f64 71.7

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 71.7%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot e^{b}}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. lower-exp.f64 46.9

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 46.9%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y + b \cdot \left(y + \frac{1}{2} \cdot \left(b \cdot y\right)\right)\right)}} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\left(b \cdot \left(y + \frac{1}{2} \cdot \left(b \cdot y\right)\right) + y\right)}} \]

       2. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(b \cdot \left(y + \frac{1}{2} \cdot \color{blue}{\left(y \cdot b\right)}\right) + y\right)} \]

       3. associate-*r* N/A

          \[\leadsto \frac{x}{a \cdot \left(b \cdot \left(y + \color{blue}{\left(\frac{1}{2} \cdot y\right) \cdot b}\right) + y\right)} \]

       4. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(b \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot y\right) \cdot b + y\right)} + y\right)} \]

       5. lower-fma.f64 N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\mathsf{fma}\left(b, \left(\frac{1}{2} \cdot y\right) \cdot b + y, y\right)}} \]

       6. associate-*r* N/A

          \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \color{blue}{\frac{1}{2} \cdot \left(y \cdot b\right)} + y, y\right)} \]

       7. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \frac{1}{2} \cdot \color{blue}{\left(b \cdot y\right)} + y, y\right)} \]

       8. lower-fma.f64 N/A

          \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(\frac{1}{2}, b \cdot y, y\right)}, y\right)} \]

       9. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{y \cdot b}, y\right), y\right)} \]

       10. lower-*.f64 44.0

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(0.5, \color{blue}{y \cdot b}, y\right), y\right)} \]

   11. Simplified 44.0%

       \[\leadsto \frac{x}{a \cdot \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(0.5, y \cdot b, y\right), y\right)}} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{x}{a \cdot \left(b \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(y \cdot b\right)} + y\right) + y\right)} \]

       2. lift-fma.f64 N/A

          \[\leadsto \frac{x}{a \cdot \left(b \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, y \cdot b, y\right)} + y\right)} \]

       3. flip-+ N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\frac{\left(b \cdot \mathsf{fma}\left(\frac{1}{2}, y \cdot b, y\right)\right) \cdot \left(b \cdot \mathsf{fma}\left(\frac{1}{2}, y \cdot b, y\right)\right) - y \cdot y}{b \cdot \mathsf{fma}\left(\frac{1}{2}, y \cdot b, y\right) - y}}} \]

       4. frac-2neg N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\frac{\mathsf{neg}\left(\left(\left(b \cdot \mathsf{fma}\left(\frac{1}{2}, y \cdot b, y\right)\right) \cdot \left(b \cdot \mathsf{fma}\left(\frac{1}{2}, y \cdot b, y\right)\right) - y \cdot y\right)\right)}{\mathsf{neg}\left(\left(b \cdot \mathsf{fma}\left(\frac{1}{2}, y \cdot b, y\right) - y\right)\right)}}} \]

       5. associate-*r/ N/A

          \[\leadsto \frac{x}{\color{blue}{\frac{a \cdot \left(\mathsf{neg}\left(\left(\left(b \cdot \mathsf{fma}\left(\frac{1}{2}, y \cdot b, y\right)\right) \cdot \left(b \cdot \mathsf{fma}\left(\frac{1}{2}, y \cdot b, y\right)\right) - y \cdot y\right)\right)\right)}{\mathsf{neg}\left(\left(b \cdot \mathsf{fma}\left(\frac{1}{2}, y \cdot b, y\right) - y\right)\right)}}} \]

       6. lower-/.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{\frac{a \cdot \left(\mathsf{neg}\left(\left(\left(b \cdot \mathsf{fma}\left(\frac{1}{2}, y \cdot b, y\right)\right) \cdot \left(b \cdot \mathsf{fma}\left(\frac{1}{2}, y \cdot b, y\right)\right) - y \cdot y\right)\right)\right)}{\mathsf{neg}\left(\left(b \cdot \mathsf{fma}\left(\frac{1}{2}, y \cdot b, y\right) - y\right)\right)}}} \]

   13. Applied egg-rr 53.2%

       \[\leadsto \frac{x}{\color{blue}{\frac{a \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b, 0.5, 1\right), y \cdot b, y\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, b \cdot 0.5, y\right), -b, y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(y, b \cdot 0.5, y\right), -b, y\right)}}} \]

   if 8e8 < b

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\left(-1 \cdot \log a + y \cdot \log z\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{e^{\left(-1 \cdot \log a + y \cdot \log z\right) - b}}{y}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \frac{e^{\left(-1 \cdot \log a + y \cdot \log z\right) - b}}{y}} \]

      3. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      5. lower-/.f64 N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      6. +-commutative N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z + -1 \cdot \log a}}}{y \cdot e^{b}} \]

      7. mul-1-neg N/A

         \[\leadsto x \cdot \frac{e^{y \cdot \log z + \color{blue}{\left(\mathsf{neg}\left(\log a\right)\right)}}}{y \cdot e^{b}} \]

      8. unsub-neg N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z - \log a}}}{y \cdot e^{b}} \]

      9. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      10. lower-/.f64 N/A

          \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      11. *-commutative N/A

          \[\leadsto x \cdot \frac{\frac{e^{\color{blue}{\log z \cdot y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      12. exp-to-pow N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      13. lower-pow.f64 N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      14. rem-exp-log N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{\color{blue}{a}}}{y \cdot e^{b}} \]

      15. lower-*.f64 N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{\color{blue}{y \cdot e^{b}}} \]

      16. lower-exp.f64 72.7

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 72.7%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot e^{b}}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. lower-exp.f64 80.9

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 80.9%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\left(1 + b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)\right)}\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\left(b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right) + 1\right)}\right)} \]

       2. lower-fma.f64 N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left(b, 1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right), 1\right)}\right)} \]

       3. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \color{blue}{b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right) + 1}, 1\right)\right)} \]

       4. lower-fma.f64 N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, \frac{1}{2} + \frac{1}{6} \cdot b, 1\right)}, 1\right)\right)} \]

       5. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\frac{1}{6} \cdot b + \frac{1}{2}}, 1\right), 1\right)\right)} \]

       6. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{b \cdot \frac{1}{6}} + \frac{1}{2}, 1\right), 1\right)\right)} \]

       7. lower-fma.f64 73.1

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, 0.16666666666666666, 0.5\right)}, 1\right), 1\right)\right)} \]

   11. Simplified 73.1%

       \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), 1\right)}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.55 \cdot 10^{+26}:\\ \;\;\;\;\mathsf{fma}\left(b, \mathsf{fma}\left(b, \left(b \cdot \frac{x}{y \cdot a}\right) \cdot -0.16666666666666666, \frac{x}{y \cdot a} \cdot \mathsf{fma}\left(b, 0.5, -1\right)\right), \frac{x}{y \cdot a}\right)\\ \mathbf{elif}\;b \leq 800000000:\\ \;\;\;\;\frac{x}{\frac{a \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b, 0.5, 1\right), y \cdot b, y\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, b \cdot 0.5, y\right), -b, y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(y, b \cdot 0.5, y\right), -b, y\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 53.8% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y \cdot b, -0.5, y\right)\\ t_2 := \frac{x}{y \cdot a}\\ \mathbf{if}\;b \leq -9 \cdot 10^{-12}:\\ \;\;\;\;\mathsf{fma}\left(b, \mathsf{fma}\left(b, \left(b \cdot t\_2\right) \cdot -0.16666666666666666, t\_2 \cdot \mathsf{fma}\left(b, 0.5, -1\right)\right), t\_2\right)\\ \mathbf{elif}\;b \leq 850000000:\\ \;\;\;\;\frac{x}{a \cdot \mathsf{fma}\left(b, \frac{\mathsf{fma}\left(y, b \cdot 0.5, y\right) \cdot t\_1}{t\_1}, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (* y b) -0.5 y)) (t_2 (/ x (* y a))))
   (if (<= b -9e-12)
     (fma
      b
      (fma b (* (* b t_2) -0.16666666666666666) (* t_2 (fma b 0.5 -1.0)))
      t_2)
     (if (<= b 850000000.0)
       (/ x (* a (fma b (/ (* (fma y (* b 0.5) y) t_1) t_1) y)))
       (/
        x
        (*
         a
         (* y (fma b (fma b (fma b 0.16666666666666666 0.5) 1.0) 1.0))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((y * b), -0.5, y);
	double t_2 = x / (y * a);
	double tmp;
	if (b <= -9e-12) {
		tmp = fma(b, fma(b, ((b * t_2) * -0.16666666666666666), (t_2 * fma(b, 0.5, -1.0))), t_2);
	} else if (b <= 850000000.0) {
		tmp = x / (a * fma(b, ((fma(y, (b * 0.5), y) * t_1) / t_1), y));
	} else {
		tmp = x / (a * (y * fma(b, fma(b, fma(b, 0.16666666666666666, 0.5), 1.0), 1.0)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(Float64(y * b), -0.5, y)
	t_2 = Float64(x / Float64(y * a))
	tmp = 0.0
	if (b <= -9e-12)
		tmp = fma(b, fma(b, Float64(Float64(b * t_2) * -0.16666666666666666), Float64(t_2 * fma(b, 0.5, -1.0))), t_2);
	elseif (b <= 850000000.0)
		tmp = Float64(x / Float64(a * fma(b, Float64(Float64(fma(y, Float64(b * 0.5), y) * t_1) / t_1), y)));
	else
		tmp = Float64(x / Float64(a * Float64(y * fma(b, fma(b, fma(b, 0.16666666666666666, 0.5), 1.0), 1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(y * b), $MachinePrecision] * -0.5 + y), $MachinePrecision]}, Block[{t$95$2 = N[(x / N[(y * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -9e-12], N[(b * N[(b * N[(N[(b * t$95$2), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + N[(t$95$2 * N[(b * 0.5 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision], If[LessEqual[b, 850000000.0], N[(x / N[(a * N[(b * N[(N[(N[(y * N[(b * 0.5), $MachinePrecision] + y), $MachinePrecision] * t$95$1), $MachinePrecision] / t$95$1), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(a * N[(y * N[(b * N[(b * N[(b * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.99999999999999962e-12

   1. Initial program 99.9%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\left(-1 \cdot \log a + y \cdot \log z\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{e^{\left(-1 \cdot \log a + y \cdot \log z\right) - b}}{y}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \frac{e^{\left(-1 \cdot \log a + y \cdot \log z\right) - b}}{y}} \]

      3. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      5. lower-/.f64 N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      6. +-commutative N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z + -1 \cdot \log a}}}{y \cdot e^{b}} \]

      7. mul-1-neg N/A

         \[\leadsto x \cdot \frac{e^{y \cdot \log z + \color{blue}{\left(\mathsf{neg}\left(\log a\right)\right)}}}{y \cdot e^{b}} \]

      8. unsub-neg N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z - \log a}}}{y \cdot e^{b}} \]

      9. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      10. lower-/.f64 N/A

          \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      11. *-commutative N/A

          \[\leadsto x \cdot \frac{\frac{e^{\color{blue}{\log z \cdot y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      12. exp-to-pow N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      13. lower-pow.f64 N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      14. rem-exp-log N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{\color{blue}{a}}}{y \cdot e^{b}} \]

      15. lower-*.f64 N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{\color{blue}{y \cdot e^{b}}} \]

      16. lower-exp.f64 67.8

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 67.8%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot e^{b}}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. lower-exp.f64 72.1

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 72.1%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \color{blue}{b \cdot \left(b \cdot \left(-1 \cdot \left(b \cdot \left(-1 \cdot \left(-1 \cdot \frac{x}{a \cdot y} + \frac{1}{2} \cdot \frac{x}{a \cdot y}\right) + \left(\frac{-1}{2} \cdot \frac{x}{a \cdot y} + \frac{1}{6} \cdot \frac{x}{a \cdot y}\right)\right)\right) - \left(-1 \cdot \frac{x}{a \cdot y} + \frac{1}{2} \cdot \frac{x}{a \cdot y}\right)\right) - \frac{x}{a \cdot y}\right) + \frac{x}{a \cdot y}} \]

   11. Simplified 59.4%

       \[\leadsto \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \left(b \cdot \frac{x}{a \cdot y}\right) \cdot -0.16666666666666666, \frac{x}{a \cdot y} \cdot \mathsf{fma}\left(b, 0.5, -1\right)\right), \frac{x}{a \cdot y}\right)} \]

   if -8.99999999999999962e-12 < b < 8.5e8

   1. Initial program 96.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\left(-1 \cdot \log a + y \cdot \log z\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{e^{\left(-1 \cdot \log a + y \cdot \log z\right) - b}}{y}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \frac{e^{\left(-1 \cdot \log a + y \cdot \log z\right) - b}}{y}} \]

      3. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      5. lower-/.f64 N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      6. +-commutative N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z + -1 \cdot \log a}}}{y \cdot e^{b}} \]

      7. mul-1-neg N/A

         \[\leadsto x \cdot \frac{e^{y \cdot \log z + \color{blue}{\left(\mathsf{neg}\left(\log a\right)\right)}}}{y \cdot e^{b}} \]

      8. unsub-neg N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z - \log a}}}{y \cdot e^{b}} \]

      9. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      10. lower-/.f64 N/A

          \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      11. *-commutative N/A

          \[\leadsto x \cdot \frac{\frac{e^{\color{blue}{\log z \cdot y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      12. exp-to-pow N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      13. lower-pow.f64 N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      14. rem-exp-log N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{\color{blue}{a}}}{y \cdot e^{b}} \]

      15. lower-*.f64 N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{\color{blue}{y \cdot e^{b}}} \]

      16. lower-exp.f64 72.6

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 72.6%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot e^{b}}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. lower-exp.f64 46.6

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 46.6%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y + b \cdot \left(y + \frac{1}{2} \cdot \left(b \cdot y\right)\right)\right)}} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\left(b \cdot \left(y + \frac{1}{2} \cdot \left(b \cdot y\right)\right) + y\right)}} \]

       2. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(b \cdot \left(y + \frac{1}{2} \cdot \color{blue}{\left(y \cdot b\right)}\right) + y\right)} \]

       3. associate-*r* N/A

          \[\leadsto \frac{x}{a \cdot \left(b \cdot \left(y + \color{blue}{\left(\frac{1}{2} \cdot y\right) \cdot b}\right) + y\right)} \]

       4. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(b \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot y\right) \cdot b + y\right)} + y\right)} \]

       5. lower-fma.f64 N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\mathsf{fma}\left(b, \left(\frac{1}{2} \cdot y\right) \cdot b + y, y\right)}} \]

       6. associate-*r* N/A

          \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \color{blue}{\frac{1}{2} \cdot \left(y \cdot b\right)} + y, y\right)} \]

       7. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \frac{1}{2} \cdot \color{blue}{\left(b \cdot y\right)} + y, y\right)} \]

       8. lower-fma.f64 N/A

          \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(\frac{1}{2}, b \cdot y, y\right)}, y\right)} \]

       9. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{y \cdot b}, y\right), y\right)} \]

       10. lower-*.f64 45.0

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(0.5, \color{blue}{y \cdot b}, y\right), y\right)} \]

   11. Simplified 45.0%

       \[\leadsto \frac{x}{a \cdot \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(0.5, y \cdot b, y\right), y\right)}} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \frac{1}{2} \cdot \color{blue}{\left(y \cdot b\right)} + y, y\right)} \]

       2. flip-+ N/A

          \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \color{blue}{\frac{\left(\frac{1}{2} \cdot \left(y \cdot b\right)\right) \cdot \left(\frac{1}{2} \cdot \left(y \cdot b\right)\right) - y \cdot y}{\frac{1}{2} \cdot \left(y \cdot b\right) - y}}, y\right)} \]

       3. frac-2neg N/A

          \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \color{blue}{\frac{\mathsf{neg}\left(\left(\left(\frac{1}{2} \cdot \left(y \cdot b\right)\right) \cdot \left(\frac{1}{2} \cdot \left(y \cdot b\right)\right) - y \cdot y\right)\right)}{\mathsf{neg}\left(\left(\frac{1}{2} \cdot \left(y \cdot b\right) - y\right)\right)}}, y\right)} \]

       4. lower-/.f64 N/A

          \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \color{blue}{\frac{\mathsf{neg}\left(\left(\left(\frac{1}{2} \cdot \left(y \cdot b\right)\right) \cdot \left(\frac{1}{2} \cdot \left(y \cdot b\right)\right) - y \cdot y\right)\right)}{\mathsf{neg}\left(\left(\frac{1}{2} \cdot \left(y \cdot b\right) - y\right)\right)}}, y\right)} \]

   13. Applied egg-rr 53.5%

       \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \color{blue}{\frac{\mathsf{fma}\left(y, b \cdot 0.5, y\right) \cdot \mathsf{fma}\left(y \cdot b, -0.5, y\right)}{\mathsf{fma}\left(y \cdot b, -0.5, y\right)}}, y\right)} \]

   if 8.5e8 < b

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\left(-1 \cdot \log a + y \cdot \log z\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{e^{\left(-1 \cdot \log a + y \cdot \log z\right) - b}}{y}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \frac{e^{\left(-1 \cdot \log a + y \cdot \log z\right) - b}}{y}} \]

      3. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      5. lower-/.f64 N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      6. +-commutative N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z + -1 \cdot \log a}}}{y \cdot e^{b}} \]

      7. mul-1-neg N/A

         \[\leadsto x \cdot \frac{e^{y \cdot \log z + \color{blue}{\left(\mathsf{neg}\left(\log a\right)\right)}}}{y \cdot e^{b}} \]

      8. unsub-neg N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z - \log a}}}{y \cdot e^{b}} \]

      9. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      10. lower-/.f64 N/A

          \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      11. *-commutative N/A

          \[\leadsto x \cdot \frac{\frac{e^{\color{blue}{\log z \cdot y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      12. exp-to-pow N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      13. lower-pow.f64 N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      14. rem-exp-log N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{\color{blue}{a}}}{y \cdot e^{b}} \]

      15. lower-*.f64 N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{\color{blue}{y \cdot e^{b}}} \]

      16. lower-exp.f64 72.7

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 72.7%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot e^{b}}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. lower-exp.f64 80.9

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 80.9%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\left(1 + b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)\right)}\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\left(b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right) + 1\right)}\right)} \]

       2. lower-fma.f64 N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left(b, 1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right), 1\right)}\right)} \]

       3. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \color{blue}{b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right) + 1}, 1\right)\right)} \]

       4. lower-fma.f64 N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, \frac{1}{2} + \frac{1}{6} \cdot b, 1\right)}, 1\right)\right)} \]

       5. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\frac{1}{6} \cdot b + \frac{1}{2}}, 1\right), 1\right)\right)} \]

       6. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{b \cdot \frac{1}{6}} + \frac{1}{2}, 1\right), 1\right)\right)} \]

       7. lower-fma.f64 73.1

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, 0.16666666666666666, 0.5\right)}, 1\right), 1\right)\right)} \]

   11. Simplified 73.1%

       \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), 1\right)}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9 \cdot 10^{-12}:\\ \;\;\;\;\mathsf{fma}\left(b, \mathsf{fma}\left(b, \left(b \cdot \frac{x}{y \cdot a}\right) \cdot -0.16666666666666666, \frac{x}{y \cdot a} \cdot \mathsf{fma}\left(b, 0.5, -1\right)\right), \frac{x}{y \cdot a}\right)\\ \mathbf{elif}\;b \leq 850000000:\\ \;\;\;\;\frac{x}{a \cdot \mathsf{fma}\left(b, \frac{\mathsf{fma}\left(y, b \cdot 0.5, y\right) \cdot \mathsf{fma}\left(y \cdot b, -0.5, y\right)}{\mathsf{fma}\left(y \cdot b, -0.5, y\right)}, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 51.4% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{y \cdot a}\\ t_2 := \mathsf{fma}\left(y \cdot b, -0.5, y\right)\\ \mathbf{if}\;b \leq -3 \cdot 10^{+111}:\\ \;\;\;\;\mathsf{fma}\left(b, t\_1 \cdot \mathsf{fma}\left(b, 0.5, -1\right), t\_1\right)\\ \mathbf{elif}\;b \leq 850000000:\\ \;\;\;\;\frac{x}{a \cdot \mathsf{fma}\left(b, \frac{\mathsf{fma}\left(y, b \cdot 0.5, y\right) \cdot t\_2}{t\_2}, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (* y a))) (t_2 (fma (* y b) -0.5 y)))
   (if (<= b -3e+111)
     (fma b (* t_1 (fma b 0.5 -1.0)) t_1)
     (if (<= b 850000000.0)
       (/ x (* a (fma b (/ (* (fma y (* b 0.5) y) t_2) t_2) y)))
       (/
        x
        (*
         a
         (* y (fma b (fma b (fma b 0.16666666666666666 0.5) 1.0) 1.0))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / (y * a);
	double t_2 = fma((y * b), -0.5, y);
	double tmp;
	if (b <= -3e+111) {
		tmp = fma(b, (t_1 * fma(b, 0.5, -1.0)), t_1);
	} else if (b <= 850000000.0) {
		tmp = x / (a * fma(b, ((fma(y, (b * 0.5), y) * t_2) / t_2), y));
	} else {
		tmp = x / (a * (y * fma(b, fma(b, fma(b, 0.16666666666666666, 0.5), 1.0), 1.0)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x / Float64(y * a))
	t_2 = fma(Float64(y * b), -0.5, y)
	tmp = 0.0
	if (b <= -3e+111)
		tmp = fma(b, Float64(t_1 * fma(b, 0.5, -1.0)), t_1);
	elseif (b <= 850000000.0)
		tmp = Float64(x / Float64(a * fma(b, Float64(Float64(fma(y, Float64(b * 0.5), y) * t_2) / t_2), y)));
	else
		tmp = Float64(x / Float64(a * Float64(y * fma(b, fma(b, fma(b, 0.16666666666666666, 0.5), 1.0), 1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x / N[(y * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y * b), $MachinePrecision] * -0.5 + y), $MachinePrecision]}, If[LessEqual[b, -3e+111], N[(b * N[(t$95$1 * N[(b * 0.5 + -1.0), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], If[LessEqual[b, 850000000.0], N[(x / N[(a * N[(b * N[(N[(N[(y * N[(b * 0.5), $MachinePrecision] + y), $MachinePrecision] * t$95$2), $MachinePrecision] / t$95$2), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(a * N[(y * N[(b * N[(b * N[(b * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -3e111

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\left(-1 \cdot \log a + y \cdot \log z\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{e^{\left(-1 \cdot \log a + y \cdot \log z\right) - b}}{y}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \frac{e^{\left(-1 \cdot \log a + y \cdot \log z\right) - b}}{y}} \]

      3. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      5. lower-/.f64 N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      6. +-commutative N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z + -1 \cdot \log a}}}{y \cdot e^{b}} \]

      7. mul-1-neg N/A

         \[\leadsto x \cdot \frac{e^{y \cdot \log z + \color{blue}{\left(\mathsf{neg}\left(\log a\right)\right)}}}{y \cdot e^{b}} \]

      8. unsub-neg N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z - \log a}}}{y \cdot e^{b}} \]

      9. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      10. lower-/.f64 N/A

          \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      11. *-commutative N/A

          \[\leadsto x \cdot \frac{\frac{e^{\color{blue}{\log z \cdot y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      12. exp-to-pow N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      13. lower-pow.f64 N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      14. rem-exp-log N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{\color{blue}{a}}}{y \cdot e^{b}} \]

      15. lower-*.f64 N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{\color{blue}{y \cdot e^{b}}} \]

      16. lower-exp.f64 77.4

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 77.4%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot e^{b}}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. lower-exp.f64 84.3

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 84.3%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \color{blue}{b \cdot \left(-1 \cdot \left(b \cdot \left(-1 \cdot \frac{x}{a \cdot y} + \frac{1}{2} \cdot \frac{x}{a \cdot y}\right)\right) - \frac{x}{a \cdot y}\right) + \frac{x}{a \cdot y}} \]
   10. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(b, -1 \cdot \left(b \cdot \left(-1 \cdot \frac{x}{a \cdot y} + \frac{1}{2} \cdot \frac{x}{a \cdot y}\right)\right) - \frac{x}{a \cdot y}, \frac{x}{a \cdot y}\right)} \]

   11. Simplified 67.1%

       \[\leadsto \color{blue}{\mathsf{fma}\left(b, \frac{x}{a \cdot y} \cdot \mathsf{fma}\left(b, 0.5, -1\right), \frac{x}{a \cdot y}\right)} \]

   if -3e111 < b < 8.5e8

   1. Initial program 96.7%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\left(-1 \cdot \log a + y \cdot \log z\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{e^{\left(-1 \cdot \log a + y \cdot \log z\right) - b}}{y}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \frac{e^{\left(-1 \cdot \log a + y \cdot \log z\right) - b}}{y}} \]

      3. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      5. lower-/.f64 N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      6. +-commutative N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z + -1 \cdot \log a}}}{y \cdot e^{b}} \]

      7. mul-1-neg N/A

         \[\leadsto x \cdot \frac{e^{y \cdot \log z + \color{blue}{\left(\mathsf{neg}\left(\log a\right)\right)}}}{y \cdot e^{b}} \]

      8. unsub-neg N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z - \log a}}}{y \cdot e^{b}} \]

      9. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      10. lower-/.f64 N/A

          \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      11. *-commutative N/A

          \[\leadsto x \cdot \frac{\frac{e^{\color{blue}{\log z \cdot y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      12. exp-to-pow N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      13. lower-pow.f64 N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      14. rem-exp-log N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{\color{blue}{a}}}{y \cdot e^{b}} \]

      15. lower-*.f64 N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{\color{blue}{y \cdot e^{b}}} \]

      16. lower-exp.f64 68.9

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 68.9%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot e^{b}}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. lower-exp.f64 48.1

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 48.1%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y + b \cdot \left(y + \frac{1}{2} \cdot \left(b \cdot y\right)\right)\right)}} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\left(b \cdot \left(y + \frac{1}{2} \cdot \left(b \cdot y\right)\right) + y\right)}} \]

       2. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(b \cdot \left(y + \frac{1}{2} \cdot \color{blue}{\left(y \cdot b\right)}\right) + y\right)} \]

       3. associate-*r* N/A

          \[\leadsto \frac{x}{a \cdot \left(b \cdot \left(y + \color{blue}{\left(\frac{1}{2} \cdot y\right) \cdot b}\right) + y\right)} \]

       4. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(b \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot y\right) \cdot b + y\right)} + y\right)} \]

       5. lower-fma.f64 N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\mathsf{fma}\left(b, \left(\frac{1}{2} \cdot y\right) \cdot b + y, y\right)}} \]

       6. associate-*r* N/A

          \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \color{blue}{\frac{1}{2} \cdot \left(y \cdot b\right)} + y, y\right)} \]

       7. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \frac{1}{2} \cdot \color{blue}{\left(b \cdot y\right)} + y, y\right)} \]

       8. lower-fma.f64 N/A

          \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(\frac{1}{2}, b \cdot y, y\right)}, y\right)} \]

       9. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{y \cdot b}, y\right), y\right)} \]

       10. lower-*.f64 43.7

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(0.5, \color{blue}{y \cdot b}, y\right), y\right)} \]

   11. Simplified 43.7%

       \[\leadsto \frac{x}{a \cdot \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(0.5, y \cdot b, y\right), y\right)}} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \frac{1}{2} \cdot \color{blue}{\left(y \cdot b\right)} + y, y\right)} \]

       2. flip-+ N/A

          \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \color{blue}{\frac{\left(\frac{1}{2} \cdot \left(y \cdot b\right)\right) \cdot \left(\frac{1}{2} \cdot \left(y \cdot b\right)\right) - y \cdot y}{\frac{1}{2} \cdot \left(y \cdot b\right) - y}}, y\right)} \]

       3. frac-2neg N/A

          \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \color{blue}{\frac{\mathsf{neg}\left(\left(\left(\frac{1}{2} \cdot \left(y \cdot b\right)\right) \cdot \left(\frac{1}{2} \cdot \left(y \cdot b\right)\right) - y \cdot y\right)\right)}{\mathsf{neg}\left(\left(\frac{1}{2} \cdot \left(y \cdot b\right) - y\right)\right)}}, y\right)} \]

       4. lower-/.f64 N/A

          \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \color{blue}{\frac{\mathsf{neg}\left(\left(\left(\frac{1}{2} \cdot \left(y \cdot b\right)\right) \cdot \left(\frac{1}{2} \cdot \left(y \cdot b\right)\right) - y \cdot y\right)\right)}{\mathsf{neg}\left(\left(\frac{1}{2} \cdot \left(y \cdot b\right) - y\right)\right)}}, y\right)} \]

   13. Applied egg-rr 50.5%

       \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \color{blue}{\frac{\mathsf{fma}\left(y, b \cdot 0.5, y\right) \cdot \mathsf{fma}\left(y \cdot b, -0.5, y\right)}{\mathsf{fma}\left(y \cdot b, -0.5, y\right)}}, y\right)} \]

   if 8.5e8 < b

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\left(-1 \cdot \log a + y \cdot \log z\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{e^{\left(-1 \cdot \log a + y \cdot \log z\right) - b}}{y}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \frac{e^{\left(-1 \cdot \log a + y \cdot \log z\right) - b}}{y}} \]

      3. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      5. lower-/.f64 N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      6. +-commutative N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z + -1 \cdot \log a}}}{y \cdot e^{b}} \]

      7. mul-1-neg N/A

         \[\leadsto x \cdot \frac{e^{y \cdot \log z + \color{blue}{\left(\mathsf{neg}\left(\log a\right)\right)}}}{y \cdot e^{b}} \]

      8. unsub-neg N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z - \log a}}}{y \cdot e^{b}} \]

      9. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      10. lower-/.f64 N/A

          \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      11. *-commutative N/A

          \[\leadsto x \cdot \frac{\frac{e^{\color{blue}{\log z \cdot y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      12. exp-to-pow N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      13. lower-pow.f64 N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      14. rem-exp-log N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{\color{blue}{a}}}{y \cdot e^{b}} \]

      15. lower-*.f64 N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{\color{blue}{y \cdot e^{b}}} \]

      16. lower-exp.f64 72.7

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 72.7%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot e^{b}}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. lower-exp.f64 80.9

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 80.9%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\left(1 + b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)\right)}\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\left(b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right) + 1\right)}\right)} \]

       2. lower-fma.f64 N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left(b, 1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right), 1\right)}\right)} \]

       3. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \color{blue}{b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right) + 1}, 1\right)\right)} \]

       4. lower-fma.f64 N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, \frac{1}{2} + \frac{1}{6} \cdot b, 1\right)}, 1\right)\right)} \]

       5. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\frac{1}{6} \cdot b + \frac{1}{2}}, 1\right), 1\right)\right)} \]

       6. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{b \cdot \frac{1}{6}} + \frac{1}{2}, 1\right), 1\right)\right)} \]

       7. lower-fma.f64 73.1

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, 0.16666666666666666, 0.5\right)}, 1\right), 1\right)\right)} \]

   11. Simplified 73.1%

       \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), 1\right)}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{+111}:\\ \;\;\;\;\mathsf{fma}\left(b, \frac{x}{y \cdot a} \cdot \mathsf{fma}\left(b, 0.5, -1\right), \frac{x}{y \cdot a}\right)\\ \mathbf{elif}\;b \leq 850000000:\\ \;\;\;\;\frac{x}{a \cdot \mathsf{fma}\left(b, \frac{\mathsf{fma}\left(y, b \cdot 0.5, y\right) \cdot \mathsf{fma}\left(y \cdot b, -0.5, y\right)}{\mathsf{fma}\left(y \cdot b, -0.5, y\right)}, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 51.7% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{y \cdot a}\\ \mathbf{if}\;b \leq -1.1 \cdot 10^{+30}:\\ \;\;\;\;\mathsf{fma}\left(b, t\_1 \cdot \mathsf{fma}\left(b, 0.5, -1\right), t\_1\right)\\ \mathbf{elif}\;b \leq 1.55 \cdot 10^{-66}:\\ \;\;\;\;\frac{x}{b \cdot \left(b \cdot \mathsf{fma}\left(\frac{y}{b}, a + \frac{a}{b}, a \cdot \left(y \cdot 0.5\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (* y a))))
   (if (<= b -1.1e+30)
     (fma b (* t_1 (fma b 0.5 -1.0)) t_1)
     (if (<= b 1.55e-66)
       (/ x (* b (* b (fma (/ y b) (+ a (/ a b)) (* a (* y 0.5))))))
       (/
        x
        (*
         a
         (* y (fma b (fma b (fma b 0.16666666666666666 0.5) 1.0) 1.0))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / (y * a);
	double tmp;
	if (b <= -1.1e+30) {
		tmp = fma(b, (t_1 * fma(b, 0.5, -1.0)), t_1);
	} else if (b <= 1.55e-66) {
		tmp = x / (b * (b * fma((y / b), (a + (a / b)), (a * (y * 0.5)))));
	} else {
		tmp = x / (a * (y * fma(b, fma(b, fma(b, 0.16666666666666666, 0.5), 1.0), 1.0)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x / Float64(y * a))
	tmp = 0.0
	if (b <= -1.1e+30)
		tmp = fma(b, Float64(t_1 * fma(b, 0.5, -1.0)), t_1);
	elseif (b <= 1.55e-66)
		tmp = Float64(x / Float64(b * Float64(b * fma(Float64(y / b), Float64(a + Float64(a / b)), Float64(a * Float64(y * 0.5))))));
	else
		tmp = Float64(x / Float64(a * Float64(y * fma(b, fma(b, fma(b, 0.16666666666666666, 0.5), 1.0), 1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x / N[(y * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.1e+30], N[(b * N[(t$95$1 * N[(b * 0.5 + -1.0), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], If[LessEqual[b, 1.55e-66], N[(x / N[(b * N[(b * N[(N[(y / b), $MachinePrecision] * N[(a + N[(a / b), $MachinePrecision]), $MachinePrecision] + N[(a * N[(y * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(a * N[(y * N[(b * N[(b * N[(b * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.1e30

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\left(-1 \cdot \log a + y \cdot \log z\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{e^{\left(-1 \cdot \log a + y \cdot \log z\right) - b}}{y}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \frac{e^{\left(-1 \cdot \log a + y \cdot \log z\right) - b}}{y}} \]

      3. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      5. lower-/.f64 N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      6. +-commutative N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z + -1 \cdot \log a}}}{y \cdot e^{b}} \]

      7. mul-1-neg N/A

         \[\leadsto x \cdot \frac{e^{y \cdot \log z + \color{blue}{\left(\mathsf{neg}\left(\log a\right)\right)}}}{y \cdot e^{b}} \]

      8. unsub-neg N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z - \log a}}}{y \cdot e^{b}} \]

      9. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      10. lower-/.f64 N/A

          \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      11. *-commutative N/A

          \[\leadsto x \cdot \frac{\frac{e^{\color{blue}{\log z \cdot y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      12. exp-to-pow N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      13. lower-pow.f64 N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      14. rem-exp-log N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{\color{blue}{a}}}{y \cdot e^{b}} \]

      15. lower-*.f64 N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{\color{blue}{y \cdot e^{b}}} \]

      16. lower-exp.f64 70.7

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 70.7%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot e^{b}}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. lower-exp.f64 77.4

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 77.4%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \color{blue}{b \cdot \left(-1 \cdot \left(b \cdot \left(-1 \cdot \frac{x}{a \cdot y} + \frac{1}{2} \cdot \frac{x}{a \cdot y}\right)\right) - \frac{x}{a \cdot y}\right) + \frac{x}{a \cdot y}} \]
   10. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(b, -1 \cdot \left(b \cdot \left(-1 \cdot \frac{x}{a \cdot y} + \frac{1}{2} \cdot \frac{x}{a \cdot y}\right)\right) - \frac{x}{a \cdot y}, \frac{x}{a \cdot y}\right)} \]

   11. Simplified 58.8%

       \[\leadsto \color{blue}{\mathsf{fma}\left(b, \frac{x}{a \cdot y} \cdot \mathsf{fma}\left(b, 0.5, -1\right), \frac{x}{a \cdot y}\right)} \]

   if -1.1e30 < b < 1.5499999999999999e-66

   1. Initial program 96.1%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\left(-1 \cdot \log a + y \cdot \log z\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{e^{\left(-1 \cdot \log a + y \cdot \log z\right) - b}}{y}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \frac{e^{\left(-1 \cdot \log a + y \cdot \log z\right) - b}}{y}} \]

      3. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      5. lower-/.f64 N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      6. +-commutative N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z + -1 \cdot \log a}}}{y \cdot e^{b}} \]

      7. mul-1-neg N/A

         \[\leadsto x \cdot \frac{e^{y \cdot \log z + \color{blue}{\left(\mathsf{neg}\left(\log a\right)\right)}}}{y \cdot e^{b}} \]

      8. unsub-neg N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z - \log a}}}{y \cdot e^{b}} \]

      9. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      10. lower-/.f64 N/A

          \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      11. *-commutative N/A

          \[\leadsto x \cdot \frac{\frac{e^{\color{blue}{\log z \cdot y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      12. exp-to-pow N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      13. lower-pow.f64 N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      14. rem-exp-log N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{\color{blue}{a}}}{y \cdot e^{b}} \]

      15. lower-*.f64 N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{\color{blue}{y \cdot e^{b}}} \]

      16. lower-exp.f64 67.9

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 67.9%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot e^{b}}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. lower-exp.f64 45.2

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 45.2%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y + b \cdot \left(y + \frac{1}{2} \cdot \left(b \cdot y\right)\right)\right)}} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\left(b \cdot \left(y + \frac{1}{2} \cdot \left(b \cdot y\right)\right) + y\right)}} \]

       2. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(b \cdot \left(y + \frac{1}{2} \cdot \color{blue}{\left(y \cdot b\right)}\right) + y\right)} \]

       3. associate-*r* N/A

          \[\leadsto \frac{x}{a \cdot \left(b \cdot \left(y + \color{blue}{\left(\frac{1}{2} \cdot y\right) \cdot b}\right) + y\right)} \]

       4. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(b \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot y\right) \cdot b + y\right)} + y\right)} \]

       5. lower-fma.f64 N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\mathsf{fma}\left(b, \left(\frac{1}{2} \cdot y\right) \cdot b + y, y\right)}} \]

       6. associate-*r* N/A

          \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \color{blue}{\frac{1}{2} \cdot \left(y \cdot b\right)} + y, y\right)} \]

       7. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \frac{1}{2} \cdot \color{blue}{\left(b \cdot y\right)} + y, y\right)} \]

       8. lower-fma.f64 N/A

          \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(\frac{1}{2}, b \cdot y, y\right)}, y\right)} \]

       9. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{y \cdot b}, y\right), y\right)} \]

       10. lower-*.f64 43.6

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(0.5, \color{blue}{y \cdot b}, y\right), y\right)} \]

   11. Simplified 43.6%

       \[\leadsto \frac{x}{a \cdot \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(0.5, y \cdot b, y\right), y\right)}} \]

   12. Taylor expanded in b around inf

       \[\leadsto \frac{x}{\color{blue}{{b}^{2} \cdot \left(\frac{1}{2} \cdot \left(a \cdot y\right) + \left(\frac{a \cdot y}{b} + \frac{a \cdot y}{{b}^{2}}\right)\right)}} \]
   13. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \frac{x}{\color{blue}{\left(b \cdot b\right)} \cdot \left(\frac{1}{2} \cdot \left(a \cdot y\right) + \left(\frac{a \cdot y}{b} + \frac{a \cdot y}{{b}^{2}}\right)\right)} \]

       2. associate-*l* N/A

          \[\leadsto \frac{x}{\color{blue}{b \cdot \left(b \cdot \left(\frac{1}{2} \cdot \left(a \cdot y\right) + \left(\frac{a \cdot y}{b} + \frac{a \cdot y}{{b}^{2}}\right)\right)\right)}} \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{b \cdot \left(b \cdot \left(\frac{1}{2} \cdot \left(a \cdot y\right) + \left(\frac{a \cdot y}{b} + \frac{a \cdot y}{{b}^{2}}\right)\right)\right)}} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{x}{b \cdot \color{blue}{\left(b \cdot \left(\frac{1}{2} \cdot \left(a \cdot y\right) + \left(\frac{a \cdot y}{b} + \frac{a \cdot y}{{b}^{2}}\right)\right)\right)}} \]

       5. +-commutative N/A

          \[\leadsto \frac{x}{b \cdot \left(b \cdot \color{blue}{\left(\left(\frac{a \cdot y}{b} + \frac{a \cdot y}{{b}^{2}}\right) + \frac{1}{2} \cdot \left(a \cdot y\right)\right)}\right)} \]

       6. associate-/l* N/A

          \[\leadsto \frac{x}{b \cdot \left(b \cdot \left(\left(\color{blue}{a \cdot \frac{y}{b}} + \frac{a \cdot y}{{b}^{2}}\right) + \frac{1}{2} \cdot \left(a \cdot y\right)\right)\right)} \]

       7. unpow2 N/A

          \[\leadsto \frac{x}{b \cdot \left(b \cdot \left(\left(a \cdot \frac{y}{b} + \frac{a \cdot y}{\color{blue}{b \cdot b}}\right) + \frac{1}{2} \cdot \left(a \cdot y\right)\right)\right)} \]

       8. times-frac N/A

          \[\leadsto \frac{x}{b \cdot \left(b \cdot \left(\left(a \cdot \frac{y}{b} + \color{blue}{\frac{a}{b} \cdot \frac{y}{b}}\right) + \frac{1}{2} \cdot \left(a \cdot y\right)\right)\right)} \]

       9. distribute-rgt-out N/A

          \[\leadsto \frac{x}{b \cdot \left(b \cdot \left(\color{blue}{\frac{y}{b} \cdot \left(a + \frac{a}{b}\right)} + \frac{1}{2} \cdot \left(a \cdot y\right)\right)\right)} \]

       10. lower-fma.f64 N/A

           \[\leadsto \frac{x}{b \cdot \left(b \cdot \color{blue}{\mathsf{fma}\left(\frac{y}{b}, a + \frac{a}{b}, \frac{1}{2} \cdot \left(a \cdot y\right)\right)}\right)} \]

       11. lower-/.f64 N/A

           \[\leadsto \frac{x}{b \cdot \left(b \cdot \mathsf{fma}\left(\color{blue}{\frac{y}{b}}, a + \frac{a}{b}, \frac{1}{2} \cdot \left(a \cdot y\right)\right)\right)} \]

       12. lower-+.f64 N/A

           \[\leadsto \frac{x}{b \cdot \left(b \cdot \mathsf{fma}\left(\frac{y}{b}, \color{blue}{a + \frac{a}{b}}, \frac{1}{2} \cdot \left(a \cdot y\right)\right)\right)} \]

       13. lower-/.f64 N/A

           \[\leadsto \frac{x}{b \cdot \left(b \cdot \mathsf{fma}\left(\frac{y}{b}, a + \color{blue}{\frac{a}{b}}, \frac{1}{2} \cdot \left(a \cdot y\right)\right)\right)} \]

       14. *-commutative N/A

           \[\leadsto \frac{x}{b \cdot \left(b \cdot \mathsf{fma}\left(\frac{y}{b}, a + \frac{a}{b}, \color{blue}{\left(a \cdot y\right) \cdot \frac{1}{2}}\right)\right)} \]

       15. associate-*l* N/A

           \[\leadsto \frac{x}{b \cdot \left(b \cdot \mathsf{fma}\left(\frac{y}{b}, a + \frac{a}{b}, \color{blue}{a \cdot \left(y \cdot \frac{1}{2}\right)}\right)\right)} \]

       16. *-commutative N/A

           \[\leadsto \frac{x}{b \cdot \left(b \cdot \mathsf{fma}\left(\frac{y}{b}, a + \frac{a}{b}, a \cdot \color{blue}{\left(\frac{1}{2} \cdot y\right)}\right)\right)} \]

       17. lower-*.f64 N/A

           \[\leadsto \frac{x}{b \cdot \left(b \cdot \mathsf{fma}\left(\frac{y}{b}, a + \frac{a}{b}, \color{blue}{a \cdot \left(\frac{1}{2} \cdot y\right)}\right)\right)} \]

       18. *-commutative N/A

           \[\leadsto \frac{x}{b \cdot \left(b \cdot \mathsf{fma}\left(\frac{y}{b}, a + \frac{a}{b}, a \cdot \color{blue}{\left(y \cdot \frac{1}{2}\right)}\right)\right)} \]

       19. lower-*.f64 50.2

           \[\leadsto \frac{x}{b \cdot \left(b \cdot \mathsf{fma}\left(\frac{y}{b}, a + \frac{a}{b}, a \cdot \color{blue}{\left(y \cdot 0.5\right)}\right)\right)} \]

   14. Simplified 50.2%

       \[\leadsto \frac{x}{\color{blue}{b \cdot \left(b \cdot \mathsf{fma}\left(\frac{y}{b}, a + \frac{a}{b}, a \cdot \left(y \cdot 0.5\right)\right)\right)}} \]

   if 1.5499999999999999e-66 < b

   1. Initial program 99.4%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\left(-1 \cdot \log a + y \cdot \log z\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{e^{\left(-1 \cdot \log a + y \cdot \log z\right) - b}}{y}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \frac{e^{\left(-1 \cdot \log a + y \cdot \log z\right) - b}}{y}} \]

      3. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      5. lower-/.f64 N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      6. +-commutative N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z + -1 \cdot \log a}}}{y \cdot e^{b}} \]

      7. mul-1-neg N/A

         \[\leadsto x \cdot \frac{e^{y \cdot \log z + \color{blue}{\left(\mathsf{neg}\left(\log a\right)\right)}}}{y \cdot e^{b}} \]

      8. unsub-neg N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z - \log a}}}{y \cdot e^{b}} \]

      9. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      10. lower-/.f64 N/A

          \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      11. *-commutative N/A

          \[\leadsto x \cdot \frac{\frac{e^{\color{blue}{\log z \cdot y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      12. exp-to-pow N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      13. lower-pow.f64 N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      14. rem-exp-log N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{\color{blue}{a}}}{y \cdot e^{b}} \]

      15. lower-*.f64 N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{\color{blue}{y \cdot e^{b}}} \]

      16. lower-exp.f64 76.2

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 76.2%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot e^{b}}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. lower-exp.f64 74.2

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 74.2%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\left(1 + b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)\right)}\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\left(b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right) + 1\right)}\right)} \]

       2. lower-fma.f64 N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left(b, 1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right), 1\right)}\right)} \]

       3. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \color{blue}{b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right) + 1}, 1\right)\right)} \]

       4. lower-fma.f64 N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, \frac{1}{2} + \frac{1}{6} \cdot b, 1\right)}, 1\right)\right)} \]

       5. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\frac{1}{6} \cdot b + \frac{1}{2}}, 1\right), 1\right)\right)} \]

       6. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{b \cdot \frac{1}{6}} + \frac{1}{2}, 1\right), 1\right)\right)} \]

       7. lower-fma.f64 66.0

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, 0.16666666666666666, 0.5\right)}, 1\right), 1\right)\right)} \]

   11. Simplified 66.0%

       \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), 1\right)}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 57.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.1 \cdot 10^{+30}:\\ \;\;\;\;\mathsf{fma}\left(b, \frac{x}{y \cdot a} \cdot \mathsf{fma}\left(b, 0.5, -1\right), \frac{x}{y \cdot a}\right)\\ \mathbf{elif}\;b \leq 1.55 \cdot 10^{-66}:\\ \;\;\;\;\frac{x}{b \cdot \left(b \cdot \mathsf{fma}\left(\frac{y}{b}, a + \frac{a}{b}, a \cdot \left(y \cdot 0.5\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 44.0% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.4 \cdot 10^{-11}:\\ \;\;\;\;\frac{x}{y \cdot a} \cdot \left(1 - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{a \cdot \left(y \cdot \left(b \cdot b\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b 1.4e-11)
   (* (/ x (* y a)) (- 1.0 b))
   (/ (* x 2.0) (* a (* y (* b b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= 1.4e-11) {
		tmp = (x / (y * a)) * (1.0 - b);
	} else {
		tmp = (x * 2.0) / (a * (y * (b * 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 (b <= 1.4d-11) then
        tmp = (x / (y * a)) * (1.0d0 - b)
    else
        tmp = (x * 2.0d0) / (a * (y * (b * 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 (b <= 1.4e-11) {
		tmp = (x / (y * a)) * (1.0 - b);
	} else {
		tmp = (x * 2.0) / (a * (y * (b * b)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if b <= 1.4e-11:
		tmp = (x / (y * a)) * (1.0 - b)
	else:
		tmp = (x * 2.0) / (a * (y * (b * b)))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= 1.4e-11)
		tmp = Float64(Float64(x / Float64(y * a)) * Float64(1.0 - b));
	else
		tmp = Float64(Float64(x * 2.0) / Float64(a * Float64(y * Float64(b * b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (b <= 1.4e-11)
		tmp = (x / (y * a)) * (1.0 - b);
	else
		tmp = (x * 2.0) / (a * (y * (b * b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, 1.4e-11], N[(N[(x / N[(y * a), $MachinePrecision]), $MachinePrecision] * N[(1.0 - b), $MachinePrecision]), $MachinePrecision], N[(N[(x * 2.0), $MachinePrecision] / N[(a * N[(y * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.4e-11

   1. Initial program 97.4%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\left(-1 \cdot \log a + y \cdot \log z\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{e^{\left(-1 \cdot \log a + y \cdot \log z\right) - b}}{y}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \frac{e^{\left(-1 \cdot \log a + y \cdot \log z\right) - b}}{y}} \]

      3. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      5. lower-/.f64 N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      6. +-commutative N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z + -1 \cdot \log a}}}{y \cdot e^{b}} \]

      7. mul-1-neg N/A

         \[\leadsto x \cdot \frac{e^{y \cdot \log z + \color{blue}{\left(\mathsf{neg}\left(\log a\right)\right)}}}{y \cdot e^{b}} \]

      8. unsub-neg N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z - \log a}}}{y \cdot e^{b}} \]

      9. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      10. lower-/.f64 N/A

          \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      11. *-commutative N/A

          \[\leadsto x \cdot \frac{\frac{e^{\color{blue}{\log z \cdot y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      12. exp-to-pow N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      13. lower-pow.f64 N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      14. rem-exp-log N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{\color{blue}{a}}}{y \cdot e^{b}} \]

      15. lower-*.f64 N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{\color{blue}{y \cdot e^{b}}} \]

      16. lower-exp.f64 70.7

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 70.7%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot e^{b}}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. lower-exp.f64 56.4

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 56.4%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{b \cdot x}{a \cdot y} + \frac{x}{a \cdot y}} \]
   10. Step-by-step derivation

       1. associate-/l* N/A

          \[\leadsto -1 \cdot \color{blue}{\left(b \cdot \frac{x}{a \cdot y}\right)} + \frac{x}{a \cdot y} \]

       2. associate-*r* N/A

          \[\leadsto \color{blue}{\left(-1 \cdot b\right) \cdot \frac{x}{a \cdot y}} + \frac{x}{a \cdot y} \]

       3. distribute-lft1-in N/A

          \[\leadsto \color{blue}{\left(-1 \cdot b + 1\right) \cdot \frac{x}{a \cdot y}} \]

       4. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(-1 \cdot b + 1\right) \cdot \frac{x}{a \cdot y}} \]

       5. lower-+.f64 N/A

          \[\leadsto \color{blue}{\left(-1 \cdot b + 1\right)} \cdot \frac{x}{a \cdot y} \]

       6. mul-1-neg N/A

          \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(b\right)\right)} + 1\right) \cdot \frac{x}{a \cdot y} \]

       7. lower-neg.f64 N/A

          \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(b\right)\right)} + 1\right) \cdot \frac{x}{a \cdot y} \]

       8. lower-/.f64 N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(b\right)\right) + 1\right) \cdot \color{blue}{\frac{x}{a \cdot y}} \]

       9. lower-*.f64 42.9

          \[\leadsto \left(\left(-b\right) + 1\right) \cdot \frac{x}{\color{blue}{a \cdot y}} \]

   11. Simplified 42.9%

       \[\leadsto \color{blue}{\left(\left(-b\right) + 1\right) \cdot \frac{x}{a \cdot y}} \]

   if 1.4e-11 < b

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\left(-1 \cdot \log a + y \cdot \log z\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{e^{\left(-1 \cdot \log a + y \cdot \log z\right) - b}}{y}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \frac{e^{\left(-1 \cdot \log a + y \cdot \log z\right) - b}}{y}} \]

      3. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      5. lower-/.f64 N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      6. +-commutative N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z + -1 \cdot \log a}}}{y \cdot e^{b}} \]

      7. mul-1-neg N/A

         \[\leadsto x \cdot \frac{e^{y \cdot \log z + \color{blue}{\left(\mathsf{neg}\left(\log a\right)\right)}}}{y \cdot e^{b}} \]

      8. unsub-neg N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z - \log a}}}{y \cdot e^{b}} \]

      9. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      10. lower-/.f64 N/A

          \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      11. *-commutative N/A

          \[\leadsto x \cdot \frac{\frac{e^{\color{blue}{\log z \cdot y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      12. exp-to-pow N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      13. lower-pow.f64 N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      14. rem-exp-log N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{\color{blue}{a}}}{y \cdot e^{b}} \]

      15. lower-*.f64 N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{\color{blue}{y \cdot e^{b}}} \]

      16. lower-exp.f64 73.0

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 73.0%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot e^{b}}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. lower-exp.f64 79.3

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 79.3%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y + b \cdot \left(y + \frac{1}{2} \cdot \left(b \cdot y\right)\right)\right)}} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\left(b \cdot \left(y + \frac{1}{2} \cdot \left(b \cdot y\right)\right) + y\right)}} \]

       2. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(b \cdot \left(y + \frac{1}{2} \cdot \color{blue}{\left(y \cdot b\right)}\right) + y\right)} \]

       3. associate-*r* N/A

          \[\leadsto \frac{x}{a \cdot \left(b \cdot \left(y + \color{blue}{\left(\frac{1}{2} \cdot y\right) \cdot b}\right) + y\right)} \]

       4. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(b \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot y\right) \cdot b + y\right)} + y\right)} \]

       5. lower-fma.f64 N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\mathsf{fma}\left(b, \left(\frac{1}{2} \cdot y\right) \cdot b + y, y\right)}} \]

       6. associate-*r* N/A

          \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \color{blue}{\frac{1}{2} \cdot \left(y \cdot b\right)} + y, y\right)} \]

       7. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \frac{1}{2} \cdot \color{blue}{\left(b \cdot y\right)} + y, y\right)} \]

       8. lower-fma.f64 N/A

          \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(\frac{1}{2}, b \cdot y, y\right)}, y\right)} \]

       9. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{y \cdot b}, y\right), y\right)} \]

       10. lower-*.f64 53.1

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(0.5, \color{blue}{y \cdot b}, y\right), y\right)} \]

   11. Simplified 53.1%

       \[\leadsto \frac{x}{a \cdot \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(0.5, y \cdot b, y\right), y\right)}} \]

   12. Taylor expanded in b around inf

       \[\leadsto \color{blue}{2 \cdot \frac{x}{a \cdot \left({b}^{2} \cdot y\right)}} \]
   13. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{2 \cdot x}{a \cdot \left({b}^{2} \cdot y\right)}} \]

       2. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{2 \cdot x}{a \cdot \left({b}^{2} \cdot y\right)}} \]

       3. *-commutative N/A

          \[\leadsto \frac{\color{blue}{x \cdot 2}}{a \cdot \left({b}^{2} \cdot y\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{x \cdot 2}}{a \cdot \left({b}^{2} \cdot y\right)} \]

       5. lower-*.f64 N/A

          \[\leadsto \frac{x \cdot 2}{\color{blue}{a \cdot \left({b}^{2} \cdot y\right)}} \]

       6. *-commutative N/A

          \[\leadsto \frac{x \cdot 2}{a \cdot \color{blue}{\left(y \cdot {b}^{2}\right)}} \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{x \cdot 2}{a \cdot \color{blue}{\left(y \cdot {b}^{2}\right)}} \]

       8. unpow2 N/A

          \[\leadsto \frac{x \cdot 2}{a \cdot \left(y \cdot \color{blue}{\left(b \cdot b\right)}\right)} \]

       9. lower-*.f64 58.8

          \[\leadsto \frac{x \cdot 2}{a \cdot \left(y \cdot \color{blue}{\left(b \cdot b\right)}\right)} \]

   14. Simplified 58.8%

       \[\leadsto \color{blue}{\frac{x \cdot 2}{a \cdot \left(y \cdot \left(b \cdot b\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.4 \cdot 10^{-11}:\\ \;\;\;\;\frac{x}{y \cdot a} \cdot \left(1 - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{a \cdot \left(y \cdot \left(b \cdot b\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 31.7% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 8.2 \cdot 10^{+87}:\\ \;\;\;\;\frac{x}{y \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t 8.2e+87) (/ x (* y a)) (/ 1.0 (/ y x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= 8.2e+87) {
		tmp = x / (y * a);
	} else {
		tmp = 1.0 / (y / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (t <= 8.2d+87) then
        tmp = x / (y * a)
    else
        tmp = 1.0d0 / (y / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= 8.2e+87) {
		tmp = x / (y * a);
	} else {
		tmp = 1.0 / (y / x);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if t <= 8.2e+87:
		tmp = x / (y * a)
	else:
		tmp = 1.0 / (y / x)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= 8.2e+87)
		tmp = Float64(x / Float64(y * a));
	else
		tmp = Float64(1.0 / Float64(y / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (t <= 8.2e+87)
		tmp = x / (y * a);
	else
		tmp = 1.0 / (y / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, 8.2e+87], N[(x / N[(y * a), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(y / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 8.1999999999999998e87

   1. Initial program 97.6%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\left(-1 \cdot \log a + y \cdot \log z\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{e^{\left(-1 \cdot \log a + y \cdot \log z\right) - b}}{y}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \frac{e^{\left(-1 \cdot \log a + y \cdot \log z\right) - b}}{y}} \]

      3. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      5. lower-/.f64 N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      6. +-commutative N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z + -1 \cdot \log a}}}{y \cdot e^{b}} \]

      7. mul-1-neg N/A

         \[\leadsto x \cdot \frac{e^{y \cdot \log z + \color{blue}{\left(\mathsf{neg}\left(\log a\right)\right)}}}{y \cdot e^{b}} \]

      8. unsub-neg N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z - \log a}}}{y \cdot e^{b}} \]

      9. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      10. lower-/.f64 N/A

          \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      11. *-commutative N/A

          \[\leadsto x \cdot \frac{\frac{e^{\color{blue}{\log z \cdot y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      12. exp-to-pow N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      13. lower-pow.f64 N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      14. rem-exp-log N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{\color{blue}{a}}}{y \cdot e^{b}} \]

      15. lower-*.f64 N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{\color{blue}{y \cdot e^{b}}} \]

      16. lower-exp.f64 81.3

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 81.3%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot e^{b}}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. lower-exp.f64 70.7

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 70.7%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot y}} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{a \cdot y}} \]

       2. lower-*.f64 37.7

          \[\leadsto \frac{x}{\color{blue}{a \cdot y}} \]

   11. Simplified 37.7%

       \[\leadsto \color{blue}{\frac{x}{a \cdot y}} \]

   if 8.1999999999999998e87 < t

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]

      2. lower-log.f64 51.1

         \[\leadsto \frac{x \cdot e^{y \cdot \color{blue}{\log z} - b}}{y} \]

   5. Simplified 51.1%

      \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]

   6. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{x \cdot {z}^{y}}{y}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot {z}^{y}}{y}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{{z}^{y} \cdot x}}{y} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{{z}^{y} \cdot x}}{y} \]

      4. lower-pow.f64 47.5

         \[\leadsto \frac{\color{blue}{{z}^{y}} \cdot x}{y} \]

   8. Simplified 47.5%

      \[\leadsto \color{blue}{\frac{{z}^{y} \cdot x}{y}} \]

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x}{y}} \]
   10. Step-by-step derivation

       1. lower-/.f64 24.9

          \[\leadsto \color{blue}{\frac{x}{y}} \]

   11. Simplified 24.9%

       \[\leadsto \color{blue}{\frac{x}{y}} \]
   12. Step-by-step derivation

       1. clear-num N/A

          \[\leadsto \color{blue}{\frac{1}{\frac{y}{x}}} \]

       2. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{\frac{y}{x}}} \]

       3. lower-/.f64 25.7

          \[\leadsto \frac{1}{\color{blue}{\frac{y}{x}}} \]

   13. Applied egg-rr 25.7%

       \[\leadsto \color{blue}{\frac{1}{\frac{y}{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8.2 \cdot 10^{+87}:\\ \;\;\;\;\frac{x}{y \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 31.6% accurate, 14.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 8.2 \cdot 10^{+87}:\\ \;\;\;\;\frac{x}{y \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t 8.2e+87) (/ x (* y a)) (/ x y)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= 8.2e+87) {
		tmp = x / (y * a);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (t <= 8.2d+87) then
        tmp = x / (y * a)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= 8.2e+87) {
		tmp = x / (y * a);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if t <= 8.2e+87:
		tmp = x / (y * a)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= 8.2e+87)
		tmp = Float64(x / Float64(y * a));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (t <= 8.2e+87)
		tmp = x / (y * a);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, 8.2e+87], N[(x / N[(y * a), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 8.1999999999999998e87

   1. Initial program 97.6%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\left(-1 \cdot \log a + y \cdot \log z\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{e^{\left(-1 \cdot \log a + y \cdot \log z\right) - b}}{y}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \frac{e^{\left(-1 \cdot \log a + y \cdot \log z\right) - b}}{y}} \]

      3. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      5. lower-/.f64 N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      6. +-commutative N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z + -1 \cdot \log a}}}{y \cdot e^{b}} \]

      7. mul-1-neg N/A

         \[\leadsto x \cdot \frac{e^{y \cdot \log z + \color{blue}{\left(\mathsf{neg}\left(\log a\right)\right)}}}{y \cdot e^{b}} \]

      8. unsub-neg N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z - \log a}}}{y \cdot e^{b}} \]

      9. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      10. lower-/.f64 N/A

          \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y \cdot e^{b}} \]

      11. *-commutative N/A

          \[\leadsto x \cdot \frac{\frac{e^{\color{blue}{\log z \cdot y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      12. exp-to-pow N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      13. lower-pow.f64 N/A

          \[\leadsto x \cdot \frac{\frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y \cdot e^{b}} \]

      14. rem-exp-log N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{\color{blue}{a}}}{y \cdot e^{b}} \]

      15. lower-*.f64 N/A

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{\color{blue}{y \cdot e^{b}}} \]

      16. lower-exp.f64 81.3

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 81.3%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot e^{b}}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. lower-exp.f64 70.7

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 70.7%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot y}} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{a \cdot y}} \]

       2. lower-*.f64 37.7

          \[\leadsto \frac{x}{\color{blue}{a \cdot y}} \]

   11. Simplified 37.7%

       \[\leadsto \color{blue}{\frac{x}{a \cdot y}} \]

   if 8.1999999999999998e87 < t

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]

      2. lower-log.f64 51.1

         \[\leadsto \frac{x \cdot e^{y \cdot \color{blue}{\log z} - b}}{y} \]

   5. Simplified 51.1%

      \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]

   6. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{x \cdot {z}^{y}}{y}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot {z}^{y}}{y}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{{z}^{y} \cdot x}}{y} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{{z}^{y} \cdot x}}{y} \]

      4. lower-pow.f64 47.5

         \[\leadsto \frac{\color{blue}{{z}^{y}} \cdot x}{y} \]

   8. Simplified 47.5%

      \[\leadsto \color{blue}{\frac{{z}^{y} \cdot x}{y}} \]

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x}{y}} \]
   10. Step-by-step derivation

       1. lower-/.f64 24.9

          \[\leadsto \color{blue}{\frac{x}{y}} \]

   11. Simplified 24.9%

       \[\leadsto \color{blue}{\frac{x}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8.2 \cdot 10^{+87}:\\ \;\;\;\;\frac{x}{y \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 16.1% accurate, 28.0× speedup

Math

\[\begin{array}{l} \\ \frac{x}{y} \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (/ x y))

C

double code(double x, double y, double z, double t, double a, double b) {
	return x / y;
}

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 / y
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return x / y;
}

Python

def code(x, y, z, t, a, b):
	return x / y

Julia

function code(x, y, z, t, a, b)
	return Float64(x / y)
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = x / y;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(x / y), $MachinePrecision]

Derivation

1. Initial program 98.1%

   \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
2. Add Preprocessing

3. Taylor expanded in y around inf

   \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]
4. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]

   2. lower-log.f64 67.6

      \[\leadsto \frac{x \cdot e^{y \cdot \color{blue}{\log z} - b}}{y} \]

5. Simplified 67.6%

   \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]

6. Taylor expanded in b around 0

   \[\leadsto \color{blue}{\frac{x \cdot {z}^{y}}{y}} \]
7. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{x \cdot {z}^{y}}{y}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\color{blue}{{z}^{y} \cdot x}}{y} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{{z}^{y} \cdot x}}{y} \]

   4. lower-pow.f64 46.5

      \[\leadsto \frac{\color{blue}{{z}^{y}} \cdot x}{y} \]

8. Simplified 46.5%

   \[\leadsto \color{blue}{\frac{{z}^{y} \cdot x}{y}} \]

9. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{x}{y}} \]
10. Step-by-step derivation

    1. lower-/.f64 17.3

       \[\leadsto \color{blue}{\frac{x}{y}} \]

11. Simplified 17.3%

    \[\leadsto \color{blue}{\frac{x}{y}} \]
12. Add Preprocessing

Developer Target 1: 71.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {a}^{\left(t - 1\right)}\\ t_2 := \frac{x \cdot \frac{t\_1}{y}}{\left(b + 1\right) - y \cdot \log z}\\ \mathbf{if}\;t < -0.8845848504127471:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t < 852031.2288374073:\\ \;\;\;\;\frac{\frac{x}{y} \cdot t\_1}{e^{b - \log z \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (pow a (- t 1.0)))
        (t_2 (/ (* x (/ t_1 y)) (- (+ b 1.0) (* y (log z))))))
   (if (< t -0.8845848504127471)
     t_2
     (if (< t 852031.2288374073)
       (/ (* (/ x y) t_1) (exp (- b (* (log z) y))))
       t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = pow(a, (t - 1.0));
	double t_2 = (x * (t_1 / y)) / ((b + 1.0) - (y * log(z)));
	double tmp;
	if (t < -0.8845848504127471) {
		tmp = t_2;
	} else if (t < 852031.2288374073) {
		tmp = ((x / y) * t_1) / exp((b - (log(z) * y)));
	} else {
		tmp = t_2;
	}
	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 = a ** (t - 1.0d0)
    t_2 = (x * (t_1 / y)) / ((b + 1.0d0) - (y * log(z)))
    if (t < (-0.8845848504127471d0)) then
        tmp = t_2
    else if (t < 852031.2288374073d0) then
        tmp = ((x / y) * t_1) / exp((b - (log(z) * y)))
    else
        tmp = t_2
    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 = Math.pow(a, (t - 1.0));
	double t_2 = (x * (t_1 / y)) / ((b + 1.0) - (y * Math.log(z)));
	double tmp;
	if (t < -0.8845848504127471) {
		tmp = t_2;
	} else if (t < 852031.2288374073) {
		tmp = ((x / y) * t_1) / Math.exp((b - (Math.log(z) * y)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = math.pow(a, (t - 1.0))
	t_2 = (x * (t_1 / y)) / ((b + 1.0) - (y * math.log(z)))
	tmp = 0
	if t < -0.8845848504127471:
		tmp = t_2
	elif t < 852031.2288374073:
		tmp = ((x / y) * t_1) / math.exp((b - (math.log(z) * y)))
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = a ^ Float64(t - 1.0)
	t_2 = Float64(Float64(x * Float64(t_1 / y)) / Float64(Float64(b + 1.0) - Float64(y * log(z))))
	tmp = 0.0
	if (t < -0.8845848504127471)
		tmp = t_2;
	elseif (t < 852031.2288374073)
		tmp = Float64(Float64(Float64(x / y) * t_1) / exp(Float64(b - Float64(log(z) * y))));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = a ^ (t - 1.0);
	t_2 = (x * (t_1 / y)) / ((b + 1.0) - (y * log(z)));
	tmp = 0.0;
	if (t < -0.8845848504127471)
		tmp = t_2;
	elseif (t < 852031.2288374073)
		tmp = ((x / y) * t_1) / exp((b - (log(z) * y)));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[Power[a, N[(t - 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * N[(t$95$1 / y), $MachinePrecision]), $MachinePrecision] / N[(N[(b + 1.0), $MachinePrecision] - N[(y * N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[t, -0.8845848504127471], t$95$2, If[Less[t, 852031.2288374073], N[(N[(N[(x / y), $MachinePrecision] * t$95$1), $MachinePrecision] / N[Exp[N[(b - N[(N[Log[z], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Reproduce

herbie shell --seed 2024207 
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:incompleteBetaWorker from math-functions-0.1.5.2, A"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< t -8845848504127471/10000000000000000) (/ (* x (/ (pow a (- t 1)) y)) (- (+ b 1) (* y (log z)))) (if (< t 8520312288374073/10000000000) (/ (* (/ x y) (pow a (- t 1))) (exp (- b (* (log z) y)))) (/ (* x (/ (pow a (- t 1)) y)) (- (+ b 1) (* y (log z)))))))

  (/ (* x (exp (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))) y))