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

Percentage Accurate: 98.6% → 98.7%

Time: 15.1s

Alternatives: 16

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-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. clear-num N/A

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

   5. un-div-inv N/A

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

   6. lower-/.f64 N/A

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

   7. div-inv N/A

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

   8. lift-exp.f64 N/A

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

   9. lift--.f64 N/A

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

   10. exp-diff N/A

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

   11. clear-num N/A

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

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: 84.5% accurate, 0.6× speedup

Math

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

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 (* y (fma 0.16666666666666666 b 0.5)) 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, (y * fma(0.16666666666666666, b, 0.5)), 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, Float64(y * fma(0.16666666666666666, b, 0.5)), 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 * N[(0.16666666666666666 * b + 0.5), $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

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

   8. Applied rewrites 79.1%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 85.0%

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

   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%

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

Alternative 3: 80.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t + -1\right) \cdot \log a\\ t_2 := \frac{x \cdot {a}^{\left(t + -1\right)}}{y}\\ \mathbf{if}\;t\_1 \leq -50000000000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -442:\\ \;\;\;\;x \cdot \frac{\frac{{z}^{y}}{a}}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, y \cdot \mathsf{fma}\left(0.16666666666666666, b, 0.5\right), y\right), y\right)}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+138}:\\ \;\;\;\;\frac{x \cdot e^{y \cdot \log z - b}}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (+ t -1.0) (log a))) (t_2 (/ (* x (pow a (+ t -1.0))) y)))
   (if (<= t_1 -50000000000000.0)
     t_2
     (if (<= t_1 -442.0)
       (*
        x
        (/
         (/ (pow z y) a)
         (fma b (fma b (* y (fma 0.16666666666666666 b 0.5)) y) y)))
       (if (<= t_1 2e+138) (/ (* x (exp (- (* y (log z)) b))) y) t_2)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t + -1.0) * log(a);
	double t_2 = (x * pow(a, (t + -1.0))) / y;
	double tmp;
	if (t_1 <= -50000000000000.0) {
		tmp = t_2;
	} else if (t_1 <= -442.0) {
		tmp = x * ((pow(z, y) / a) / fma(b, fma(b, (y * fma(0.16666666666666666, b, 0.5)), y), y));
	} else if (t_1 <= 2e+138) {
		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 * (a ^ Float64(t + -1.0))) / 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, Float64(y * fma(0.16666666666666666, b, 0.5)), y), y)));
	elseif (t_1 <= 2e+138)
		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[Power[a, N[(t + -1.0), $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 * N[(0.16666666666666666 * b + 0.5), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+138], 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 2.0000000000000001e138 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. exp-diff N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. exp-prod N/A

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

      8. lower-pow.f64 N/A

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

      9. rem-exp-log N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-exp.f64 74.5

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

   5. Applied rewrites 74.5%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 91.6%

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

   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

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

   8. Applied rewrites 79.1%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 85.0%

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

   if -442 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < 2.0000000000000001e138

   1. Initial program 98.1%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 81.7

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

   5. Applied rewrites 81.7%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(t + -1\right) \cdot \log a \leq -50000000000000:\\ \;\;\;\;\frac{x \cdot {a}^{\left(t + -1\right)}}{y}\\ \mathbf{elif}\;\left(t + -1\right) \cdot \log a \leq -442:\\ \;\;\;\;x \cdot \frac{\frac{{z}^{y}}{a}}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, y \cdot \mathsf{fma}\left(0.16666666666666666, b, 0.5\right), y\right), y\right)}\\ \mathbf{elif}\;\left(t + -1\right) \cdot \log a \leq 2 \cdot 10^{+138}:\\ \;\;\;\;\frac{x \cdot e^{y \cdot \log z - b}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot {a}^{\left(t + -1\right)}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t + -1\right) \cdot \log a\\ t_2 := \frac{x \cdot {a}^{\left(t + -1\right)}}{y}\\ \mathbf{if}\;t\_1 \leq -50000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -298:\\ \;\;\;\;x \cdot \frac{\frac{1}{a}}{y \cdot e^{b}}\\ \mathbf{elif}\;t\_1 \leq 10^{+40}:\\ \;\;\;\;x \cdot \frac{{z}^{y}}{y \cdot a}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (+ t -1.0) (log a))) (t_2 (/ (* x (pow a (+ t -1.0))) y)))
   (if (<= t_1 -50000000.0)
     t_2
     (if (<= t_1 -298.0)
       (* x (/ (/ 1.0 a) (* y (exp b))))
       (if (<= t_1 1e+40) (* x (/ (pow z y) (* y a))) t_2)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t + -1.0) * log(a);
	double t_2 = (x * pow(a, (t + -1.0))) / y;
	double tmp;
	if (t_1 <= -50000000.0) {
		tmp = t_2;
	} else if (t_1 <= -298.0) {
		tmp = x * ((1.0 / a) / (y * exp(b)));
	} else if (t_1 <= 1e+40) {
		tmp = x * (pow(z, y) / (y * a));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (t + (-1.0d0)) * log(a)
    t_2 = (x * (a ** (t + (-1.0d0)))) / y
    if (t_1 <= (-50000000.0d0)) then
        tmp = t_2
    else if (t_1 <= (-298.0d0)) then
        tmp = x * ((1.0d0 / a) / (y * exp(b)))
    else if (t_1 <= 1d+40) then
        tmp = x * ((z ** y) / (y * a))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t + -1.0) * Math.log(a);
	double t_2 = (x * Math.pow(a, (t + -1.0))) / y;
	double tmp;
	if (t_1 <= -50000000.0) {
		tmp = t_2;
	} else if (t_1 <= -298.0) {
		tmp = x * ((1.0 / a) / (y * Math.exp(b)));
	} else if (t_1 <= 1e+40) {
		tmp = x * (Math.pow(z, y) / (y * a));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (t + -1.0) * math.log(a)
	t_2 = (x * math.pow(a, (t + -1.0))) / y
	tmp = 0
	if t_1 <= -50000000.0:
		tmp = t_2
	elif t_1 <= -298.0:
		tmp = x * ((1.0 / a) / (y * math.exp(b)))
	elif t_1 <= 1e+40:
		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(Float64(x * (a ^ Float64(t + -1.0))) / y)
	tmp = 0.0
	if (t_1 <= -50000000.0)
		tmp = t_2;
	elseif (t_1 <= -298.0)
		tmp = Float64(x * Float64(Float64(1.0 / a) / Float64(y * exp(b))));
	elseif (t_1 <= 1e+40)
		tmp = Float64(x * Float64((z ^ y) / Float64(y * a)));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (t + -1.0) * log(a);
	t_2 = (x * (a ^ (t + -1.0))) / y;
	tmp = 0.0;
	if (t_1 <= -50000000.0)
		tmp = t_2;
	elseif (t_1 <= -298.0)
		tmp = x * ((1.0 / a) / (y * exp(b)));
	elseif (t_1 <= 1e+40)
		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[(N[(x * N[Power[a, N[(t + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$1, -50000000.0], t$95$2, If[LessEqual[t$95$1, -298.0], N[(x * N[(N[(1.0 / a), $MachinePrecision] / N[(y * N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+40], N[(x * N[(N[Power[z, y], $MachinePrecision] / N[(y * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < -5e7 or 1.00000000000000003e40 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. exp-diff N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. exp-prod N/A

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

      8. lower-pow.f64 N/A

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

      9. rem-exp-log N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-exp.f64 70.9

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

   5. Applied rewrites 70.9%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 86.1%

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

   if -5e7 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < -298

   1. Initial program 89.3%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   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 91.0

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

   5. Applied rewrites 91.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 73.5%

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

   if -298 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < 1.00000000000000003e40

   1. Initial program 98.2%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. exp-diff N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

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

      9. exp-diff N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. exp-to-pow N/A

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

      13. lower-pow.f64 N/A

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

      14. rem-exp-log N/A

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

      15. lower-*.f64 N/A

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

      16. lower-exp.f64 77.7

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

   5. Applied rewrites 77.7%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 75.9%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(t + -1\right) \cdot \log a \leq -50000000:\\ \;\;\;\;\frac{x \cdot {a}^{\left(t + -1\right)}}{y}\\ \mathbf{elif}\;\left(t + -1\right) \cdot \log a \leq -298:\\ \;\;\;\;x \cdot \frac{\frac{1}{a}}{y \cdot e^{b}}\\ \mathbf{elif}\;\left(t + -1\right) \cdot \log a \leq 10^{+40}:\\ \;\;\;\;x \cdot \frac{{z}^{y}}{y \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot {a}^{\left(t + -1\right)}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t + -1\right) \cdot \log a\\ t_2 := \frac{x \cdot {a}^{\left(t + -1\right)}}{y}\\ \mathbf{if}\;t\_1 \leq -50000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -298:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot e^{b}\right)}\\ \mathbf{elif}\;t\_1 \leq 10^{+40}:\\ \;\;\;\;x \cdot \frac{{z}^{y}}{y \cdot a}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (+ t -1.0) (log a))) (t_2 (/ (* x (pow a (+ t -1.0))) y)))
   (if (<= t_1 -50000000.0)
     t_2
     (if (<= t_1 -298.0)
       (/ x (* a (* y (exp b))))
       (if (<= t_1 1e+40) (* x (/ (pow z y) (* y a))) t_2)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t + -1.0) * log(a);
	double t_2 = (x * pow(a, (t + -1.0))) / y;
	double tmp;
	if (t_1 <= -50000000.0) {
		tmp = t_2;
	} else if (t_1 <= -298.0) {
		tmp = x / (a * (y * exp(b)));
	} else if (t_1 <= 1e+40) {
		tmp = x * (pow(z, y) / (y * a));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (t + (-1.0d0)) * log(a)
    t_2 = (x * (a ** (t + (-1.0d0)))) / y
    if (t_1 <= (-50000000.0d0)) then
        tmp = t_2
    else if (t_1 <= (-298.0d0)) then
        tmp = x / (a * (y * exp(b)))
    else if (t_1 <= 1d+40) then
        tmp = x * ((z ** y) / (y * a))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t + -1.0) * Math.log(a);
	double t_2 = (x * Math.pow(a, (t + -1.0))) / y;
	double tmp;
	if (t_1 <= -50000000.0) {
		tmp = t_2;
	} else if (t_1 <= -298.0) {
		tmp = x / (a * (y * Math.exp(b)));
	} else if (t_1 <= 1e+40) {
		tmp = x * (Math.pow(z, y) / (y * a));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (t + -1.0) * math.log(a)
	t_2 = (x * math.pow(a, (t + -1.0))) / y
	tmp = 0
	if t_1 <= -50000000.0:
		tmp = t_2
	elif t_1 <= -298.0:
		tmp = x / (a * (y * math.exp(b)))
	elif t_1 <= 1e+40:
		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(Float64(x * (a ^ Float64(t + -1.0))) / y)
	tmp = 0.0
	if (t_1 <= -50000000.0)
		tmp = t_2;
	elseif (t_1 <= -298.0)
		tmp = Float64(x / Float64(a * Float64(y * exp(b))));
	elseif (t_1 <= 1e+40)
		tmp = Float64(x * Float64((z ^ y) / Float64(y * a)));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (t + -1.0) * log(a);
	t_2 = (x * (a ^ (t + -1.0))) / y;
	tmp = 0.0;
	if (t_1 <= -50000000.0)
		tmp = t_2;
	elseif (t_1 <= -298.0)
		tmp = x / (a * (y * exp(b)));
	elseif (t_1 <= 1e+40)
		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[(N[(x * N[Power[a, N[(t + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$1, -50000000.0], t$95$2, If[LessEqual[t$95$1, -298.0], N[(x / N[(a * N[(y * N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+40], N[(x * N[(N[Power[z, y], $MachinePrecision] / N[(y * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < -5e7 or 1.00000000000000003e40 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. exp-diff N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. exp-prod N/A

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

      8. lower-pow.f64 N/A

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

      9. rem-exp-log N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-exp.f64 70.9

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

   5. Applied rewrites 70.9%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 86.1%

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

   if -5e7 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < -298

   1. Initial program 89.3%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   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 91.0

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

   5. Applied rewrites 91.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 73.4%

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

   if -298 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < 1.00000000000000003e40

   1. Initial program 98.2%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. exp-diff N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

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

      9. exp-diff N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. exp-to-pow N/A

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

      13. lower-pow.f64 N/A

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

      14. rem-exp-log N/A

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

      15. lower-*.f64 N/A

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

      16. lower-exp.f64 77.7

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

   5. Applied rewrites 77.7%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 75.9%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(t + -1\right) \cdot \log a \leq -50000000:\\ \;\;\;\;\frac{x \cdot {a}^{\left(t + -1\right)}}{y}\\ \mathbf{elif}\;\left(t + -1\right) \cdot \log a \leq -298:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot e^{b}\right)}\\ \mathbf{elif}\;\left(t + -1\right) \cdot \log a \leq 10^{+40}:\\ \;\;\;\;x \cdot \frac{{z}^{y}}{y \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot {a}^{\left(t + -1\right)}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 32.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x \cdot e^{\left(y \cdot \log z + \left(t + -1\right) \cdot \log a\right) - b}}{y} \leq -50000000:\\ \;\;\;\;\frac{x \cdot \frac{1}{a}}{\mathsf{fma}\left(y, b, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} \cdot \left(x \cdot \frac{1}{\mathsf{fma}\left(y, b, y\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<=
      (/ (* x (exp (- (+ (* y (log z)) (* (+ t -1.0) (log a))) b))) y)
      -50000000.0)
   (/ (* x (/ 1.0 a)) (fma y b y))
   (* (/ 1.0 a) (* x (/ 1.0 (fma y b y))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((x * exp((((y * log(z)) + ((t + -1.0) * log(a))) - b))) / y) <= -50000000.0) {
		tmp = (x * (1.0 / a)) / fma(y, b, y);
	} else {
		tmp = (1.0 / a) * (x * (1.0 / fma(y, b, y)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(Float64(x * exp(Float64(Float64(Float64(y * log(z)) + Float64(Float64(t + -1.0) * log(a))) - b))) / y) <= -50000000.0)
		tmp = Float64(Float64(x * Float64(1.0 / a)) / fma(y, b, y));
	else
		tmp = Float64(Float64(1.0 / a) * Float64(x * Float64(1.0 / fma(y, b, y))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[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], -50000000.0], N[(N[(x * N[(1.0 / a), $MachinePrecision]), $MachinePrecision] / N[(y * b + y), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / a), $MachinePrecision] * N[(x * N[(1.0 / N[(y * b + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

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) < -5e7

   1. Initial program 99.4%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. exp-diff N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

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

      9. exp-diff N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. exp-to-pow N/A

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

      13. lower-pow.f64 N/A

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

      14. rem-exp-log N/A

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

      15. lower-*.f64 N/A

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

      16. lower-exp.f64 68.7

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

   5. Applied rewrites 68.7%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 47.4%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 16.4%

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

   13. Applied rewrites 21.2%

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

   if -5e7 < (/.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 96.3%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. exp-diff N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

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

      9. exp-diff N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. exp-to-pow N/A

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

      13. lower-pow.f64 N/A

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

      14. rem-exp-log N/A

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

      15. lower-*.f64 N/A

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

      16. lower-exp.f64 67.6

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

   5. Applied rewrites 67.6%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 55.7%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 37.9%

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

   13. Applied rewrites 40.3%

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

4. Final simplification 34.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot e^{\left(y \cdot \log z + \left(t + -1\right) \cdot \log a\right) - b}}{y} \leq -50000000:\\ \;\;\;\;\frac{x \cdot \frac{1}{a}}{\mathsf{fma}\left(y, b, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} \cdot \left(x \cdot \frac{1}{\mathsf{fma}\left(y, b, y\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t + -1\right) \cdot \log a\\ t_2 := \frac{x \cdot {a}^{\left(t + -1\right)}}{y}\\ \mathbf{if}\;t\_1 \leq -2000000000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+122}:\\ \;\;\;\;x \cdot \left(\frac{{z}^{y}}{y} \cdot \frac{1}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (+ t -1.0) (log a))) (t_2 (/ (* x (pow a (+ t -1.0))) y)))
   (if (<= t_1 -2000000000000.0)
     t_2
     (if (<= t_1 5e+122) (* x (* (/ (pow z y) y) (/ 1.0 a))) t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t + -1.0) * log(a);
	double t_2 = (x * pow(a, (t + -1.0))) / y;
	double tmp;
	if (t_1 <= -2000000000000.0) {
		tmp = t_2;
	} else if (t_1 <= 5e+122) {
		tmp = x * ((pow(z, y) / y) * (1.0 / a));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (t + (-1.0d0)) * log(a)
    t_2 = (x * (a ** (t + (-1.0d0)))) / y
    if (t_1 <= (-2000000000000.0d0)) then
        tmp = t_2
    else if (t_1 <= 5d+122) then
        tmp = x * (((z ** y) / y) * (1.0d0 / a))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t + -1.0) * Math.log(a);
	double t_2 = (x * Math.pow(a, (t + -1.0))) / y;
	double tmp;
	if (t_1 <= -2000000000000.0) {
		tmp = t_2;
	} else if (t_1 <= 5e+122) {
		tmp = x * ((Math.pow(z, y) / y) * (1.0 / a));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (t + -1.0) * math.log(a)
	t_2 = (x * math.pow(a, (t + -1.0))) / y
	tmp = 0
	if t_1 <= -2000000000000.0:
		tmp = t_2
	elif t_1 <= 5e+122:
		tmp = x * ((math.pow(z, y) / y) * (1.0 / 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(Float64(x * (a ^ Float64(t + -1.0))) / y)
	tmp = 0.0
	if (t_1 <= -2000000000000.0)
		tmp = t_2;
	elseif (t_1 <= 5e+122)
		tmp = Float64(x * Float64(Float64((z ^ y) / y) * Float64(1.0 / a)));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (t + -1.0) * log(a);
	t_2 = (x * (a ^ (t + -1.0))) / y;
	tmp = 0.0;
	if (t_1 <= -2000000000000.0)
		tmp = t_2;
	elseif (t_1 <= 5e+122)
		tmp = x * (((z ^ y) / y) * (1.0 / 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[(N[(x * N[Power[a, N[(t + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$1, -2000000000000.0], t$95$2, If[LessEqual[t$95$1, 5e+122], N[(x * N[(N[(N[Power[z, y], $MachinePrecision] / y), $MachinePrecision] * N[(1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < -2e12 or 4.99999999999999989e122 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. exp-diff N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. exp-prod N/A

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

      8. lower-pow.f64 N/A

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

      9. rem-exp-log N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-exp.f64 74.3

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

   5. Applied rewrites 74.3%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 90.9%

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

   if -2e12 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < 4.99999999999999989e122

   1. Initial program 95.6%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. exp-diff N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

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

      9. exp-diff N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. exp-to-pow N/A

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

      13. lower-pow.f64 N/A

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

      14. rem-exp-log N/A

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

      15. lower-*.f64 N/A

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

      16. lower-exp.f64 81.1

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

   5. Applied rewrites 81.1%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 60.9%

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

   10. Applied rewrites 60.9%

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

   11. Taylor expanded in b around 0

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

   13. Applied rewrites 75.4%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(t + -1\right) \cdot \log a \leq -2000000000000:\\ \;\;\;\;\frac{x \cdot {a}^{\left(t + -1\right)}}{y}\\ \mathbf{elif}\;\left(t + -1\right) \cdot \log a \leq 5 \cdot 10^{+122}:\\ \;\;\;\;x \cdot \left(\frac{{z}^{y}}{y} \cdot \frac{1}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot {a}^{\left(t + -1\right)}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t + -1\right) \cdot \log a\\ t_2 := \frac{x \cdot {a}^{\left(t + -1\right)}}{y}\\ \mathbf{if}\;t\_1 \leq -50000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+138}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot e^{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (+ t -1.0) (log a))) (t_2 (/ (* x (pow a (+ t -1.0))) y)))
   (if (<= t_1 -50000000.0)
     t_2
     (if (<= t_1 2e+138) (/ x (* a (* y (exp b)))) t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t + -1.0) * log(a);
	double t_2 = (x * pow(a, (t + -1.0))) / y;
	double tmp;
	if (t_1 <= -50000000.0) {
		tmp = t_2;
	} else if (t_1 <= 2e+138) {
		tmp = x / (a * (y * exp(b)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (t + (-1.0d0)) * log(a)
    t_2 = (x * (a ** (t + (-1.0d0)))) / y
    if (t_1 <= (-50000000.0d0)) then
        tmp = t_2
    else if (t_1 <= 2d+138) then
        tmp = x / (a * (y * exp(b)))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t + -1.0) * Math.log(a);
	double t_2 = (x * Math.pow(a, (t + -1.0))) / y;
	double tmp;
	if (t_1 <= -50000000.0) {
		tmp = t_2;
	} else if (t_1 <= 2e+138) {
		tmp = x / (a * (y * Math.exp(b)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (t + -1.0) * math.log(a)
	t_2 = (x * math.pow(a, (t + -1.0))) / y
	tmp = 0
	if t_1 <= -50000000.0:
		tmp = t_2
	elif t_1 <= 2e+138:
		tmp = x / (a * (y * math.exp(b)))
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t + -1.0) * log(a))
	t_2 = Float64(Float64(x * (a ^ Float64(t + -1.0))) / y)
	tmp = 0.0
	if (t_1 <= -50000000.0)
		tmp = t_2;
	elseif (t_1 <= 2e+138)
		tmp = Float64(x / Float64(a * Float64(y * exp(b))));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (t + -1.0) * log(a);
	t_2 = (x * (a ^ (t + -1.0))) / y;
	tmp = 0.0;
	if (t_1 <= -50000000.0)
		tmp = t_2;
	elseif (t_1 <= 2e+138)
		tmp = x / (a * (y * exp(b)));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t + -1.0), $MachinePrecision] * N[Log[a], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * N[Power[a, N[(t + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$1, -50000000.0], t$95$2, If[LessEqual[t$95$1, 2e+138], N[(x / N[(a * N[(y * N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < -5e7 or 2.0000000000000001e138 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. exp-diff N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. exp-prod N/A

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

      8. lower-pow.f64 N/A

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

      9. rem-exp-log N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-exp.f64 75.0

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

   5. Applied rewrites 75.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 91.8%

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

   if -5e7 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < 2.0000000000000001e138

   1. Initial program 95.6%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   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 80.1

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

   5. Applied rewrites 80.1%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 64.5%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(t + -1\right) \cdot \log a \leq -50000000:\\ \;\;\;\;\frac{x \cdot {a}^{\left(t + -1\right)}}{y}\\ \mathbf{elif}\;\left(t + -1\right) \cdot \log a \leq 2 \cdot 10^{+138}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot e^{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot {a}^{\left(t + -1\right)}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 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-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. div-inv N/A

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

      8. lift-exp.f64 N/A

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

      9. lift--.f64 N/A

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

      10. exp-diff N/A

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

      11. clear-num N/A

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

   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 10: 87.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot e^{y \cdot \log z - b}}{y}\\ \mathbf{if}\;y \leq -3.4 \cdot 10^{+44}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 0.106:\\ \;\;\;\;x \cdot \frac{{a}^{\left(t + -1\right)}}{y \cdot e^{b}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (* x (exp (- (* y (log z)) b))) y)))
   (if (<= y -3.4e+44)
     t_1
     (if (<= y 0.106) (* x (/ (pow a (+ t -1.0)) (* y (exp b)))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x * exp(((y * log(z)) - b))) / y;
	double tmp;
	if (y <= -3.4e+44) {
		tmp = t_1;
	} else if (y <= 0.106) {
		tmp = x * (pow(a, (t + -1.0)) / (y * exp(b)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x * exp(((y * log(z)) - b))) / y
    if (y <= (-3.4d+44)) then
        tmp = t_1
    else if (y <= 0.106d0) then
        tmp = x * ((a ** (t + (-1.0d0))) / (y * exp(b)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x * Math.exp(((y * Math.log(z)) - b))) / y;
	double tmp;
	if (y <= -3.4e+44) {
		tmp = t_1;
	} else if (y <= 0.106) {
		tmp = x * (Math.pow(a, (t + -1.0)) / (y * Math.exp(b)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (x * math.exp(((y * math.log(z)) - b))) / y
	tmp = 0
	if y <= -3.4e+44:
		tmp = t_1
	elif y <= 0.106:
		tmp = x * (math.pow(a, (t + -1.0)) / (y * math.exp(b)))
	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 <= -3.4e+44)
		tmp = t_1;
	elseif (y <= 0.106)
		tmp = Float64(x * Float64((a ^ Float64(t + -1.0)) / Float64(y * exp(b))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (x * exp(((y * log(z)) - b))) / y;
	tmp = 0.0;
	if (y <= -3.4e+44)
		tmp = t_1;
	elseif (y <= 0.106)
		tmp = x * ((a ^ (t + -1.0)) / (y * exp(b)));
	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, -3.4e+44], t$95$1, If[LessEqual[y, 0.106], N[(x * N[(N[Power[a, N[(t + -1.0), $MachinePrecision]], $MachinePrecision] / N[(y * N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.4e44 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 -3.4e44 < 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-/.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. metadata-eval N/A

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

      5. pow-to-exp N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-*.f64 N/A

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

   4. Applied rewrites 37.1%

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

   5. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. exp-to-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-exp.f64 84.2

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

   7. Applied rewrites 84.2%

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

Alternative 11: 72.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \frac{e^{-b}}{y}\\ \mathbf{if}\;b \leq -2.5 \cdot 10^{+134}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 1.65 \cdot 10^{+85}:\\ \;\;\;\;\frac{x \cdot {a}^{\left(t + -1\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (/ (exp (- b)) y))))
   (if (<= b -2.5e+134)
     t_1
     (if (<= b 1.65e+85) (/ (* x (pow a (+ t -1.0))) y) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * (exp(-b) / y);
	double tmp;
	if (b <= -2.5e+134) {
		tmp = t_1;
	} else if (b <= 1.65e+85) {
		tmp = (x * pow(a, (t + -1.0))) / y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * (exp(-b) / y)
    if (b <= (-2.5d+134)) then
        tmp = t_1
    else if (b <= 1.65d+85) then
        tmp = (x * (a ** (t + (-1.0d0)))) / y
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * (Math.exp(-b) / y);
	double tmp;
	if (b <= -2.5e+134) {
		tmp = t_1;
	} else if (b <= 1.65e+85) {
		tmp = (x * Math.pow(a, (t + -1.0))) / y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x * (math.exp(-b) / y)
	tmp = 0
	if b <= -2.5e+134:
		tmp = t_1
	elif b <= 1.65e+85:
		tmp = (x * math.pow(a, (t + -1.0))) / y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x * Float64(exp(Float64(-b)) / y))
	tmp = 0.0
	if (b <= -2.5e+134)
		tmp = t_1;
	elseif (b <= 1.65e+85)
		tmp = Float64(Float64(x * (a ^ Float64(t + -1.0))) / y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x * (exp(-b) / y);
	tmp = 0.0;
	if (b <= -2.5e+134)
		tmp = t_1;
	elseif (b <= 1.65e+85)
		tmp = (x * (a ^ (t + -1.0))) / 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[Exp[(-b)], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -2.5e+134], t$95$1, If[LessEqual[b, 1.65e+85], N[(N[(x * N[Power[a, N[(t + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if b < -2.4999999999999999e134 or 1.65e85 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in 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 95.8

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

   5. Applied rewrites 95.8%

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

   6. Taylor expanded in b around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{-1 \cdot b}}}{y} \]
   7. Step-by-step derivation

      1. neg-mul-1 N/A

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

      2. lower-neg.f64 90.3

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

   8. Applied rewrites 90.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 90.3

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

   10. Applied rewrites 90.3%

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

   if -2.4999999999999999e134 < b < 1.65e85

   1. Initial program 96.2%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. exp-diff N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. exp-prod N/A

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

      8. lower-pow.f64 N/A

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

      9. rem-exp-log N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-exp.f64 62.9

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

   5. Applied rewrites 62.9%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 68.2%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.5 \cdot 10^{+134}:\\ \;\;\;\;x \cdot \frac{e^{-b}}{y}\\ \mathbf{elif}\;b \leq 1.65 \cdot 10^{+85}:\\ \;\;\;\;\frac{x \cdot {a}^{\left(t + -1\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{e^{-b}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 59.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \frac{e^{-b}}{y}\\ \mathbf{if}\;b \leq -0.031:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 1.3:\\ \;\;\;\;\frac{1}{a} \cdot \left(x \cdot \frac{1}{\mathsf{fma}\left(y, b, y\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (/ (exp (- b)) y))))
   (if (<= b -0.031)
     t_1
     (if (<= b 1.3) (* (/ 1.0 a) (* x (/ 1.0 (fma y b y)))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * (exp(-b) / y);
	double tmp;
	if (b <= -0.031) {
		tmp = t_1;
	} else if (b <= 1.3) {
		tmp = (1.0 / a) * (x * (1.0 / fma(y, b, y)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x * Float64(exp(Float64(-b)) / y))
	tmp = 0.0
	if (b <= -0.031)
		tmp = t_1;
	elseif (b <= 1.3)
		tmp = Float64(Float64(1.0 / a) * Float64(x * Float64(1.0 / fma(y, b, y))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x * N[(N[Exp[(-b)], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -0.031], t$95$1, If[LessEqual[b, 1.3], N[(N[(1.0 / a), $MachinePrecision] * N[(x * N[(1.0 / N[(y * b + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if b < -0.031 or 1.30000000000000004 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in 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 89.8

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

   5. Applied rewrites 89.8%

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

   6. Taylor expanded in b around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{-1 \cdot b}}}{y} \]
   7. Step-by-step derivation

      1. neg-mul-1 N/A

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

      2. lower-neg.f64 74.8

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

   8. Applied rewrites 74.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 74.8

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

   10. Applied rewrites 74.8%

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

   if -0.031 < b < 1.30000000000000004

   1. Initial program 94.6%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   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 73.8

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

   5. Applied rewrites 73.8%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 73.2%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 35.4%

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

   13. Applied rewrites 35.6%

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

4. Final simplification 54.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.031:\\ \;\;\;\;x \cdot \frac{e^{-b}}{y}\\ \mathbf{elif}\;b \leq 1.3:\\ \;\;\;\;\frac{1}{a} \cdot \left(x \cdot \frac{1}{\mathsf{fma}\left(y, b, y\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{e^{-b}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 40.1% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.75 \cdot 10^{+44}:\\ \;\;\;\;x \cdot \frac{\frac{1}{a}}{\mathsf{fma}\left(b, \mathsf{fma}\left(y, b \cdot 0.5, y\right), y\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{1}{a}}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, y \cdot \mathsf{fma}\left(0.16666666666666666, b, 0.5\right), y\right), y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b 2.75e+44)
   (* x (/ (/ 1.0 a) (fma b (fma y (* b 0.5) y) y)))
   (*
    x
    (/ (/ 1.0 a) (fma b (fma b (* y (fma 0.16666666666666666 b 0.5)) y) y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= 2.75e+44) {
		tmp = x * ((1.0 / a) / fma(b, fma(y, (b * 0.5), y), y));
	} else {
		tmp = x * ((1.0 / a) / fma(b, fma(b, (y * fma(0.16666666666666666, b, 0.5)), y), y));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= 2.75e+44)
		tmp = Float64(x * Float64(Float64(1.0 / a) / fma(b, fma(y, Float64(b * 0.5), y), y)));
	else
		tmp = Float64(x * Float64(Float64(1.0 / a) / fma(b, fma(b, Float64(y * fma(0.16666666666666666, b, 0.5)), y), y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, 2.75e+44], N[(x * N[(N[(1.0 / a), $MachinePrecision] / N[(b * N[(y * N[(b * 0.5), $MachinePrecision] + y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(1.0 / a), $MachinePrecision] / N[(b * N[(b * N[(y * N[(0.16666666666666666 * b + 0.5), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.75e44

   1. Initial program 96.6%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   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 67.7

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

   5. Applied rewrites 67.7%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 56.8%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 29.4%

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

   12. Taylor expanded in b around 0

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

   14. Applied rewrites 33.0%

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

   if 2.75e44 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. exp-diff N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

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

      9. exp-diff N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. exp-to-pow N/A

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

      13. lower-pow.f64 N/A

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

      14. rem-exp-log N/A

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

      15. lower-*.f64 N/A

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

      16. lower-exp.f64 68.7

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

   5. Applied rewrites 68.7%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 38.6%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 39.3%

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

   12. Taylor expanded in b around 0

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

   14. Applied rewrites 65.9%

       \[\leadsto x \cdot \frac{\frac{1}{a}}{\mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, y \cdot \mathsf{fma}\left(0.16666666666666666, b, 0.5\right), y\right)}, y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 39.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.75 \cdot 10^{+44}:\\ \;\;\;\;x \cdot \frac{\frac{1}{a}}{\mathsf{fma}\left(b, \mathsf{fma}\left(y, b \cdot 0.5, y\right), y\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{1}{a}}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, y \cdot \mathsf{fma}\left(0.16666666666666666, b, 0.5\right), y\right), y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 37.7% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ x \cdot \frac{\frac{1}{a}}{\mathsf{fma}\left(b, \mathsf{fma}\left(y, b \cdot 0.5, y\right), y\right)} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (* x (/ (/ 1.0 a) (fma b (fma y (* b 0.5) y) y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return x * ((1.0 / a) / fma(b, fma(y, (b * 0.5), y), y));
}

Julia

function code(x, y, z, t, a, b)
	return Float64(x * Float64(Float64(1.0 / a) / fma(b, fma(y, Float64(b * 0.5), y), y)))
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(x * N[(N[(1.0 / a), $MachinePrecision] / N[(b * N[(y * N[(b * 0.5), $MachinePrecision] + y), $MachinePrecision] + y), $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. 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

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

8. Applied rewrites 53.2%

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

9. Taylor expanded in y around 0

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

11. Applied rewrites 31.4%

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

12. Taylor expanded in b around 0

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

14. Applied rewrites 37.9%

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

15. Final simplification 37.9%

    \[\leadsto x \cdot \frac{\frac{1}{a}}{\mathsf{fma}\left(b, \mathsf{fma}\left(y, b \cdot 0.5, y\right), y\right)} \]
16. Add Preprocessing

Alternative 15: 31.5% accurate, 9.9× speedup

Math

\[\begin{array}{l} \\ x \cdot \frac{\frac{1}{a}}{\mathsf{fma}\left(y, b, y\right)} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

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

8. Applied rewrites 53.2%

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

9. Taylor expanded in y around 0

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

11. Applied rewrites 31.4%

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

Alternative 16: 31.9% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot 1\right) \cdot \frac{1}{a \cdot \mathsf{fma}\left(y, b, y\right)} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

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

8. Applied rewrites 53.2%

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

10. Applied rewrites 48.0%

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

11. Taylor expanded in y around 0

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

13. Applied rewrites 31.3%

    \[\leadsto \left(x \cdot 1\right) \cdot \frac{1}{a \cdot \mathsf{fma}\left(y, b, y\right)} \]
14. 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))