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

Percentage Accurate: 98.6% → 98.7%

Time: 16.6s

Alternatives: 22

Speedup: 1.0×

• 47.0% 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: 98.6% 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: 98.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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. Step-by-step derivation

   1. lift-log.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-log.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. lift--.f64 N/A

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

   8. lift-exp.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. remove-double-neg N/A

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

4. Applied rewrites 97.8%

   \[\leadsto \color{blue}{\frac{x}{y \cdot e^{b - \mathsf{fma}\left(y, \log z, \left(t + -1\right) \cdot \log a\right)}}} \]
5. Add Preprocessing

Alternative 2: 48.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot e^{\left(y \cdot \log z + \left(t + -1\right) \cdot \log a\right) - b}}{y}\\ t_2 := \frac{\frac{x \cdot 2}{y \cdot a} + \frac{x}{y \cdot \left(b \cdot a\right)} \cdot \left(-4 - \frac{-4}{b}\right)}{b \cdot b}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+283}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(b, b \cdot 0.5, b\right), a\right) \cdot \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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 2.0) (* y a)) (* (/ x (* y (* b a))) (- -4.0 (/ -4.0 b))))
          (* b b))))
   (if (<= t_1 (- INFINITY))
     t_2
     (if (<= t_1 5e+283)
       (/ 1.0 (* (fma a (fma b (* b 0.5) b) a) (/ y x)))
       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 * 2.0) / (y * a)) + ((x / (y * (b * a))) * (-4.0 - (-4.0 / b)))) / (b * b);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_1 <= 5e+283) {
		tmp = 1.0 / (fma(a, fma(b, (b * 0.5), b), a) * (y / x));
	} 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(Float64(Float64(x * 2.0) / Float64(y * a)) + Float64(Float64(x / Float64(y * Float64(b * a))) * Float64(-4.0 - Float64(-4.0 / b)))) / Float64(b * b))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_1 <= 5e+283)
		tmp = Float64(1.0 / Float64(fma(a, fma(b, Float64(b * 0.5), b), a) * Float64(y / x)));
	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[(N[(N[(x * 2.0), $MachinePrecision] / N[(y * a), $MachinePrecision]), $MachinePrecision] + N[(N[(x / N[(y * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-4.0 - N[(-4.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$2, If[LessEqual[t$95$1, 5e+283], N[(1.0 / N[(N[(a * N[(b * N[(b * 0.5), $MachinePrecision] + b), $MachinePrecision] + a), $MachinePrecision] * N[(y / x), $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 5.0000000000000004e283 < (/.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 68.3

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

   5. Applied rewrites 68.3%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 49.8

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

   8. Applied rewrites 49.8%

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

   9. Taylor expanded in y around 0

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

       1. distribute-rgt-in N/A

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

       2. *-lft-identity N/A

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

       3. associate-*l* N/A

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

       4. +-commutative N/A

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

       5. distribute-rgt1-in N/A

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

       6. associate-*r* N/A

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

       7. lower-/.f64 N/A

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

       8. *-commutative N/A

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

       9. *-lft-identity N/A

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

       10. associate-*r* N/A

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

       11. distribute-rgt1-in N/A

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

       12. +-commutative N/A

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

       13. associate-*l* N/A

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

       14. distribute-rgt-in N/A

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

   11. Applied rewrites 17.7%

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

   12. Taylor expanded in b around inf

       \[\leadsto \color{blue}{\frac{\left(-1 \cdot \frac{-8 \cdot \frac{x}{a \cdot y} + 4 \cdot \frac{x}{a \cdot y}}{{b}^{2}} + 2 \cdot \frac{x}{a \cdot y}\right) - 4 \cdot \frac{x}{a \cdot \left(b \cdot y\right)}}{{b}^{2}}} \]

   14. Applied rewrites 44.1%

       \[\leadsto \color{blue}{\frac{\frac{x \cdot 2}{a \cdot y} - \frac{x}{y \cdot \left(a \cdot b\right)} \cdot \left(\frac{-4}{b} - -4\right)}{b \cdot b}} \]

   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) < 5.0000000000000004e283

   1. Initial program 94.8%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   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 67.5

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

   5. Applied rewrites 67.5%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 65.1

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

   8. Applied rewrites 65.1%

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

   9. Taylor expanded in y around 0

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

       1. distribute-rgt-in N/A

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

       2. *-lft-identity N/A

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

       3. associate-*l* N/A

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

       4. +-commutative N/A

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

       5. distribute-rgt1-in N/A

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

       6. associate-*r* N/A

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

       7. lower-/.f64 N/A

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

       8. *-commutative N/A

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

       9. *-lft-identity N/A

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

       10. associate-*r* N/A

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

       11. distribute-rgt1-in N/A

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

       12. +-commutative N/A

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

       13. associate-*l* N/A

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

       14. distribute-rgt-in N/A

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

   11. Applied rewrites 56.7%

       \[\leadsto \color{blue}{\frac{x}{y \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(b, b \cdot 0.5, b\right), a\right)}} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-fma.f64 N/A

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

       3. lift-fma.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. remove-double-neg N/A

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

       6. *-rgt-identity N/A

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

       7. remove-double-neg N/A

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

       8. lift-*.f64 N/A

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

       9. frac-times N/A

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

       10. clear-num N/A

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

       11. frac-times N/A

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

       12. metadata-eval N/A

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

       13. lower-/.f64 N/A

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

       14. lower-*.f64 N/A

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

       15. lower-/.f64 57.9

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

   13. Applied rewrites 57.9%

       \[\leadsto \color{blue}{\frac{1}{\frac{y}{x} \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(b, b \cdot 0.5, b\right), a\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot e^{\left(y \cdot \log z + \left(t + -1\right) \cdot \log a\right) - b}}{y} \leq -\infty:\\ \;\;\;\;\frac{\frac{x \cdot 2}{y \cdot a} + \frac{x}{y \cdot \left(b \cdot a\right)} \cdot \left(-4 - \frac{-4}{b}\right)}{b \cdot b}\\ \mathbf{elif}\;\frac{x \cdot e^{\left(y \cdot \log z + \left(t + -1\right) \cdot \log a\right) - b}}{y} \leq 5 \cdot 10^{+283}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(b, b \cdot 0.5, b\right), a\right) \cdot \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot 2}{y \cdot a} + \frac{x}{y \cdot \left(b \cdot a\right)} \cdot \left(-4 - \frac{-4}{b}\right)}{b \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 47.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot e^{\left(y \cdot \log z + \left(t + -1\right) \cdot \log a\right) - b}}{y}\\ t_2 := \frac{x}{y \cdot \left(0.5 \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+283}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(b, b \cdot 0.5, b\right), a\right) \cdot \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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 (* 0.5 (* a (* b b)))))))
   (if (<= t_1 (- INFINITY))
     t_2
     (if (<= t_1 5e+283)
       (/ 1.0 (* (fma a (fma b (* b 0.5) b) a) (/ y x)))
       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 * (0.5 * (a * (b * b))));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_1 <= 5e+283) {
		tmp = 1.0 / (fma(a, fma(b, (b * 0.5), b), a) * (y / x));
	} 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 * Float64(0.5 * Float64(a * Float64(b * b)))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_1 <= 5e+283)
		tmp = Float64(1.0 / Float64(fma(a, fma(b, Float64(b * 0.5), b), a) * Float64(y / x)));
	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 * N[(0.5 * N[(a * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$2, If[LessEqual[t$95$1, 5e+283], N[(1.0 / N[(N[(a * N[(b * N[(b * 0.5), $MachinePrecision] + b), $MachinePrecision] + a), $MachinePrecision] * N[(y / x), $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 5.0000000000000004e283 < (/.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 68.3

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

   5. Applied rewrites 68.3%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 49.8

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

   8. Applied rewrites 49.8%

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

   9. Taylor expanded in y around 0

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

       1. distribute-rgt-in N/A

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

       2. *-lft-identity N/A

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

       3. associate-*l* N/A

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

       4. +-commutative N/A

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

       5. distribute-rgt1-in N/A

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

       6. associate-*r* N/A

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

       7. lower-/.f64 N/A

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

       8. *-commutative N/A

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

       9. *-lft-identity N/A

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

       10. associate-*r* N/A

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

       11. distribute-rgt1-in N/A

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

       12. +-commutative N/A

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

       13. associate-*l* N/A

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

       14. distribute-rgt-in N/A

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

   11. Applied rewrites 17.7%

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

   12. Taylor expanded in b around inf

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 41.0

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

   14. Applied rewrites 41.0%

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

   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) < 5.0000000000000004e283

   1. Initial program 94.8%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   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 67.5

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

   5. Applied rewrites 67.5%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 65.1

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

   8. Applied rewrites 65.1%

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

   9. Taylor expanded in y around 0

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

       1. distribute-rgt-in N/A

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

       2. *-lft-identity N/A

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

       3. associate-*l* N/A

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

       4. +-commutative N/A

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

       5. distribute-rgt1-in N/A

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

       6. associate-*r* N/A

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

       7. lower-/.f64 N/A

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

       8. *-commutative N/A

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

       9. *-lft-identity N/A

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

       10. associate-*r* N/A

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

       11. distribute-rgt1-in N/A

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

       12. +-commutative N/A

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

       13. associate-*l* N/A

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

       14. distribute-rgt-in N/A

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

   11. Applied rewrites 56.7%

       \[\leadsto \color{blue}{\frac{x}{y \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(b, b \cdot 0.5, b\right), a\right)}} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-fma.f64 N/A

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

       3. lift-fma.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. remove-double-neg N/A

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

       6. *-rgt-identity N/A

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

       7. remove-double-neg N/A

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

       8. lift-*.f64 N/A

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

       9. frac-times N/A

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

       10. clear-num N/A

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

       11. frac-times N/A

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

       12. metadata-eval N/A

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

       13. lower-/.f64 N/A

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

       14. lower-*.f64 N/A

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

       15. lower-/.f64 57.9

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

   13. Applied rewrites 57.9%

       \[\leadsto \color{blue}{\frac{1}{\frac{y}{x} \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(b, b \cdot 0.5, b\right), a\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot e^{\left(y \cdot \log z + \left(t + -1\right) \cdot \log a\right) - b}}{y} \leq -\infty:\\ \;\;\;\;\frac{x}{y \cdot \left(0.5 \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)}\\ \mathbf{elif}\;\frac{x \cdot e^{\left(y \cdot \log z + \left(t + -1\right) \cdot \log a\right) - b}}{y} \leq 5 \cdot 10^{+283}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(b, b \cdot 0.5, b\right), a\right) \cdot \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(0.5 \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 49.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot e^{\left(y \cdot \log z + \left(t + -1\right) \cdot \log a\right) - b}}{y}\\ t_2 := \frac{x}{y \cdot \left(0.5 \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+283}:\\ \;\;\;\;\frac{x}{y \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(b, b \cdot 0.5, b\right), a\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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 (* 0.5 (* a (* b b)))))))
   (if (<= t_1 (- INFINITY))
     t_2
     (if (<= t_1 5e+283) (/ x (* y (fma a (fma b (* b 0.5) b) a))) 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 * (0.5 * (a * (b * b))));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_1 <= 5e+283) {
		tmp = x / (y * fma(a, fma(b, (b * 0.5), b), a));
	} 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 * Float64(0.5 * Float64(a * Float64(b * b)))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_1 <= 5e+283)
		tmp = Float64(x / Float64(y * fma(a, fma(b, Float64(b * 0.5), b), a)));
	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 * N[(0.5 * N[(a * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$2, If[LessEqual[t$95$1, 5e+283], N[(x / N[(y * N[(a * N[(b * N[(b * 0.5), $MachinePrecision] + b), $MachinePrecision] + a), $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 5.0000000000000004e283 < (/.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 68.3

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

   5. Applied rewrites 68.3%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 49.8

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

   8. Applied rewrites 49.8%

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

   9. Taylor expanded in y around 0

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

       1. distribute-rgt-in N/A

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

       2. *-lft-identity N/A

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

       3. associate-*l* N/A

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

       4. +-commutative N/A

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

       5. distribute-rgt1-in N/A

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

       6. associate-*r* N/A

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

       7. lower-/.f64 N/A

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

       8. *-commutative N/A

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

       9. *-lft-identity N/A

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

       10. associate-*r* N/A

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

       11. distribute-rgt1-in N/A

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

       12. +-commutative N/A

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

       13. associate-*l* N/A

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

       14. distribute-rgt-in N/A

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

   11. Applied rewrites 17.7%

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

   12. Taylor expanded in b around inf

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 41.0

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

   14. Applied rewrites 41.0%

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

   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) < 5.0000000000000004e283

   1. Initial program 94.8%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   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 67.5

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

   5. Applied rewrites 67.5%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 65.1

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

   8. Applied rewrites 65.1%

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

   9. Taylor expanded in y around 0

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

       1. distribute-rgt-in N/A

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

       2. *-lft-identity N/A

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

       3. associate-*l* N/A

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

       4. +-commutative N/A

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

       5. distribute-rgt1-in N/A

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

       6. associate-*r* N/A

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

       7. lower-/.f64 N/A

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

       8. *-commutative N/A

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

       9. *-lft-identity N/A

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

       10. associate-*r* N/A

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

       11. distribute-rgt1-in N/A

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

       12. +-commutative N/A

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

       13. associate-*l* N/A

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

       14. distribute-rgt-in N/A

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

   11. Applied rewrites 56.7%

       \[\leadsto \color{blue}{\frac{x}{y \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(b, b \cdot 0.5, b\right), a\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot e^{\left(y \cdot \log z + \left(t + -1\right) \cdot \log a\right) - b}}{y} \leq -\infty:\\ \;\;\;\;\frac{x}{y \cdot \left(0.5 \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)}\\ \mathbf{elif}\;\frac{x \cdot e^{\left(y \cdot \log z + \left(t + -1\right) \cdot \log a\right) - b}}{y} \leq 5 \cdot 10^{+283}:\\ \;\;\;\;\frac{x}{y \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(b, b \cdot 0.5, b\right), a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(0.5 \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t + -1\right) \cdot \log a\\ t_2 := \frac{x \cdot e^{t \cdot \log a - b}}{y}\\ \mathbf{if}\;t\_1 \leq -50000000000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -442:\\ \;\;\;\;x \cdot \frac{\frac{{z}^{y}}{a}}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(y, 0.5, 0.16666666666666666 \cdot \left(y \cdot b\right)\right), y\right), y\right)}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+51}:\\ \;\;\;\;\frac{x \cdot e^{y \cdot \log z - b}}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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 -50000000000000.0)
     t_2
     (if (<= t_1 -442.0)
       (*
        x
        (/
         (/ (pow z y) a)
         (fma b (fma b (fma y 0.5 (* 0.16666666666666666 (* y b))) y) y)))
       (if (<= t_1 5e+51) (/ (* 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 <= -50000000000000.0) {
		tmp = t_2;
	} else if (t_1 <= -442.0) {
		tmp = x * ((pow(z, y) / a) / fma(b, fma(b, fma(y, 0.5, (0.16666666666666666 * (y * b))), y), y));
	} else if (t_1 <= 5e+51) {
		tmp = (x * exp(((y * 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 <= -50000000000000.0)
		tmp = t_2;
	elseif (t_1 <= -442.0)
		tmp = Float64(x * Float64(Float64((z ^ y) / a) / fma(b, fma(b, fma(y, 0.5, Float64(0.16666666666666666 * Float64(y * b))), y), y)));
	elseif (t_1 <= 5e+51)
		tmp = Float64(Float64(x * exp(Float64(Float64(y * log(z)) - b))) / y);
	else
		tmp = t_2;
	end
	return 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, -50000000000000.0], t$95$2, If[LessEqual[t$95$1, -442.0], N[(x * N[(N[(N[Power[z, y], $MachinePrecision] / a), $MachinePrecision] / N[(b * N[(b * N[(y * 0.5 + N[(0.16666666666666666 * N[(y * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+51], 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)) < -5e13 or 5e51 < (*.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 94.5

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

   5. Applied rewrites 94.5%

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

   if -5e13 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < -442

   1. Initial program 86.6%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   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 95.6

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

   5. Applied rewrites 95.6%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 85.0

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

   8. Applied rewrites 85.0%

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

   if -442 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < 5e51

   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 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 83.8

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

   5. Applied rewrites 83.8%

      \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(t + -1\right) \cdot \log a \leq -50000000000000:\\ \;\;\;\;\frac{x \cdot e^{t \cdot \log a - b}}{y}\\ \mathbf{elif}\;\left(t + -1\right) \cdot \log a \leq -442:\\ \;\;\;\;x \cdot \frac{\frac{{z}^{y}}{a}}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(y, 0.5, 0.16666666666666666 \cdot \left(y \cdot b\right)\right), y\right), y\right)}\\ \mathbf{elif}\;\left(t + -1\right) \cdot \log a \leq 5 \cdot 10^{+51}:\\ \;\;\;\;\frac{x \cdot e^{y \cdot \log z - b}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot e^{t \cdot \log a - b}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t + -1\right) \cdot \log a\\ \mathbf{if}\;t\_1 \leq -50000000000000:\\ \;\;\;\;\frac{x}{y \cdot {a}^{\left(1 - t\right)}}\\ \mathbf{elif}\;t\_1 \leq -315:\\ \;\;\;\;x \cdot \frac{\frac{{z}^{y}}{a}}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(y, 0.5, 0.16666666666666666 \cdot \left(y \cdot b\right)\right), y\right), y\right)}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+51}:\\ \;\;\;\;\frac{\frac{x \cdot {z}^{y}}{y}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot e^{\log a \cdot \left(-t\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (+ t -1.0) (log a))))
   (if (<= t_1 -50000000000000.0)
     (/ x (* y (pow a (- 1.0 t))))
     (if (<= t_1 -315.0)
       (*
        x
        (/
         (/ (pow z y) a)
         (fma b (fma b (fma y 0.5 (* 0.16666666666666666 (* y b))) y) y)))
       (if (<= t_1 5e+51)
         (/ (/ (* x (pow z y)) y) a)
         (/ x (* y (exp (* (log a) (- t))))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t + -1.0) * log(a);
	double tmp;
	if (t_1 <= -50000000000000.0) {
		tmp = x / (y * pow(a, (1.0 - t)));
	} else if (t_1 <= -315.0) {
		tmp = x * ((pow(z, y) / a) / fma(b, fma(b, fma(y, 0.5, (0.16666666666666666 * (y * b))), y), y));
	} else if (t_1 <= 5e+51) {
		tmp = ((x * pow(z, y)) / y) / a;
	} else {
		tmp = x / (y * exp((log(a) * -t)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t + -1.0) * log(a))
	tmp = 0.0
	if (t_1 <= -50000000000000.0)
		tmp = Float64(x / Float64(y * (a ^ Float64(1.0 - t))));
	elseif (t_1 <= -315.0)
		tmp = Float64(x * Float64(Float64((z ^ y) / a) / fma(b, fma(b, fma(y, 0.5, Float64(0.16666666666666666 * Float64(y * b))), y), y)));
	elseif (t_1 <= 5e+51)
		tmp = Float64(Float64(Float64(x * (z ^ y)) / y) / a);
	else
		tmp = Float64(x / Float64(y * exp(Float64(log(a) * Float64(-t)))));
	end
	return 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]}, If[LessEqual[t$95$1, -50000000000000.0], N[(x / N[(y * N[Power[a, N[(1.0 - t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -315.0], N[(x * N[(N[(N[Power[z, y], $MachinePrecision] / a), $MachinePrecision] / N[(b * N[(b * N[(y * 0.5 + N[(0.16666666666666666 * N[(y * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+51], N[(N[(N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] / a), $MachinePrecision], N[(x / N[(y * N[Exp[N[(N[Log[a], $MachinePrecision] * (-t)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

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

   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. Step-by-step derivation

      1. lift-log.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. remove-double-neg N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-+.f64 96.5

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

   7. Applied rewrites 96.5%

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

   8. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. exp-to-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. distribute-neg-in N/A

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

      9. mul-1-neg N/A

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. mul-1-neg N/A

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

      13. unsub-neg N/A

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

      14. lower--.f64 89.6

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

   10. Applied rewrites 89.6%

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

   if -5e13 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < -315

   1. Initial program 89.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   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 90.8

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

   5. Applied rewrites 90.8%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 79.5

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

   8. Applied rewrites 79.5%

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

   if -315 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < 5e51

   1. Initial program 98.3%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   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 77.3

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

   5. Applied rewrites 77.3%

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

   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 72.7

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

   8. Applied rewrites 72.7%

      \[\leadsto x \cdot \color{blue}{\frac{{z}^{y}}{y \cdot a}} \]
   9. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. div-inv N/A

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

      8. lift-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-/.f64 N/A

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

      13. div-inv N/A

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

      14. lower-/.f64 80.0

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

   10. Applied rewrites 80.0%

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

   if 5e51 < (*.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. Step-by-step derivation

      1. lift-log.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. remove-double-neg N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-log.f64 84.3

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

   7. Applied rewrites 84.3%

      \[\leadsto \frac{x}{y \cdot e^{\color{blue}{-\log a \cdot t}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(t + -1\right) \cdot \log a \leq -50000000000000:\\ \;\;\;\;\frac{x}{y \cdot {a}^{\left(1 - t\right)}}\\ \mathbf{elif}\;\left(t + -1\right) \cdot \log a \leq -315:\\ \;\;\;\;x \cdot \frac{\frac{{z}^{y}}{a}}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(y, 0.5, 0.16666666666666666 \cdot \left(y \cdot b\right)\right), y\right), y\right)}\\ \mathbf{elif}\;\left(t + -1\right) \cdot \log a \leq 5 \cdot 10^{+51}:\\ \;\;\;\;\frac{\frac{x \cdot {z}^{y}}{y}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot e^{\log a \cdot \left(-t\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t + -1\right) \cdot \log a\\ t_2 := \frac{x}{y \cdot {a}^{\left(1 - t\right)}}\\ \mathbf{if}\;t\_1 \leq -50000000000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -315:\\ \;\;\;\;x \cdot \frac{\frac{{z}^{y}}{a}}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(y, 0.5, 0.16666666666666666 \cdot \left(y \cdot b\right)\right), y\right), y\right)}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+51}:\\ \;\;\;\;\frac{\frac{x \cdot {z}^{y}}{y}}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (+ t -1.0) (log a))) (t_2 (/ x (* y (pow a (- 1.0 t))))))
   (if (<= t_1 -50000000000000.0)
     t_2
     (if (<= t_1 -315.0)
       (*
        x
        (/
         (/ (pow z y) a)
         (fma b (fma b (fma y 0.5 (* 0.16666666666666666 (* y b))) y) y)))
       (if (<= t_1 5e+51) (/ (/ (* 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 / (y * pow(a, (1.0 - t)));
	double tmp;
	if (t_1 <= -50000000000000.0) {
		tmp = t_2;
	} else if (t_1 <= -315.0) {
		tmp = x * ((pow(z, y) / a) / fma(b, fma(b, fma(y, 0.5, (0.16666666666666666 * (y * b))), y), y));
	} else if (t_1 <= 5e+51) {
		tmp = ((x * 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(y * (a ^ Float64(1.0 - t))))
	tmp = 0.0
	if (t_1 <= -50000000000000.0)
		tmp = t_2;
	elseif (t_1 <= -315.0)
		tmp = Float64(x * Float64(Float64((z ^ y) / a) / fma(b, fma(b, fma(y, 0.5, Float64(0.16666666666666666 * Float64(y * b))), y), y)));
	elseif (t_1 <= 5e+51)
		tmp = Float64(Float64(Float64(x * (z ^ y)) / y) / a);
	else
		tmp = t_2;
	end
	return 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[(y * N[Power[a, N[(1.0 - t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -50000000000000.0], t$95$2, If[LessEqual[t$95$1, -315.0], N[(x * N[(N[(N[Power[z, y], $MachinePrecision] / a), $MachinePrecision] / N[(b * N[(b * N[(y * 0.5 + N[(0.16666666666666666 * N[(y * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+51], N[(N[(N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] / a), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < -5e13 or 5e51 < (*.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. Step-by-step derivation

      1. lift-log.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. remove-double-neg N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-+.f64 94.5

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

   7. Applied rewrites 94.5%

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

   8. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. exp-to-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. distribute-neg-in N/A

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

      9. mul-1-neg N/A

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. mul-1-neg N/A

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

      13. unsub-neg N/A

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

      14. lower--.f64 87.1

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

   10. Applied rewrites 87.1%

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

   if -5e13 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < -315

   1. Initial program 89.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   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 90.8

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

   5. Applied rewrites 90.8%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 79.5

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

   8. Applied rewrites 79.5%

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

   if -315 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < 5e51

   1. Initial program 98.3%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   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 77.3

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

   5. Applied rewrites 77.3%

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

   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 72.7

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

   8. Applied rewrites 72.7%

      \[\leadsto x \cdot \color{blue}{\frac{{z}^{y}}{y \cdot a}} \]
   9. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. div-inv N/A

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

      8. lift-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-/.f64 N/A

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

      13. div-inv N/A

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

      14. lower-/.f64 80.0

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

   10. Applied rewrites 80.0%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(t + -1\right) \cdot \log a \leq -50000000000000:\\ \;\;\;\;\frac{x}{y \cdot {a}^{\left(1 - t\right)}}\\ \mathbf{elif}\;\left(t + -1\right) \cdot \log a \leq -315:\\ \;\;\;\;x \cdot \frac{\frac{{z}^{y}}{a}}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(y, 0.5, 0.16666666666666666 \cdot \left(y \cdot b\right)\right), y\right), y\right)}\\ \mathbf{elif}\;\left(t + -1\right) \cdot \log a \leq 5 \cdot 10^{+51}:\\ \;\;\;\;\frac{\frac{x \cdot {z}^{y}}{y}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot {a}^{\left(1 - t\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t + -1\right) \cdot \log a\\ t_2 := \frac{x}{y \cdot {a}^{\left(1 - t\right)}}\\ \mathbf{if}\;t\_1 \leq -50000000000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -315:\\ \;\;\;\;x \cdot \frac{\frac{{z}^{y}}{a}}{\mathsf{fma}\left(b, \mathsf{fma}\left(0.5, y \cdot b, y\right), y\right)}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+51}:\\ \;\;\;\;\frac{\frac{x \cdot {z}^{y}}{y}}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (+ t -1.0) (log a))) (t_2 (/ x (* y (pow a (- 1.0 t))))))
   (if (<= t_1 -50000000000000.0)
     t_2
     (if (<= t_1 -315.0)
       (* x (/ (/ (pow z y) a) (fma b (fma 0.5 (* y b) y) y)))
       (if (<= t_1 5e+51) (/ (/ (* 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 / (y * pow(a, (1.0 - t)));
	double tmp;
	if (t_1 <= -50000000000000.0) {
		tmp = t_2;
	} else if (t_1 <= -315.0) {
		tmp = x * ((pow(z, y) / a) / fma(b, fma(0.5, (y * b), y), y));
	} else if (t_1 <= 5e+51) {
		tmp = ((x * 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(y * (a ^ Float64(1.0 - t))))
	tmp = 0.0
	if (t_1 <= -50000000000000.0)
		tmp = t_2;
	elseif (t_1 <= -315.0)
		tmp = Float64(x * Float64(Float64((z ^ y) / a) / fma(b, fma(0.5, Float64(y * b), y), y)));
	elseif (t_1 <= 5e+51)
		tmp = Float64(Float64(Float64(x * (z ^ y)) / y) / a);
	else
		tmp = t_2;
	end
	return 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[(y * N[Power[a, N[(1.0 - t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -50000000000000.0], t$95$2, If[LessEqual[t$95$1, -315.0], N[(x * N[(N[(N[Power[z, y], $MachinePrecision] / a), $MachinePrecision] / N[(b * N[(0.5 * N[(y * b), $MachinePrecision] + y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+51], N[(N[(N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] / a), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < -5e13 or 5e51 < (*.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. Step-by-step derivation

      1. lift-log.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. remove-double-neg N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-+.f64 94.5

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

   7. Applied rewrites 94.5%

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

   8. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. exp-to-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. distribute-neg-in N/A

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

      9. mul-1-neg N/A

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. mul-1-neg N/A

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

      13. unsub-neg N/A

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

      14. lower--.f64 87.1

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

   10. Applied rewrites 87.1%

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

   if -5e13 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < -315

   1. Initial program 89.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   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 90.8

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

   5. Applied rewrites 90.8%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 77.4

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

   8. Applied rewrites 77.4%

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

   if -315 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < 5e51

   1. Initial program 98.3%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   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 77.3

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

   5. Applied rewrites 77.3%

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

   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 72.7

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

   8. Applied rewrites 72.7%

      \[\leadsto x \cdot \color{blue}{\frac{{z}^{y}}{y \cdot a}} \]
   9. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. div-inv N/A

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

      8. lift-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-/.f64 N/A

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

      13. div-inv N/A

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

      14. lower-/.f64 80.0

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

   10. Applied rewrites 80.0%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(t + -1\right) \cdot \log a \leq -50000000000000:\\ \;\;\;\;\frac{x}{y \cdot {a}^{\left(1 - t\right)}}\\ \mathbf{elif}\;\left(t + -1\right) \cdot \log a \leq -315:\\ \;\;\;\;x \cdot \frac{\frac{{z}^{y}}{a}}{\mathsf{fma}\left(b, \mathsf{fma}\left(0.5, y \cdot b, y\right), y\right)}\\ \mathbf{elif}\;\left(t + -1\right) \cdot \log a \leq 5 \cdot 10^{+51}:\\ \;\;\;\;\frac{\frac{x \cdot {z}^{y}}{y}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot {a}^{\left(1 - t\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 75.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t + -1\right) \cdot \log a\\ t_2 := \frac{x}{y \cdot {a}^{\left(1 - t\right)}}\\ \mathbf{if}\;t\_1 \leq -2000000000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+51}:\\ \;\;\;\;\frac{\frac{x \cdot {z}^{y}}{y}}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (+ t -1.0) (log a))) (t_2 (/ x (* y (pow a (- 1.0 t))))))
   (if (<= t_1 -2000000000000.0)
     t_2
     (if (<= t_1 5e+51) (/ (/ (* 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 / (y * pow(a, (1.0 - t)));
	double tmp;
	if (t_1 <= -2000000000000.0) {
		tmp = t_2;
	} else if (t_1 <= 5e+51) {
		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 / (y * (a ** (1.0d0 - t)))
    if (t_1 <= (-2000000000000.0d0)) then
        tmp = t_2
    else if (t_1 <= 5d+51) 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 / (y * Math.pow(a, (1.0 - t)));
	double tmp;
	if (t_1 <= -2000000000000.0) {
		tmp = t_2;
	} else if (t_1 <= 5e+51) {
		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 / (y * math.pow(a, (1.0 - t)))
	tmp = 0
	if t_1 <= -2000000000000.0:
		tmp = t_2
	elif t_1 <= 5e+51:
		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(y * (a ^ Float64(1.0 - t))))
	tmp = 0.0
	if (t_1 <= -2000000000000.0)
		tmp = t_2;
	elseif (t_1 <= 5e+51)
		tmp = Float64(Float64(Float64(x * (z ^ y)) / 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 / (y * (a ^ (1.0 - t)));
	tmp = 0.0;
	if (t_1 <= -2000000000000.0)
		tmp = t_2;
	elseif (t_1 <= 5e+51)
		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[(y * N[Power[a, N[(1.0 - t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2000000000000.0], t$95$2, If[LessEqual[t$95$1, 5e+51], N[(N[(N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] / a), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < -2e12 or 5e51 < (*.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. Step-by-step derivation

      1. lift-log.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. remove-double-neg N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-+.f64 94.5

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

   7. Applied rewrites 94.5%

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

   8. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. exp-to-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. distribute-neg-in N/A

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

      9. mul-1-neg N/A

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. mul-1-neg N/A

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

      13. unsub-neg N/A

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

      14. lower--.f64 87.2

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

   10. Applied rewrites 87.2%

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

   if -2e12 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < 5e51

   1. Initial program 95.3%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   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 81.6

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

   5. Applied rewrites 81.6%

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

   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 67.3

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

   8. Applied rewrites 67.3%

      \[\leadsto x \cdot \color{blue}{\frac{{z}^{y}}{y \cdot a}} \]
   9. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. div-inv N/A

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

      8. lift-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-/.f64 N/A

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

      13. div-inv N/A

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

      14. lower-/.f64 75.7

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

   10. Applied rewrites 75.7%

       \[\leadsto \color{blue}{\frac{\frac{x \cdot {z}^{y}}{y}}{a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(t + -1\right) \cdot \log a \leq -2000000000000:\\ \;\;\;\;\frac{x}{y \cdot {a}^{\left(1 - t\right)}}\\ \mathbf{elif}\;\left(t + -1\right) \cdot \log a \leq 5 \cdot 10^{+51}:\\ \;\;\;\;\frac{\frac{x \cdot {z}^{y}}{y}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot {a}^{\left(1 - t\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 75.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t + -1\right) \cdot \log a\\ t_2 := \frac{x}{y \cdot {a}^{\left(1 - t\right)}}\\ \mathbf{if}\;t\_1 \leq -2000000000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+51}:\\ \;\;\;\;\frac{\frac{x \cdot {z}^{y}}{a}}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (+ t -1.0) (log a))) (t_2 (/ x (* y (pow a (- 1.0 t))))))
   (if (<= t_1 -2000000000000.0)
     t_2
     (if (<= t_1 5e+51) (/ (/ (* x (pow z y)) a) 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 / (y * pow(a, (1.0 - t)));
	double tmp;
	if (t_1 <= -2000000000000.0) {
		tmp = t_2;
	} else if (t_1 <= 5e+51) {
		tmp = ((x * pow(z, y)) / a) / 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 / (y * (a ** (1.0d0 - t)))
    if (t_1 <= (-2000000000000.0d0)) then
        tmp = t_2
    else if (t_1 <= 5d+51) then
        tmp = ((x * (z ** y)) / a) / 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 / (y * Math.pow(a, (1.0 - t)));
	double tmp;
	if (t_1 <= -2000000000000.0) {
		tmp = t_2;
	} else if (t_1 <= 5e+51) {
		tmp = ((x * Math.pow(z, y)) / a) / 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 / (y * math.pow(a, (1.0 - t)))
	tmp = 0
	if t_1 <= -2000000000000.0:
		tmp = t_2
	elif t_1 <= 5e+51:
		tmp = ((x * math.pow(z, y)) / a) / 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(y * (a ^ Float64(1.0 - t))))
	tmp = 0.0
	if (t_1 <= -2000000000000.0)
		tmp = t_2;
	elseif (t_1 <= 5e+51)
		tmp = Float64(Float64(Float64(x * (z ^ y)) / a) / 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 / (y * (a ^ (1.0 - t)));
	tmp = 0.0;
	if (t_1 <= -2000000000000.0)
		tmp = t_2;
	elseif (t_1 <= 5e+51)
		tmp = ((x * (z ^ y)) / a) / 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[(y * N[Power[a, N[(1.0 - t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2000000000000.0], t$95$2, If[LessEqual[t$95$1, 5e+51], N[(N[(N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] / y), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < -2e12 or 5e51 < (*.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. Step-by-step derivation

      1. lift-log.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. remove-double-neg N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-+.f64 94.5

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

   7. Applied rewrites 94.5%

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

   8. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. exp-to-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. distribute-neg-in N/A

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

      9. mul-1-neg N/A

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. mul-1-neg N/A

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

      13. unsub-neg N/A

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

      14. lower--.f64 87.2

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

   10. Applied rewrites 87.2%

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

   if -2e12 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < 5e51

   1. Initial program 95.3%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   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 81.6

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

   5. Applied rewrites 81.6%

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

   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 67.3

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

   8. Applied rewrites 67.3%

      \[\leadsto x \cdot \color{blue}{\frac{{z}^{y}}{y \cdot a}} \]
   9. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lift-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. associate-*r/ N/A

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

      14. lift-*.f64 N/A

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

      15. lower-/.f64 75.1

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

   10. Applied rewrites 75.1%

       \[\leadsto \color{blue}{\frac{\frac{x \cdot {z}^{y}}{a}}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(t + -1\right) \cdot \log a \leq -2000000000000:\\ \;\;\;\;\frac{x}{y \cdot {a}^{\left(1 - t\right)}}\\ \mathbf{elif}\;\left(t + -1\right) \cdot \log a \leq 5 \cdot 10^{+51}:\\ \;\;\;\;\frac{\frac{x \cdot {z}^{y}}{a}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot {a}^{\left(1 - t\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 92.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot e^{y \cdot \log z - b}}{y}\\ \mathbf{if}\;y \leq -3.5 \cdot 10^{+47}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 0.106:\\ \;\;\;\;\frac{x}{y \cdot e^{b - \left(t + -1\right) \cdot \log a}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (* x (exp (- (* y (log z)) b))) y)))
   (if (<= y -3.5e+47)
     t_1
     (if (<= y 0.106) (/ x (* y (exp (- b (* (+ t -1.0) (log a)))))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x * exp(((y * log(z)) - b))) / y;
	double tmp;
	if (y <= -3.5e+47) {
		tmp = t_1;
	} else if (y <= 0.106) {
		tmp = x / (y * exp((b - ((t + -1.0) * log(a)))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x * exp(((y * log(z)) - b))) / y
    if (y <= (-3.5d+47)) then
        tmp = t_1
    else if (y <= 0.106d0) then
        tmp = x / (y * exp((b - ((t + (-1.0d0)) * log(a)))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x * Math.exp(((y * Math.log(z)) - b))) / y;
	double tmp;
	if (y <= -3.5e+47) {
		tmp = t_1;
	} else if (y <= 0.106) {
		tmp = x / (y * Math.exp((b - ((t + -1.0) * Math.log(a)))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (x * math.exp(((y * math.log(z)) - b))) / y
	tmp = 0
	if y <= -3.5e+47:
		tmp = t_1
	elif y <= 0.106:
		tmp = x / (y * math.exp((b - ((t + -1.0) * math.log(a)))))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x * exp(Float64(Float64(y * log(z)) - b))) / y)
	tmp = 0.0
	if (y <= -3.5e+47)
		tmp = t_1;
	elseif (y <= 0.106)
		tmp = Float64(x / Float64(y * exp(Float64(b - Float64(Float64(t + -1.0) * log(a))))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (x * exp(((y * log(z)) - b))) / y;
	tmp = 0.0;
	if (y <= -3.5e+47)
		tmp = t_1;
	elseif (y <= 0.106)
		tmp = x / (y * exp((b - ((t + -1.0) * log(a)))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x * N[Exp[N[(N[(y * N[Log[z], $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[y, -3.5e+47], t$95$1, If[LessEqual[y, 0.106], N[(x / N[(y * N[Exp[N[(b - N[(N[(t + -1.0), $MachinePrecision] * N[Log[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.50000000000000015e47 or 0.105999999999999997 < 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 91.3

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

   5. Applied rewrites 91.3%

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

   if -3.50000000000000015e47 < y < 0.105999999999999997

   1. Initial program 94.1%

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

      1. lift-log.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. remove-double-neg N/A

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

   4. Applied rewrites 95.3%

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

   5. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-+.f64 95.3

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

   7. Applied rewrites 95.3%

      \[\leadsto \frac{x}{y \cdot e^{b - \color{blue}{\log a \cdot \left(t + -1\right)}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{+47}:\\ \;\;\;\;\frac{x \cdot e^{y \cdot \log z - b}}{y}\\ \mathbf{elif}\;y \leq 0.106:\\ \;\;\;\;\frac{x}{y \cdot e^{b - \left(t + -1\right) \cdot \log a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot e^{y \cdot \log z - b}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 81.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot e^{y \cdot \log z - b}}{y}\\ \mathbf{if}\;y \leq -50:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 0.036:\\ \;\;\;\;\frac{x}{y \cdot {a}^{\left(1 - t\right)}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (* x (exp (- (* y (log z)) b))) y)))
   (if (<= y -50.0) t_1 (if (<= y 0.036) (/ x (* y (pow a (- 1.0 t)))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x * exp(((y * log(z)) - b))) / y;
	double tmp;
	if (y <= -50.0) {
		tmp = t_1;
	} else if (y <= 0.036) {
		tmp = x / (y * pow(a, (1.0 - t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x * exp(((y * log(z)) - b))) / y
    if (y <= (-50.0d0)) then
        tmp = t_1
    else if (y <= 0.036d0) then
        tmp = x / (y * (a ** (1.0d0 - t)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x * Math.exp(((y * Math.log(z)) - b))) / y;
	double tmp;
	if (y <= -50.0) {
		tmp = t_1;
	} else if (y <= 0.036) {
		tmp = x / (y * Math.pow(a, (1.0 - t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (x * math.exp(((y * math.log(z)) - b))) / y
	tmp = 0
	if y <= -50.0:
		tmp = t_1
	elif y <= 0.036:
		tmp = x / (y * math.pow(a, (1.0 - t)))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x * exp(Float64(Float64(y * log(z)) - b))) / y)
	tmp = 0.0
	if (y <= -50.0)
		tmp = t_1;
	elseif (y <= 0.036)
		tmp = Float64(x / Float64(y * (a ^ Float64(1.0 - t))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (x * exp(((y * log(z)) - b))) / y;
	tmp = 0.0;
	if (y <= -50.0)
		tmp = t_1;
	elseif (y <= 0.036)
		tmp = x / (y * (a ^ (1.0 - t)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x * N[Exp[N[(N[(y * N[Log[z], $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[y, -50.0], t$95$1, If[LessEqual[y, 0.036], N[(x / N[(y * N[Power[a, N[(1.0 - t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -50 or 0.0359999999999999973 < 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 91.6

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

   5. Applied rewrites 91.6%

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

   if -50 < y < 0.0359999999999999973

   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. Step-by-step derivation

      1. lift-log.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. remove-double-neg N/A

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

   4. Applied rewrites 95.1%

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

   5. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-+.f64 95.1

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

   7. Applied rewrites 95.1%

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

   8. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. exp-to-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. distribute-neg-in N/A

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

      9. mul-1-neg N/A

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. mul-1-neg N/A

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

      13. unsub-neg N/A

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

      14. lower--.f64 76.5

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

   10. Applied rewrites 76.5%

       \[\leadsto \color{blue}{\frac{x}{y \cdot {a}^{\left(1 - t\right)}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 73.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \frac{{z}^{y}}{y}\\ \mathbf{if}\;y \leq -1.95 \cdot 10^{+143}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 0.65:\\ \;\;\;\;\frac{x}{y \cdot {a}^{\left(1 - t\right)}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (/ (pow z y) y))))
   (if (<= y -1.95e+143)
     t_1
     (if (<= y 0.65) (/ x (* y (pow a (- 1.0 t)))) 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 <= -1.95e+143) {
		tmp = t_1;
	} else if (y <= 0.65) {
		tmp = x / (y * pow(a, (1.0 - t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * ((z ** y) / y)
    if (y <= (-1.95d+143)) then
        tmp = t_1
    else if (y <= 0.65d0) then
        tmp = x / (y * (a ** (1.0d0 - t)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * (Math.pow(z, y) / y);
	double tmp;
	if (y <= -1.95e+143) {
		tmp = t_1;
	} else if (y <= 0.65) {
		tmp = x / (y * Math.pow(a, (1.0 - t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x * (math.pow(z, y) / y)
	tmp = 0
	if y <= -1.95e+143:
		tmp = t_1
	elif y <= 0.65:
		tmp = x / (y * math.pow(a, (1.0 - t)))
	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 <= -1.95e+143)
		tmp = t_1;
	elseif (y <= 0.65)
		tmp = Float64(x / Float64(y * (a ^ Float64(1.0 - t))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x * ((z ^ y) / y);
	tmp = 0.0;
	if (y <= -1.95e+143)
		tmp = t_1;
	elseif (y <= 0.65)
		tmp = x / (y * (a ^ (1.0 - t)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x * N[(N[Power[z, y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.95e+143], t$95$1, If[LessEqual[y, 0.65], N[(x / N[(y * N[Power[a, N[(1.0 - t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.9499999999999999e143 or 0.650000000000000022 < 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 93.8

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

   5. Applied rewrites 93.8%

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

   6. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{x \cdot {z}^{y}}{y}} \]
   7. Step-by-step derivation

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-pow.f64 85.8

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

   8. Applied rewrites 85.8%

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

   if -1.9499999999999999e143 < y < 0.650000000000000022

   1. Initial program 95.2%

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

      1. lift-log.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. remove-double-neg N/A

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

   4. Applied rewrites 96.1%

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

   5. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-+.f64 92.0

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

   7. Applied rewrites 92.0%

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

   8. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. exp-to-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. distribute-neg-in N/A

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

      9. mul-1-neg N/A

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. mul-1-neg N/A

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

      13. unsub-neg N/A

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

      14. lower--.f64 73.8

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

   10. Applied rewrites 73.8%

       \[\leadsto \color{blue}{\frac{x}{y \cdot {a}^{\left(1 - t\right)}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 69.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \frac{{z}^{y}}{y}\\ \mathbf{if}\;y \leq -1.52 \cdot 10^{+143}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 0.65:\\ \;\;\;\;{a}^{\left(t + -1\right)} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (/ (pow z y) y))))
   (if (<= y -1.52e+143)
     t_1
     (if (<= y 0.65) (* (pow a (+ t -1.0)) (/ x 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 <= -1.52e+143) {
		tmp = t_1;
	} else if (y <= 0.65) {
		tmp = pow(a, (t + -1.0)) * (x / y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * ((z ** y) / y)
    if (y <= (-1.52d+143)) then
        tmp = t_1
    else if (y <= 0.65d0) then
        tmp = (a ** (t + (-1.0d0))) * (x / y)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * (Math.pow(z, y) / y);
	double tmp;
	if (y <= -1.52e+143) {
		tmp = t_1;
	} else if (y <= 0.65) {
		tmp = Math.pow(a, (t + -1.0)) * (x / y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x * (math.pow(z, y) / y)
	tmp = 0
	if y <= -1.52e+143:
		tmp = t_1
	elif y <= 0.65:
		tmp = math.pow(a, (t + -1.0)) * (x / 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 <= -1.52e+143)
		tmp = t_1;
	elseif (y <= 0.65)
		tmp = Float64((a ^ Float64(t + -1.0)) * Float64(x / y));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x * ((z ^ y) / y);
	tmp = 0.0;
	if (y <= -1.52e+143)
		tmp = t_1;
	elseif (y <= 0.65)
		tmp = (a ^ (t + -1.0)) * (x / y);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x * N[(N[Power[z, y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.52e+143], t$95$1, If[LessEqual[y, 0.65], N[(N[Power[a, N[(t + -1.0), $MachinePrecision]], $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.51999999999999996e143 or 0.650000000000000022 < 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 93.8

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

   5. Applied rewrites 93.8%

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

   6. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{x \cdot {z}^{y}}{y}} \]
   7. Step-by-step derivation

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-pow.f64 85.8

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

   8. Applied rewrites 85.8%

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

   if -1.51999999999999996e143 < y < 0.650000000000000022

   1. Initial program 95.2%

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

   4. Applied rewrites 75.5%

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

   5. Taylor expanded in b around 0

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

   7. Applied rewrites 70.7%

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

   8. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. exp-to-pow N/A

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

      4. lower-pow.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 72.2

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

   10. Applied rewrites 72.2%

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

       1. lift-+.f64 N/A

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

       2. lift-pow.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. remove-double-neg N/A

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

       5. remove-double-neg N/A

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

       6. lift-*.f64 N/A

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

       7. *-commutative N/A

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

       8. associate-/l* N/A

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

       9. lift-/.f64 N/A

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

       10. lower-*.f64 68.9

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

       11. lift-+.f64 N/A

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

       12. +-commutative N/A

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

       13. lower-+.f64 68.9

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

   12. Applied rewrites 68.9%

       \[\leadsto \color{blue}{{a}^{\left(t + -1\right)} \cdot \frac{x}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 61.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \frac{{z}^{y}}{y}\\ \mathbf{if}\;y \leq -1.66 \cdot 10^{+44}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 0.65:\\ \;\;\;\;\frac{x}{\frac{1}{\frac{1}{y \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(b, b \cdot 0.5, b\right), a\right)}}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (/ (pow z y) y))))
   (if (<= y -1.66e+44)
     t_1
     (if (<= y 0.65)
       (/ x (/ 1.0 (/ 1.0 (* y (fma a (fma b (* b 0.5) b) a)))))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * (pow(z, y) / y);
	double tmp;
	if (y <= -1.66e+44) {
		tmp = t_1;
	} else if (y <= 0.65) {
		tmp = x / (1.0 / (1.0 / (y * fma(a, fma(b, (b * 0.5), b), a))));
	} 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 <= -1.66e+44)
		tmp = t_1;
	elseif (y <= 0.65)
		tmp = Float64(x / Float64(1.0 / Float64(1.0 / Float64(y * fma(a, fma(b, Float64(b * 0.5), b), a)))));
	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, -1.66e+44], t$95$1, If[LessEqual[y, 0.65], N[(x / N[(1.0 / N[(1.0 / N[(y * N[(a * N[(b * N[(b * 0.5), $MachinePrecision] + b), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.65999999999999992e44 or 0.650000000000000022 < 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 91.3

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

   5. Applied rewrites 91.3%

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

   6. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{x \cdot {z}^{y}}{y}} \]
   7. Step-by-step derivation

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-pow.f64 81.8

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

   8. Applied rewrites 81.8%

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

   if -1.65999999999999992e44 < y < 0.650000000000000022

   1. Initial program 94.2%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   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 68.4

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

   5. Applied rewrites 68.4%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 46.2

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

   8. Applied rewrites 46.2%

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

   9. Taylor expanded in y around 0

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

       1. distribute-rgt-in N/A

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

       2. *-lft-identity N/A

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

       3. associate-*l* N/A

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

       4. +-commutative N/A

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

       5. distribute-rgt1-in N/A

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

       6. associate-*r* N/A

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

       7. lower-/.f64 N/A

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

       8. *-commutative N/A

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

       9. *-lft-identity N/A

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

       10. associate-*r* N/A

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

       11. distribute-rgt1-in N/A

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

       12. +-commutative N/A

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

       13. associate-*l* N/A

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

       14. distribute-rgt-in N/A

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

   11. Applied rewrites 50.9%

       \[\leadsto \color{blue}{\frac{x}{y \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(b, b \cdot 0.5, b\right), a\right)}} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-fma.f64 N/A

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

       3. lift-fma.f64 N/A

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

       4. lift-*.f64 50.9

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

       5. /-rgt-identity N/A

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

       6. clear-num N/A

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

       7. lower-/.f64 N/A

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

       8. lower-/.f64 51.7

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

   13. Applied rewrites 51.7%

       \[\leadsto \frac{x}{\color{blue}{\frac{1}{\frac{1}{y \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(b, b \cdot 0.5, b\right), a\right)}}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 35.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log a \leq -42:\\ \;\;\;\;\frac{\frac{x}{a}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \mathsf{fma}\left(a, b, a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (log a) -42.0) (/ (/ x a) y) (/ x (* y (fma a b a)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (log(a) <= -42.0) {
		tmp = (x / a) / y;
	} else {
		tmp = x / (y * fma(a, b, a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (log(a) <= -42.0)
		tmp = Float64(Float64(x / a) / y);
	else
		tmp = Float64(x / Float64(y * fma(a, b, a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[Log[a], $MachinePrecision], -42.0], N[(N[(x / a), $MachinePrecision] / y), $MachinePrecision], N[(x / N[(y * N[(a * b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (log.f64 a) < -42

   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

   4. Applied rewrites 65.4%

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

   5. Taylor expanded in b around 0

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

   7. Applied rewrites 70.7%

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

   8. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. exp-to-pow N/A

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

      4. lower-pow.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 63.9

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

   10. Applied rewrites 63.9%

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

   11. Taylor expanded in t around 0

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

       1. lower-/.f64 30.4

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

   13. Applied rewrites 30.4%

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

   if -42 < (log.f64 a)

   1. Initial program 95.1%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   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 71.0

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

   5. Applied rewrites 71.0%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 65.5

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

   8. Applied rewrites 65.5%

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

   9. Taylor expanded in y around 0

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

       1. distribute-rgt-in N/A

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

       2. *-lft-identity N/A

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

       3. associate-*l* N/A

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

       4. +-commutative N/A

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

       5. distribute-rgt1-in N/A

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

       6. associate-*r* N/A

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

       7. lower-/.f64 N/A

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

       8. *-commutative N/A

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

       9. *-lft-identity N/A

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

       10. associate-*r* N/A

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

       11. distribute-rgt1-in N/A

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

       12. +-commutative N/A

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

       13. associate-*l* N/A

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

       14. distribute-rgt-in N/A

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

   11. Applied rewrites 44.5%

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

   12. Taylor expanded in b around 0

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

       1. associate-*r* N/A

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

       2. distribute-rgt-out N/A

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

       3. lower-*.f64 N/A

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

       4. +-commutative N/A

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

       5. lower-fma.f64 41.5

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

   14. Applied rewrites 41.5%

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

Alternative 17: 34.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log a \leq -42:\\ \;\;\;\;x \cdot \frac{1}{y \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \mathsf{fma}\left(a, b, a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (log a) -42.0) (* x (/ 1.0 (* y a))) (/ x (* y (fma a b a)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (log(a) <= -42.0) {
		tmp = x * (1.0 / (y * a));
	} else {
		tmp = x / (y * fma(a, b, a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (log(a) <= -42.0)
		tmp = Float64(x * Float64(1.0 / Float64(y * a)));
	else
		tmp = Float64(x / Float64(y * fma(a, b, a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[Log[a], $MachinePrecision], -42.0], N[(x * N[(1.0 / N[(y * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * N[(a * b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (log.f64 a) < -42

   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 64.6

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

   5. Applied rewrites 64.6%

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

   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 60.9

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

   8. Applied rewrites 60.9%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 26.9%

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

   if -42 < (log.f64 a)

   1. Initial program 95.1%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   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 71.0

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

   5. Applied rewrites 71.0%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 65.5

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

   8. Applied rewrites 65.5%

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

   9. Taylor expanded in y around 0

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

       1. distribute-rgt-in N/A

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

       2. *-lft-identity N/A

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

       3. associate-*l* N/A

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

       4. +-commutative N/A

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

       5. distribute-rgt1-in N/A

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

       6. associate-*r* N/A

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

       7. lower-/.f64 N/A

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

       8. *-commutative N/A

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

       9. *-lft-identity N/A

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

       10. associate-*r* N/A

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

       11. distribute-rgt1-in N/A

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

       12. +-commutative N/A

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

       13. associate-*l* N/A

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

       14. distribute-rgt-in N/A

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

   11. Applied rewrites 44.5%

       \[\leadsto \color{blue}{\frac{x}{y \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(b, b \cdot 0.5, b\right), a\right)}} \]

   12. Taylor expanded in b around 0

       \[\leadsto \frac{x}{\color{blue}{a \cdot y + a \cdot \left(b \cdot y\right)}} \]
   13. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \frac{x}{a \cdot y + \color{blue}{\left(a \cdot b\right) \cdot y}} \]

       2. distribute-rgt-out N/A

          \[\leadsto \frac{x}{\color{blue}{y \cdot \left(a + a \cdot b\right)}} \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{y \cdot \left(a + a \cdot b\right)}} \]

       4. +-commutative N/A

          \[\leadsto \frac{x}{y \cdot \color{blue}{\left(a \cdot b + a\right)}} \]

       5. lower-fma.f64 41.5

          \[\leadsto \frac{x}{y \cdot \color{blue}{\mathsf{fma}\left(a, b, a\right)}} \]

   14. Applied rewrites 41.5%

       \[\leadsto \frac{x}{\color{blue}{y \cdot \mathsf{fma}\left(a, b, a\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 44.6% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.65 \cdot 10^{-173}:\\ \;\;\;\;\left(1 - b\right) \cdot \frac{x}{y \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(0.5 \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -1.65e-173)
   (* (- 1.0 b) (/ x (* y a)))
   (/ x (* y (* 0.5 (* a (* b b)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -1.65e-173) {
		tmp = (1.0 - b) * (x / (y * a));
	} else {
		tmp = x / (y * (0.5 * (a * (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 <= (-1.65d-173)) then
        tmp = (1.0d0 - b) * (x / (y * a))
    else
        tmp = x / (y * (0.5d0 * (a * (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 <= -1.65e-173) {
		tmp = (1.0 - b) * (x / (y * a));
	} else {
		tmp = x / (y * (0.5 * (a * (b * b))));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if b <= -1.65e-173:
		tmp = (1.0 - b) * (x / (y * a))
	else:
		tmp = x / (y * (0.5 * (a * (b * b))))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -1.65e-173)
		tmp = Float64(Float64(1.0 - b) * Float64(x / Float64(y * a)));
	else
		tmp = Float64(x / Float64(y * Float64(0.5 * Float64(a * Float64(b * b)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (b <= -1.65e-173)
		tmp = (1.0 - b) * (x / (y * a));
	else
		tmp = x / (y * (0.5 * (a * (b * b))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -1.65e-173], N[(N[(1.0 - b), $MachinePrecision] * N[(x / N[(y * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * N[(0.5 * N[(a * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.6500000000000001e-173

   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 67.9

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot \color{blue}{e^{b}}} \]

   5. Applied rewrites 67.9%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot e^{b}}} \]

   6. Taylor expanded in b around 0

      \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{\color{blue}{y + b \cdot \left(y + \frac{1}{2} \cdot \left(b \cdot y\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{\color{blue}{b \cdot \left(y + \frac{1}{2} \cdot \left(b \cdot y\right)\right) + y}} \]

      2. lower-fma.f64 N/A

         \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{\color{blue}{\mathsf{fma}\left(b, y + \frac{1}{2} \cdot \left(b \cdot y\right), y\right)}} \]

      3. +-commutative N/A

         \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{\mathsf{fma}\left(b, \color{blue}{\frac{1}{2} \cdot \left(b \cdot y\right) + y}, y\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{\mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(\frac{1}{2}, b \cdot y, y\right)}, y\right)} \]

      5. *-commutative N/A

         \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{\mathsf{fma}\left(b, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{y \cdot b}, y\right), y\right)} \]

      6. lower-*.f64 52.8

         \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{\mathsf{fma}\left(b, \mathsf{fma}\left(0.5, \color{blue}{y \cdot b}, y\right), y\right)} \]

   8. Applied rewrites 52.8%

      \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{\color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(0.5, y \cdot b, y\right), y\right)}} \]

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot \left(1 + b \cdot \left(1 + \frac{1}{2} \cdot b\right)\right)\right)}} \]
   10. Step-by-step derivation

       1. distribute-rgt-in N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\left(1 \cdot y + \left(b \cdot \left(1 + \frac{1}{2} \cdot b\right)\right) \cdot y\right)}} \]

       2. *-lft-identity N/A

          \[\leadsto \frac{x}{a \cdot \left(\color{blue}{y} + \left(b \cdot \left(1 + \frac{1}{2} \cdot b\right)\right) \cdot y\right)} \]

       3. associate-*l* N/A

          \[\leadsto \frac{x}{a \cdot \left(y + \color{blue}{b \cdot \left(\left(1 + \frac{1}{2} \cdot b\right) \cdot y\right)}\right)} \]

       4. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(y + b \cdot \left(\color{blue}{\left(\frac{1}{2} \cdot b + 1\right)} \cdot y\right)\right)} \]

       5. distribute-rgt1-in N/A

          \[\leadsto \frac{x}{a \cdot \left(y + b \cdot \color{blue}{\left(y + \left(\frac{1}{2} \cdot b\right) \cdot y\right)}\right)} \]

       6. associate-*r* N/A

          \[\leadsto \frac{x}{a \cdot \left(y + b \cdot \left(y + \color{blue}{\frac{1}{2} \cdot \left(b \cdot y\right)}\right)\right)} \]

       7. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y + b \cdot \left(y + \frac{1}{2} \cdot \left(b \cdot y\right)\right)\right)}} \]

       8. *-commutative N/A

          \[\leadsto \frac{x}{\color{blue}{\left(y + b \cdot \left(y + \frac{1}{2} \cdot \left(b \cdot y\right)\right)\right) \cdot a}} \]

       9. *-lft-identity N/A

          \[\leadsto \frac{x}{\left(\color{blue}{1 \cdot y} + b \cdot \left(y + \frac{1}{2} \cdot \left(b \cdot y\right)\right)\right) \cdot a} \]

       10. associate-*r* N/A

           \[\leadsto \frac{x}{\left(1 \cdot y + b \cdot \left(y + \color{blue}{\left(\frac{1}{2} \cdot b\right) \cdot y}\right)\right) \cdot a} \]

       11. distribute-rgt1-in N/A

           \[\leadsto \frac{x}{\left(1 \cdot y + b \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot b + 1\right) \cdot y\right)}\right) \cdot a} \]

       12. +-commutative N/A

           \[\leadsto \frac{x}{\left(1 \cdot y + b \cdot \left(\color{blue}{\left(1 + \frac{1}{2} \cdot b\right)} \cdot y\right)\right) \cdot a} \]

       13. associate-*l* N/A

           \[\leadsto \frac{x}{\left(1 \cdot y + \color{blue}{\left(b \cdot \left(1 + \frac{1}{2} \cdot b\right)\right) \cdot y}\right) \cdot a} \]

       14. distribute-rgt-in N/A

           \[\leadsto \frac{x}{\color{blue}{\left(y \cdot \left(1 + b \cdot \left(1 + \frac{1}{2} \cdot b\right)\right)\right)} \cdot a} \]

   11. Applied rewrites 35.0%

       \[\leadsto \color{blue}{\frac{x}{y \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(b, b \cdot 0.5, b\right), a\right)}} \]

   12. Taylor expanded in b around 0

       \[\leadsto \color{blue}{-1 \cdot \frac{b \cdot x}{a \cdot y} + \frac{x}{a \cdot y}} \]
   13. 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 43.0

          \[\leadsto \left(\left(-b\right) + 1\right) \cdot \frac{x}{\color{blue}{a \cdot y}} \]

   14. Applied rewrites 43.0%

       \[\leadsto \color{blue}{\left(\left(-b\right) + 1\right) \cdot \frac{x}{a \cdot y}} \]

   if -1.6500000000000001e-173 < b

   1. Initial program 97.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   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 67.9

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot \color{blue}{e^{b}}} \]

   5. Applied rewrites 67.9%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot e^{b}}} \]

   6. Taylor expanded in b around 0

      \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{\color{blue}{y + b \cdot \left(y + \frac{1}{2} \cdot \left(b \cdot y\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{\color{blue}{b \cdot \left(y + \frac{1}{2} \cdot \left(b \cdot y\right)\right) + y}} \]

      2. lower-fma.f64 N/A

         \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{\color{blue}{\mathsf{fma}\left(b, y + \frac{1}{2} \cdot \left(b \cdot y\right), y\right)}} \]

      3. +-commutative N/A

         \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{\mathsf{fma}\left(b, \color{blue}{\frac{1}{2} \cdot \left(b \cdot y\right) + y}, y\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{\mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(\frac{1}{2}, b \cdot y, y\right)}, y\right)} \]

      5. *-commutative N/A

         \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{\mathsf{fma}\left(b, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{y \cdot b}, y\right), y\right)} \]

      6. lower-*.f64 61.2

         \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{\mathsf{fma}\left(b, \mathsf{fma}\left(0.5, \color{blue}{y \cdot b}, y\right), y\right)} \]

   8. Applied rewrites 61.2%

      \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{\color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(0.5, y \cdot b, y\right), y\right)}} \]

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot \left(1 + b \cdot \left(1 + \frac{1}{2} \cdot b\right)\right)\right)}} \]
   10. Step-by-step derivation

       1. distribute-rgt-in N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\left(1 \cdot y + \left(b \cdot \left(1 + \frac{1}{2} \cdot b\right)\right) \cdot y\right)}} \]

       2. *-lft-identity N/A

          \[\leadsto \frac{x}{a \cdot \left(\color{blue}{y} + \left(b \cdot \left(1 + \frac{1}{2} \cdot b\right)\right) \cdot y\right)} \]

       3. associate-*l* N/A

          \[\leadsto \frac{x}{a \cdot \left(y + \color{blue}{b \cdot \left(\left(1 + \frac{1}{2} \cdot b\right) \cdot y\right)}\right)} \]

       4. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(y + b \cdot \left(\color{blue}{\left(\frac{1}{2} \cdot b + 1\right)} \cdot y\right)\right)} \]

       5. distribute-rgt1-in N/A

          \[\leadsto \frac{x}{a \cdot \left(y + b \cdot \color{blue}{\left(y + \left(\frac{1}{2} \cdot b\right) \cdot y\right)}\right)} \]

       6. associate-*r* N/A

          \[\leadsto \frac{x}{a \cdot \left(y + b \cdot \left(y + \color{blue}{\frac{1}{2} \cdot \left(b \cdot y\right)}\right)\right)} \]

       7. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y + b \cdot \left(y + \frac{1}{2} \cdot \left(b \cdot y\right)\right)\right)}} \]

       8. *-commutative N/A

          \[\leadsto \frac{x}{\color{blue}{\left(y + b \cdot \left(y + \frac{1}{2} \cdot \left(b \cdot y\right)\right)\right) \cdot a}} \]

       9. *-lft-identity N/A

          \[\leadsto \frac{x}{\left(\color{blue}{1 \cdot y} + b \cdot \left(y + \frac{1}{2} \cdot \left(b \cdot y\right)\right)\right) \cdot a} \]

       10. associate-*r* N/A

           \[\leadsto \frac{x}{\left(1 \cdot y + b \cdot \left(y + \color{blue}{\left(\frac{1}{2} \cdot b\right) \cdot y}\right)\right) \cdot a} \]

       11. distribute-rgt1-in N/A

           \[\leadsto \frac{x}{\left(1 \cdot y + b \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot b + 1\right) \cdot y\right)}\right) \cdot a} \]

       12. +-commutative N/A

           \[\leadsto \frac{x}{\left(1 \cdot y + b \cdot \left(\color{blue}{\left(1 + \frac{1}{2} \cdot b\right)} \cdot y\right)\right) \cdot a} \]

       13. associate-*l* N/A

           \[\leadsto \frac{x}{\left(1 \cdot y + \color{blue}{\left(b \cdot \left(1 + \frac{1}{2} \cdot b\right)\right) \cdot y}\right) \cdot a} \]

       14. distribute-rgt-in N/A

           \[\leadsto \frac{x}{\color{blue}{\left(y \cdot \left(1 + b \cdot \left(1 + \frac{1}{2} \cdot b\right)\right)\right)} \cdot a} \]

   11. Applied rewrites 40.4%

       \[\leadsto \color{blue}{\frac{x}{y \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(b, b \cdot 0.5, b\right), a\right)}} \]

   12. Taylor expanded in b around inf

       \[\leadsto \frac{x}{y \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)}} \]
   13. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \frac{x}{y \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)}} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{x}{y \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(a \cdot {b}^{2}\right)}\right)} \]

       3. unpow2 N/A

          \[\leadsto \frac{x}{y \cdot \left(\frac{1}{2} \cdot \left(a \cdot \color{blue}{\left(b \cdot b\right)}\right)\right)} \]

       4. lower-*.f64 46.2

          \[\leadsto \frac{x}{y \cdot \left(0.5 \cdot \left(a \cdot \color{blue}{\left(b \cdot b\right)}\right)\right)} \]

   14. Applied rewrites 46.2%

       \[\leadsto \frac{x}{y \cdot \color{blue}{\left(0.5 \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.65 \cdot 10^{-173}:\\ \;\;\;\;\left(1 - b\right) \cdot \frac{x}{y \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(0.5 \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 31.4% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t + -1 \leq 3.3 \cdot 10^{+87}:\\ \;\;\;\;x \cdot \frac{1}{y \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (+ t -1.0) 3.3e+87) (* x (/ 1.0 (* 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 + -1.0) <= 3.3e+87) {
		tmp = x * (1.0 / (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 + (-1.0d0)) <= 3.3d+87) then
        tmp = x * (1.0d0 / (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 + -1.0) <= 3.3e+87) {
		tmp = x * (1.0 / (y * a));
	} else {
		tmp = 1.0 / (y / x);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (t + -1.0) <= 3.3e+87:
		tmp = x * (1.0 / (y * a))
	else:
		tmp = 1.0 / (y / x)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(t + -1.0) <= 3.3e+87)
		tmp = Float64(x * Float64(1.0 / 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 + -1.0) <= 3.3e+87)
		tmp = x * (1.0 / (y * a));
	else
		tmp = 1.0 / (y / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(t + -1.0), $MachinePrecision], 3.3e+87], N[(x * N[(1.0 / N[(y * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(y / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 t #s(literal 1 binary64)) < 3.3000000000000001e87

   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 72.6

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot \color{blue}{e^{b}}} \]

   5. Applied rewrites 72.6%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot e^{b}}} \]

   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 62.6

         \[\leadsto x \cdot \frac{{z}^{y}}{\color{blue}{y \cdot a}} \]

   8. Applied rewrites 62.6%

      \[\leadsto x \cdot \color{blue}{\frac{{z}^{y}}{y \cdot a}} \]

   9. Taylor expanded in y around 0

      \[\leadsto x \cdot \frac{\color{blue}{1}}{y \cdot a} \]
   10. Step-by-step derivation

   11. Applied rewrites 35.7%

       \[\leadsto x \cdot \frac{\color{blue}{1}}{y \cdot a} \]

   if 3.3000000000000001e87 < (-.f64 t #s(literal 1 binary64))

   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 57.6

         \[\leadsto \frac{x \cdot e^{y \cdot \color{blue}{\log z} - b}}{y} \]

   5. Applied rewrites 57.6%

      \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]

   6. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{x \cdot {z}^{y}}{y}} \]
   7. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{{z}^{y}}{y}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \frac{{z}^{y}}{y}} \]

      3. lower-/.f64 N/A

         \[\leadsto x \cdot \color{blue}{\frac{{z}^{y}}{y}} \]

      4. lower-pow.f64 42.8

         \[\leadsto x \cdot \frac{\color{blue}{{z}^{y}}}{y} \]

   8. Applied rewrites 42.8%

      \[\leadsto \color{blue}{x \cdot \frac{{z}^{y}}{y}} \]

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x}{y}} \]
   10. Step-by-step derivation

       1. lower-/.f64 20.8

          \[\leadsto \color{blue}{\frac{x}{y}} \]

   11. Applied rewrites 20.8%

       \[\leadsto \color{blue}{\frac{x}{y}} \]
   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 22.0

          \[\leadsto \frac{1}{\color{blue}{\frac{y}{x}}} \]

   13. Applied rewrites 22.0%

       \[\leadsto \color{blue}{\frac{1}{\frac{y}{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t + -1 \leq 3.3 \cdot 10^{+87}:\\ \;\;\;\;x \cdot \frac{1}{y \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 31.3% accurate, 10.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t + -1 \leq 3.3 \cdot 10^{+87}:\\ \;\;\;\;x \cdot \frac{1}{y \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (+ t -1.0) 3.3e+87) (* x (/ 1.0 (* y a))) (/ x y)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t + -1.0) <= 3.3e+87) {
		tmp = x * (1.0 / (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 + (-1.0d0)) <= 3.3d+87) then
        tmp = x * (1.0d0 / (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 + -1.0) <= 3.3e+87) {
		tmp = x * (1.0 / (y * a));
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (t + -1.0) <= 3.3e+87:
		tmp = x * (1.0 / (y * a))
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(t + -1.0) <= 3.3e+87)
		tmp = Float64(x * Float64(1.0 / 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 + -1.0) <= 3.3e+87)
		tmp = x * (1.0 / (y * a));
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(t + -1.0), $MachinePrecision], 3.3e+87], N[(x * N[(1.0 / N[(y * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 t #s(literal 1 binary64)) < 3.3000000000000001e87

   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 72.6

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot \color{blue}{e^{b}}} \]

   5. Applied rewrites 72.6%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot e^{b}}} \]

   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 62.6

         \[\leadsto x \cdot \frac{{z}^{y}}{\color{blue}{y \cdot a}} \]

   8. Applied rewrites 62.6%

      \[\leadsto x \cdot \color{blue}{\frac{{z}^{y}}{y \cdot a}} \]

   9. Taylor expanded in y around 0

      \[\leadsto x \cdot \frac{\color{blue}{1}}{y \cdot a} \]
   10. Step-by-step derivation

   11. Applied rewrites 35.7%

       \[\leadsto x \cdot \frac{\color{blue}{1}}{y \cdot a} \]

   if 3.3000000000000001e87 < (-.f64 t #s(literal 1 binary64))

   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 57.6

         \[\leadsto \frac{x \cdot e^{y \cdot \color{blue}{\log z} - b}}{y} \]

   5. Applied rewrites 57.6%

      \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]

   6. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{x \cdot {z}^{y}}{y}} \]
   7. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{{z}^{y}}{y}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \frac{{z}^{y}}{y}} \]

      3. lower-/.f64 N/A

         \[\leadsto x \cdot \color{blue}{\frac{{z}^{y}}{y}} \]

      4. lower-pow.f64 42.8

         \[\leadsto x \cdot \frac{\color{blue}{{z}^{y}}}{y} \]

   8. Applied rewrites 42.8%

      \[\leadsto \color{blue}{x \cdot \frac{{z}^{y}}{y}} \]

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x}{y}} \]
   10. Step-by-step derivation

       1. lower-/.f64 20.8

          \[\leadsto \color{blue}{\frac{x}{y}} \]

   11. Applied rewrites 20.8%

       \[\leadsto \color{blue}{\frac{x}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t + -1 \leq 3.3 \cdot 10^{+87}:\\ \;\;\;\;x \cdot \frac{1}{y \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 31.3% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t + -1 \leq 3.3 \cdot 10^{+87}:\\ \;\;\;\;\frac{x}{y \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (+ t -1.0) 3.3e+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 + -1.0) <= 3.3e+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 + (-1.0d0)) <= 3.3d+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 + -1.0) <= 3.3e+87) {
		tmp = x / (y * a);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (t + -1.0) <= 3.3e+87:
		tmp = x / (y * a)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(t + -1.0) <= 3.3e+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 + -1.0) <= 3.3e+87)
		tmp = x / (y * a);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(t + -1.0), $MachinePrecision], 3.3e+87], N[(x / N[(y * a), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 t #s(literal 1 binary64)) < 3.3000000000000001e87

   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 72.6

          \[\leadsto x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot \color{blue}{e^{b}}} \]

   5. Applied rewrites 72.6%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot e^{b}}} \]

   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 62.6

         \[\leadsto x \cdot \frac{{z}^{y}}{\color{blue}{y \cdot a}} \]

   8. Applied rewrites 62.6%

      \[\leadsto x \cdot \color{blue}{\frac{{z}^{y}}{y \cdot a}} \]

   9. Taylor expanded in y 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 35.2

          \[\leadsto \frac{x}{\color{blue}{a \cdot y}} \]

   11. Applied rewrites 35.2%

       \[\leadsto \color{blue}{\frac{x}{a \cdot y}} \]

   if 3.3000000000000001e87 < (-.f64 t #s(literal 1 binary64))

   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 57.6

         \[\leadsto \frac{x \cdot e^{y \cdot \color{blue}{\log z} - b}}{y} \]

   5. Applied rewrites 57.6%

      \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]

   6. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{x \cdot {z}^{y}}{y}} \]
   7. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{{z}^{y}}{y}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \frac{{z}^{y}}{y}} \]

      3. lower-/.f64 N/A

         \[\leadsto x \cdot \color{blue}{\frac{{z}^{y}}{y}} \]

      4. lower-pow.f64 42.8

         \[\leadsto x \cdot \frac{\color{blue}{{z}^{y}}}{y} \]

   8. Applied rewrites 42.8%

      \[\leadsto \color{blue}{x \cdot \frac{{z}^{y}}{y}} \]

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x}{y}} \]
   10. Step-by-step derivation

       1. lower-/.f64 20.8

          \[\leadsto \color{blue}{\frac{x}{y}} \]

   11. Applied rewrites 20.8%

       \[\leadsto \color{blue}{\frac{x}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 32.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t + -1 \leq 3.3 \cdot 10^{+87}:\\ \;\;\;\;\frac{x}{y \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 16.0% 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 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 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 70.7

      \[\leadsto \frac{x \cdot e^{y \cdot \color{blue}{\log z} - b}}{y} \]

5. Applied rewrites 70.7%

   \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]

6. Taylor expanded in b around 0

   \[\leadsto \color{blue}{\frac{x \cdot {z}^{y}}{y}} \]
7. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \color{blue}{x \cdot \frac{{z}^{y}}{y}} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{x \cdot \frac{{z}^{y}}{y}} \]

   3. lower-/.f64 N/A

      \[\leadsto x \cdot \color{blue}{\frac{{z}^{y}}{y}} \]

   4. lower-pow.f64 51.4

      \[\leadsto x \cdot \frac{\color{blue}{{z}^{y}}}{y} \]

8. Applied rewrites 51.4%

   \[\leadsto \color{blue}{x \cdot \frac{{z}^{y}}{y}} \]

9. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{x}{y}} \]
10. Step-by-step derivation

    1. lower-/.f64 15.0

       \[\leadsto \color{blue}{\frac{x}{y}} \]

11. Applied rewrites 15.0%

    \[\leadsto \color{blue}{\frac{x}{y}} \]
12. Add Preprocessing

Developer Target 1: 71.4% 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 2024219 
(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))