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

Percentage Accurate: 98.4% → 98.4%

Time: 5.9s

Alternatives: 18

Speedup: 1.0×

• 47.4% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    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: 98.4% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    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: 98.4% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.4%

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

Alternative 2: 47.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\\ t_2 := \frac{x \cdot \frac{1}{a}}{y}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{-195}:\\ \;\;\;\;\frac{x}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 1\right) \cdot y\right) \cdot a}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (* x (exp (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))) y))
        (t_2 (/ (* x (/ 1.0 a)) y)))
   (if (<= t_1 (- INFINITY))
     t_2
     (if (<= t_1 1e-195)
       (/
        x
        (* (* (fma (fma (fma 0.16666666666666666 b 0.5) b 1.0) b 1.0) y) a))
       t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x * exp((((y * log(z)) + ((t - 1.0) * log(a))) - b))) / y;
	double t_2 = (x * (1.0 / a)) / y;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_1 <= 1e-195) {
		tmp = x / ((fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 1.0) * y) * a);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x * exp(Float64(Float64(Float64(y * log(z)) + Float64(Float64(t - 1.0) * log(a))) - b))) / y)
	t_2 = Float64(Float64(x * Float64(1.0 / a)) / y)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_1 <= 1e-195)
		tmp = Float64(x / Float64(Float64(fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 1.0) * y) * a));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x * N[Exp[N[(N[(N[(y * N[Log[z], $MachinePrecision]), $MachinePrecision] + N[(N[(t - 1.0), $MachinePrecision] * N[Log[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * N[(1.0 / a), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$2, If[LessEqual[t$95$1, 1e-195], N[(x / N[(N[(N[(N[(N[(0.16666666666666666 * b + 0.5), $MachinePrecision] * b + 1.0), $MachinePrecision] * b + 1.0), $MachinePrecision] * y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (exp.f64 (-.f64 (+.f64 (*.f64 y (log.f64 z)) (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))) b))) y) < -inf.0 or 1.0000000000000001e-195 < (/.f64 (*.f64 x (exp.f64 (-.f64 (+.f64 (*.f64 y (log.f64 z)) (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))) b))) y)

   1. Initial program 99.3%

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

   2. Taylor expanded in b around 0

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

      1. exp-sum N/A

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

      2. *-commutative N/A

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

      3. pow-to-exp N/A

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

      4. pow-to-exp N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift--.f64 72.5

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

   4. Applied rewrites 72.5%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lift-pow.f64 64.5

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

   7. Applied rewrites 64.5%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 33.2%

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

   if -inf.0 < (/.f64 (*.f64 x (exp.f64 (-.f64 (+.f64 (*.f64 y (log.f64 z)) (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))) b))) y) < 1.0000000000000001e-195

   1. Initial program 97.5%

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

   2. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. div-exp N/A

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

      5. pow-to-exp N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-exp.f64 71.4

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

   4. Applied rewrites 71.4%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-exp.f64 60.7

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

   7. Applied rewrites 60.7%

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

   8. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 62.9

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

   10. Applied rewrites 62.9%

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

Alternative 3: 45.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\\ t_2 := \frac{x \cdot \frac{1}{a}}{y}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{-195}:\\ \;\;\;\;\frac{x}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right), b, 1\right) \cdot y\right) \cdot a}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (* x (exp (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))) y))
        (t_2 (/ (* x (/ 1.0 a)) y)))
   (if (<= t_1 (- INFINITY))
     t_2
     (if (<= t_1 1e-195) (/ x (* (* (fma (fma 0.5 b 1.0) b 1.0) y) a)) t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x * exp((((y * log(z)) + ((t - 1.0) * log(a))) - b))) / y;
	double t_2 = (x * (1.0 / a)) / y;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_1 <= 1e-195) {
		tmp = x / ((fma(fma(0.5, b, 1.0), b, 1.0) * y) * a);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x * exp(Float64(Float64(Float64(y * log(z)) + Float64(Float64(t - 1.0) * log(a))) - b))) / y)
	t_2 = Float64(Float64(x * Float64(1.0 / a)) / y)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_1 <= 1e-195)
		tmp = Float64(x / Float64(Float64(fma(fma(0.5, b, 1.0), b, 1.0) * y) * a));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x * N[Exp[N[(N[(N[(y * N[Log[z], $MachinePrecision]), $MachinePrecision] + N[(N[(t - 1.0), $MachinePrecision] * N[Log[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * N[(1.0 / a), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$2, If[LessEqual[t$95$1, 1e-195], N[(x / N[(N[(N[(N[(0.5 * b + 1.0), $MachinePrecision] * b + 1.0), $MachinePrecision] * y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (exp.f64 (-.f64 (+.f64 (*.f64 y (log.f64 z)) (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))) b))) y) < -inf.0 or 1.0000000000000001e-195 < (/.f64 (*.f64 x (exp.f64 (-.f64 (+.f64 (*.f64 y (log.f64 z)) (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))) b))) y)

   1. Initial program 99.3%

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

   2. Taylor expanded in b around 0

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

      1. exp-sum N/A

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

      2. *-commutative N/A

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

      3. pow-to-exp N/A

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

      4. pow-to-exp N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift--.f64 72.5

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

   4. Applied rewrites 72.5%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lift-pow.f64 64.5

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

   7. Applied rewrites 64.5%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 33.2%

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

   if -inf.0 < (/.f64 (*.f64 x (exp.f64 (-.f64 (+.f64 (*.f64 y (log.f64 z)) (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))) b))) y) < 1.0000000000000001e-195

   1. Initial program 97.5%

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

   2. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. div-exp N/A

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

      5. pow-to-exp N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-exp.f64 71.4

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

   4. Applied rewrites 71.4%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-exp.f64 60.7

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

   7. Applied rewrites 60.7%

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

   8. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 59.1

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

   10. Applied rewrites 59.1%

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

Alternative 4: 44.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\\ t_2 := \frac{x \cdot \frac{1}{a}}{y}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{-195}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(b \cdot y, 0.5, y\right), b, y\right) \cdot a}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (* x (exp (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))) y))
        (t_2 (/ (* x (/ 1.0 a)) y)))
   (if (<= t_1 (- INFINITY))
     t_2
     (if (<= t_1 1e-195) (/ x (* (fma (fma (* b y) 0.5 y) b y) a)) t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x * exp((((y * log(z)) + ((t - 1.0) * log(a))) - b))) / y;
	double t_2 = (x * (1.0 / a)) / y;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_1 <= 1e-195) {
		tmp = x / (fma(fma((b * y), 0.5, y), b, y) * a);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x * exp(Float64(Float64(Float64(y * log(z)) + Float64(Float64(t - 1.0) * log(a))) - b))) / y)
	t_2 = Float64(Float64(x * Float64(1.0 / a)) / y)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_1 <= 1e-195)
		tmp = Float64(x / Float64(fma(fma(Float64(b * y), 0.5, y), b, y) * a));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x * N[Exp[N[(N[(N[(y * N[Log[z], $MachinePrecision]), $MachinePrecision] + N[(N[(t - 1.0), $MachinePrecision] * N[Log[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * N[(1.0 / a), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$2, If[LessEqual[t$95$1, 1e-195], N[(x / N[(N[(N[(N[(b * y), $MachinePrecision] * 0.5 + y), $MachinePrecision] * b + y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (exp.f64 (-.f64 (+.f64 (*.f64 y (log.f64 z)) (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))) b))) y) < -inf.0 or 1.0000000000000001e-195 < (/.f64 (*.f64 x (exp.f64 (-.f64 (+.f64 (*.f64 y (log.f64 z)) (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))) b))) y)

   1. Initial program 99.3%

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

   2. Taylor expanded in b around 0

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

      1. exp-sum N/A

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

      2. *-commutative N/A

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

      3. pow-to-exp N/A

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

      4. pow-to-exp N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift--.f64 72.5

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

   4. Applied rewrites 72.5%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lift-pow.f64 64.5

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

   7. Applied rewrites 64.5%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 33.2%

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

   if -inf.0 < (/.f64 (*.f64 x (exp.f64 (-.f64 (+.f64 (*.f64 y (log.f64 z)) (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))) b))) y) < 1.0000000000000001e-195

   1. Initial program 97.5%

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

   2. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. div-exp N/A

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

      5. pow-to-exp N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-exp.f64 71.4

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

   4. Applied rewrites 71.4%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-exp.f64 60.7

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

   7. Applied rewrites 60.7%

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

   8. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 55.8

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

   10. Applied rewrites 55.8%

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

Alternative 5: 67.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t - 1\right) \cdot \log a\\ t_2 := x \cdot \frac{{a}^{t}}{y}\\ \mathbf{if}\;t\_1 \leq -680:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -129:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(b \cdot y, 0.5, y\right), b, y\right) \cdot a}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+43}:\\ \;\;\;\;x \cdot \frac{e^{-b}}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- t 1.0) (log a))) (t_2 (* x (/ (pow a t) y))))
   (if (<= t_1 -680.0)
     t_2
     (if (<= t_1 -129.0)
       (/ x (* (fma (fma (* b y) 0.5 y) b y) a))
       (if (<= t_1 5e+43) (* x (/ (exp (- 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) / y);
	double tmp;
	if (t_1 <= -680.0) {
		tmp = t_2;
	} else if (t_1 <= -129.0) {
		tmp = x / (fma(fma((b * y), 0.5, y), b, y) * a);
	} else if (t_1 <= 5e+43) {
		tmp = x * (exp(-b) / y);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - 1.0) * log(a))
	t_2 = Float64(x * Float64((a ^ t) / y))
	tmp = 0.0
	if (t_1 <= -680.0)
		tmp = t_2;
	elseif (t_1 <= -129.0)
		tmp = Float64(x / Float64(fma(fma(Float64(b * y), 0.5, y), b, y) * a));
	elseif (t_1 <= 5e+43)
		tmp = Float64(x * Float64(exp(Float64(-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[(x * N[(N[Power[a, t], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -680.0], t$95$2, If[LessEqual[t$95$1, -129.0], N[(x / N[(N[(N[(N[(b * y), $MachinePrecision] * 0.5 + y), $MachinePrecision] * b + y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+43], N[(x * N[(N[Exp[(-b)], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < -680 or 5.0000000000000004e43 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. div-exp N/A

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

      5. pow-to-exp N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-exp.f64 68.1

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

   4. Applied rewrites 68.1%

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

   5. Taylor expanded in b around 0

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

      1. pow-to-exp N/A

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

      2. lift-pow.f64 N/A

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

      3. lift--.f64 81.4

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

   7. Applied rewrites 81.4%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 80.6%

       \[\leadsto x \cdot \frac{{a}^{t}}{y} \]

   if -680 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < -129

   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. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. div-exp N/A

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

      5. pow-to-exp N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-exp.f64 77.5

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

   4. Applied rewrites 77.5%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-exp.f64 77.4

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

   7. Applied rewrites 77.4%

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

   8. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 54.5

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

   10. Applied rewrites 54.5%

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

   if -129 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < 5.0000000000000004e43

   1. Initial program 98.8%

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

   2. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 54.3

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

   4. Applied rewrites 54.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 54.3

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

   6. Applied rewrites 54.3%

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

Alternative 6: 76.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t - 1\right) \cdot \log a\\ \mathbf{if}\;t\_1 \leq -500000:\\ \;\;\;\;x \cdot \frac{{a}^{t}}{y}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+43}:\\ \;\;\;\;\frac{\frac{x}{e^{b} \cdot y}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot {a}^{\left(t - 1\right)}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- t 1.0) (log a))))
   (if (<= t_1 -500000.0)
     (* x (/ (pow a t) y))
     (if (<= t_1 5e+43)
       (/ (/ x (* (exp b) y)) a)
       (/ (* x (pow a (- t 1.0))) y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - 1.0) * log(a);
	double tmp;
	if (t_1 <= -500000.0) {
		tmp = x * (pow(a, t) / y);
	} else if (t_1 <= 5e+43) {
		tmp = (x / (exp(b) * y)) / a;
	} else {
		tmp = (x * pow(a, (t - 1.0))) / y;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    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 = (t - 1.0d0) * log(a)
    if (t_1 <= (-500000.0d0)) then
        tmp = x * ((a ** t) / y)
    else if (t_1 <= 5d+43) then
        tmp = (x / (exp(b) * y)) / a
    else
        tmp = (x * (a ** (t - 1.0d0))) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - 1.0) * Math.log(a);
	double tmp;
	if (t_1 <= -500000.0) {
		tmp = x * (Math.pow(a, t) / y);
	} else if (t_1 <= 5e+43) {
		tmp = (x / (Math.exp(b) * y)) / a;
	} else {
		tmp = (x * Math.pow(a, (t - 1.0))) / y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (t - 1.0) * math.log(a)
	tmp = 0
	if t_1 <= -500000.0:
		tmp = x * (math.pow(a, t) / y)
	elif t_1 <= 5e+43:
		tmp = (x / (math.exp(b) * y)) / a
	else:
		tmp = (x * math.pow(a, (t - 1.0))) / y
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - 1.0) * log(a))
	tmp = 0.0
	if (t_1 <= -500000.0)
		tmp = Float64(x * Float64((a ^ t) / y));
	elseif (t_1 <= 5e+43)
		tmp = Float64(Float64(x / Float64(exp(b) * y)) / a);
	else
		tmp = Float64(Float64(x * (a ^ Float64(t - 1.0))) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (t - 1.0) * log(a);
	tmp = 0.0;
	if (t_1 <= -500000.0)
		tmp = x * ((a ^ t) / y);
	elseif (t_1 <= 5e+43)
		tmp = (x / (exp(b) * y)) / a;
	else
		tmp = (x * (a ^ (t - 1.0))) / y;
	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]}, If[LessEqual[t$95$1, -500000.0], N[(x * N[(N[Power[a, t], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+43], N[(N[(x / N[(N[Exp[b], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(x * N[Power[a, N[(t - 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < -5e5

   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. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. div-exp N/A

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

      5. pow-to-exp N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-exp.f64 67.3

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

   4. Applied rewrites 67.3%

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

   5. Taylor expanded in b around 0

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

      1. pow-to-exp N/A

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

      2. lift-pow.f64 N/A

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

      3. lift--.f64 81.1

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

   7. Applied rewrites 81.1%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 81.1%

       \[\leadsto x \cdot \frac{{a}^{t}}{y} \]

   if -5e5 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < 5.0000000000000004e43

   1. Initial program 97.0%

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

   2. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. div-exp N/A

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

      5. pow-to-exp N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-exp.f64 71.5

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

   4. Applied rewrites 71.5%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-exp.f64 71.1

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

   7. Applied rewrites 71.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-exp.f64 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-exp.f64 N/A

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

      11. lift-*.f64 70.8

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

   9. Applied rewrites 70.8%

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

   if 5.0000000000000004e43 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))

   1. Initial program 100.0%

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

   2. Taylor expanded in b around 0

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

      1. exp-sum N/A

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

      2. *-commutative N/A

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

      3. pow-to-exp N/A

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

      4. pow-to-exp N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift--.f64 69.4

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

   4. Applied rewrites 69.4%

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

   5. Taylor expanded in y around 0

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

      1. pow-to-exp N/A

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

      2. lift-pow.f64 N/A

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

      3. lift--.f64 83.3

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

   7. Applied rewrites 83.3%

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

Alternative 7: 76.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t - 1\right) \cdot \log a\\ \mathbf{if}\;t\_1 \leq -500000:\\ \;\;\;\;x \cdot \frac{{a}^{t}}{y}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+43}:\\ \;\;\;\;\frac{x}{\left(e^{b} \cdot y\right) \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot {a}^{\left(t - 1\right)}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- t 1.0) (log a))))
   (if (<= t_1 -500000.0)
     (* x (/ (pow a t) y))
     (if (<= t_1 5e+43)
       (/ x (* (* (exp b) y) a))
       (/ (* x (pow a (- t 1.0))) y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - 1.0) * log(a);
	double tmp;
	if (t_1 <= -500000.0) {
		tmp = x * (pow(a, t) / y);
	} else if (t_1 <= 5e+43) {
		tmp = x / ((exp(b) * y) * a);
	} else {
		tmp = (x * pow(a, (t - 1.0))) / y;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    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 = (t - 1.0d0) * log(a)
    if (t_1 <= (-500000.0d0)) then
        tmp = x * ((a ** t) / y)
    else if (t_1 <= 5d+43) then
        tmp = x / ((exp(b) * y) * a)
    else
        tmp = (x * (a ** (t - 1.0d0))) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - 1.0) * Math.log(a);
	double tmp;
	if (t_1 <= -500000.0) {
		tmp = x * (Math.pow(a, t) / y);
	} else if (t_1 <= 5e+43) {
		tmp = x / ((Math.exp(b) * y) * a);
	} else {
		tmp = (x * Math.pow(a, (t - 1.0))) / y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (t - 1.0) * math.log(a)
	tmp = 0
	if t_1 <= -500000.0:
		tmp = x * (math.pow(a, t) / y)
	elif t_1 <= 5e+43:
		tmp = x / ((math.exp(b) * y) * a)
	else:
		tmp = (x * math.pow(a, (t - 1.0))) / y
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - 1.0) * log(a))
	tmp = 0.0
	if (t_1 <= -500000.0)
		tmp = Float64(x * Float64((a ^ t) / y));
	elseif (t_1 <= 5e+43)
		tmp = Float64(x / Float64(Float64(exp(b) * y) * a));
	else
		tmp = Float64(Float64(x * (a ^ Float64(t - 1.0))) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (t - 1.0) * log(a);
	tmp = 0.0;
	if (t_1 <= -500000.0)
		tmp = x * ((a ^ t) / y);
	elseif (t_1 <= 5e+43)
		tmp = x / ((exp(b) * y) * a);
	else
		tmp = (x * (a ^ (t - 1.0))) / y;
	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]}, If[LessEqual[t$95$1, -500000.0], N[(x * N[(N[Power[a, t], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+43], N[(x / N[(N[(N[Exp[b], $MachinePrecision] * y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[Power[a, N[(t - 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < -5e5

   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. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. div-exp N/A

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

      5. pow-to-exp N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-exp.f64 67.3

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

   4. Applied rewrites 67.3%

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

   5. Taylor expanded in b around 0

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

      1. pow-to-exp N/A

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

      2. lift-pow.f64 N/A

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

      3. lift--.f64 81.1

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

   7. Applied rewrites 81.1%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 81.1%

       \[\leadsto x \cdot \frac{{a}^{t}}{y} \]

   if -5e5 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < 5.0000000000000004e43

   1. Initial program 97.0%

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

   2. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. div-exp N/A

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

      5. pow-to-exp N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-exp.f64 71.5

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

   4. Applied rewrites 71.5%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-exp.f64 71.1

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

   7. Applied rewrites 71.1%

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

   if 5.0000000000000004e43 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))

   1. Initial program 100.0%

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

   2. Taylor expanded in b around 0

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

      1. exp-sum N/A

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

      2. *-commutative N/A

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

      3. pow-to-exp N/A

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

      4. pow-to-exp N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift--.f64 69.4

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

   4. Applied rewrites 69.4%

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

   5. Taylor expanded in y around 0

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

      1. pow-to-exp N/A

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

      2. lift-pow.f64 N/A

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

      3. lift--.f64 83.3

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

   7. Applied rewrites 83.3%

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

Alternative 8: 64.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t - 1\right) \cdot \log a\\ t_2 := x \cdot \frac{{a}^{t}}{y}\\ \mathbf{if}\;t\_1 \leq -680:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+25}:\\ \;\;\;\;\frac{x}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 1\right) \cdot y\right) \cdot a}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- t 1.0) (log a))) (t_2 (* x (/ (pow a t) y))))
   (if (<= t_1 -680.0)
     t_2
     (if (<= t_1 2e+25)
       (/
        x
        (* (* (fma (fma (fma 0.16666666666666666 b 0.5) b 1.0) b 1.0) 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) / y);
	double tmp;
	if (t_1 <= -680.0) {
		tmp = t_2;
	} else if (t_1 <= 2e+25) {
		tmp = x / ((fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 1.0) * y) * a);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - 1.0) * log(a))
	t_2 = Float64(x * Float64((a ^ t) / y))
	tmp = 0.0
	if (t_1 <= -680.0)
		tmp = t_2;
	elseif (t_1 <= 2e+25)
		tmp = Float64(x / Float64(Float64(fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 1.0) * y) * a));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - 1.0), $MachinePrecision] * N[Log[a], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x * N[(N[Power[a, t], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -680.0], t$95$2, If[LessEqual[t$95$1, 2e+25], N[(x / N[(N[(N[(N[(N[(0.16666666666666666 * b + 0.5), $MachinePrecision] * b + 1.0), $MachinePrecision] * b + 1.0), $MachinePrecision] * y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < -680 or 2.00000000000000018e25 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. div-exp N/A

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

      5. pow-to-exp N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-exp.f64 67.9

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

   4. Applied rewrites 67.9%

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

   5. Taylor expanded in b around 0

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

      1. pow-to-exp N/A

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

      2. lift-pow.f64 N/A

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

      3. lift--.f64 81.1

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

   7. Applied rewrites 81.1%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 80.2%

       \[\leadsto x \cdot \frac{{a}^{t}}{y} \]

   if -680 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < 2.00000000000000018e25

   1. Initial program 96.9%

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

   2. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. div-exp N/A

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

      5. pow-to-exp N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-exp.f64 71.6

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

   4. Applied rewrites 71.6%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-exp.f64 71.4

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

   7. Applied rewrites 71.4%

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

   8. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 48.6

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

   10. Applied rewrites 48.6%

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

Alternative 9: 89.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot \frac{{z}^{y}}{a}}{y}\\ \mathbf{if}\;y \leq -6.5 \cdot 10^{+140}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+119}:\\ \;\;\;\;\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (* x (/ (pow z y) a)) y)))
   (if (<= y -6.5e+140)
     t_1
     (if (<= y 5e+119) (/ (* x (exp (- (* (log a) (- t 1.0)) b))) y) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x * (pow(z, y) / a)) / y;
	double tmp;
	if (y <= -6.5e+140) {
		tmp = t_1;
	} else if (y <= 5e+119) {
		tmp = (x * exp(((log(a) * (t - 1.0)) - b))) / y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x * ((z ** y) / a)) / y
    if (y <= (-6.5d+140)) then
        tmp = t_1
    else if (y <= 5d+119) then
        tmp = (x * exp(((log(a) * (t - 1.0d0)) - b))) / y
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x * (Math.pow(z, y) / a)) / y;
	double tmp;
	if (y <= -6.5e+140) {
		tmp = t_1;
	} else if (y <= 5e+119) {
		tmp = (x * Math.exp(((Math.log(a) * (t - 1.0)) - b))) / y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (x * (math.pow(z, y) / a)) / y
	tmp = 0
	if y <= -6.5e+140:
		tmp = t_1
	elif y <= 5e+119:
		tmp = (x * math.exp(((math.log(a) * (t - 1.0)) - b))) / y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x * Float64((z ^ y) / a)) / y)
	tmp = 0.0
	if (y <= -6.5e+140)
		tmp = t_1;
	elseif (y <= 5e+119)
		tmp = Float64(Float64(x * exp(Float64(Float64(log(a) * Float64(t - 1.0)) - b))) / y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (x * ((z ^ y) / a)) / y;
	tmp = 0.0;
	if (y <= -6.5e+140)
		tmp = t_1;
	elseif (y <= 5e+119)
		tmp = (x * exp(((log(a) * (t - 1.0)) - b))) / y;
	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[(N[Power[z, y], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[y, -6.5e+140], t$95$1, If[LessEqual[y, 5e+119], N[(N[(x * N[Exp[N[(N[(N[Log[a], $MachinePrecision] * N[(t - 1.0), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -6.4999999999999999e140 or 4.9999999999999999e119 < 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. Taylor expanded in b around 0

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

      1. exp-sum N/A

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

      2. *-commutative N/A

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

      3. pow-to-exp N/A

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

      4. pow-to-exp N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift--.f64 70.4

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

   4. Applied rewrites 70.4%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lift-pow.f64 86.9

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

   7. Applied rewrites 86.9%

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

   if -6.4999999999999999e140 < y < 4.9999999999999999e119

   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. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift--.f64 90.0

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

   4. Applied rewrites 90.0%

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

Alternative 10: 81.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \frac{e^{-b}}{y}\\ \mathbf{if}\;b \leq -7.2 \cdot 10^{+88}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 22000000000:\\ \;\;\;\;x \cdot \frac{{z}^{y} \cdot {a}^{\left(t - 1\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (/ (exp (- b)) y))))
   (if (<= b -7.2e+88)
     t_1
     (if (<= b 22000000000.0)
       (* x (/ (* (pow z y) (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 <= -7.2e+88) {
		tmp = t_1;
	} else if (b <= 22000000000.0) {
		tmp = x * ((pow(z, y) * pow(a, (t - 1.0))) / y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    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 <= (-7.2d+88)) then
        tmp = t_1
    else if (b <= 22000000000.0d0) then
        tmp = x * (((z ** y) * (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 <= -7.2e+88) {
		tmp = t_1;
	} else if (b <= 22000000000.0) {
		tmp = x * ((Math.pow(z, y) * 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 <= -7.2e+88:
		tmp = t_1
	elif b <= 22000000000.0:
		tmp = x * ((math.pow(z, y) * 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 <= -7.2e+88)
		tmp = t_1;
	elseif (b <= 22000000000.0)
		tmp = Float64(x * Float64(Float64((z ^ y) * (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 <= -7.2e+88)
		tmp = t_1;
	elseif (b <= 22000000000.0)
		tmp = x * (((z ^ y) * (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, -7.2e+88], t$95$1, If[LessEqual[b, 22000000000.0], N[(x * N[(N[(N[Power[z, y], $MachinePrecision] * N[Power[a, N[(t - 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if b < -7.2000000000000004e88 or 2.2e10 < 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. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 80.1

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

   4. Applied rewrites 80.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 80.1

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

   6. Applied rewrites 80.1%

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

   if -7.2000000000000004e88 < b < 2.2e10

   1. Initial program 97.2%

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

   2. Taylor expanded in b around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. exp-sum N/A

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

      5. *-commutative N/A

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

      6. pow-to-exp N/A

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

      7. pow-to-exp N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lift--.f64 83.1

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

   4. Applied rewrites 83.1%

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

Alternative 11: 74.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \frac{e^{-b}}{y}\\ \mathbf{if}\;b \leq -6.2 \cdot 10^{+88}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 6200000000:\\ \;\;\;\;\frac{x \cdot {a}^{\left(t - 1\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (/ (exp (- b)) y))))
   (if (<= b -6.2e+88)
     t_1
     (if (<= b 6200000000.0) (/ (* 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 <= -6.2e+88) {
		tmp = t_1;
	} else if (b <= 6200000000.0) {
		tmp = (x * pow(a, (t - 1.0))) / y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    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 <= (-6.2d+88)) then
        tmp = t_1
    else if (b <= 6200000000.0d0) 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 <= -6.2e+88) {
		tmp = t_1;
	} else if (b <= 6200000000.0) {
		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 <= -6.2e+88:
		tmp = t_1
	elif b <= 6200000000.0:
		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 <= -6.2e+88)
		tmp = t_1;
	elseif (b <= 6200000000.0)
		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 <= -6.2e+88)
		tmp = t_1;
	elseif (b <= 6200000000.0)
		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, -6.2e+88], t$95$1, If[LessEqual[b, 6200000000.0], 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 < -6.2000000000000003e88 or 6.2e9 < 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. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 80.1

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

   4. Applied rewrites 80.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 80.1

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

   6. Applied rewrites 80.1%

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

   if -6.2000000000000003e88 < b < 6.2e9

   1. Initial program 97.2%

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

   2. Taylor expanded in b around 0

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

      1. exp-sum N/A

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

      2. *-commutative N/A

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

      3. pow-to-exp N/A

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

      4. pow-to-exp N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift--.f64 83.2

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

   4. Applied rewrites 83.2%

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

   5. Taylor expanded in y around 0

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

      1. pow-to-exp N/A

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

      2. lift-pow.f64 N/A

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

      3. lift--.f64 70.3

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

   7. Applied rewrites 70.3%

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

Alternative 12: 74.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \frac{e^{-b}}{y}\\ \mathbf{if}\;b \leq -7.2 \cdot 10^{+50}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 5800000000:\\ \;\;\;\;x \cdot \frac{{a}^{\left(t - 1\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (/ (exp (- b)) y))))
   (if (<= b -7.2e+50)
     t_1
     (if (<= b 5800000000.0) (* 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 <= -7.2e+50) {
		tmp = t_1;
	} else if (b <= 5800000000.0) {
		tmp = x * (pow(a, (t - 1.0)) / y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    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 <= (-7.2d+50)) then
        tmp = t_1
    else if (b <= 5800000000.0d0) 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 <= -7.2e+50) {
		tmp = t_1;
	} else if (b <= 5800000000.0) {
		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 <= -7.2e+50:
		tmp = t_1
	elif b <= 5800000000.0:
		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 <= -7.2e+50)
		tmp = t_1;
	elseif (b <= 5800000000.0)
		tmp = Float64(x * Float64((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 <= -7.2e+50)
		tmp = t_1;
	elseif (b <= 5800000000.0)
		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, -7.2e+50], t$95$1, If[LessEqual[b, 5800000000.0], N[(x * N[(N[Power[a, N[(t - 1.0), $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if b < -7.19999999999999972e50 or 5.8e9 < 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. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 79.7

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

   4. Applied rewrites 79.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 79.7

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

   6. Applied rewrites 79.7%

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

   if -7.19999999999999972e50 < b < 5.8e9

   1. Initial program 97.1%

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

   2. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. div-exp N/A

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

      5. pow-to-exp N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-exp.f64 71.8

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

   4. Applied rewrites 71.8%

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

   5. Taylor expanded in b around 0

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

      1. pow-to-exp N/A

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

      2. lift-pow.f64 N/A

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

      3. lift--.f64 71.0

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

   7. Applied rewrites 71.0%

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

Alternative 13: 49.5% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{a \cdot y}\\ \mathbf{if}\;b \leq -0.78:\\ \;\;\;\;\mathsf{fma}\left(\left(-b\right) \cdot \left(t\_1 \cdot -0.5\right) - t\_1, b, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 1\right) \cdot y\right) \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (* a y))))
   (if (<= b -0.78)
     (fma (- (* (- b) (* t_1 -0.5)) t_1) b t_1)
     (/
      x
      (* (* (fma (fma (fma 0.16666666666666666 b 0.5) b 1.0) b 1.0) y) a)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / (a * y);
	double tmp;
	if (b <= -0.78) {
		tmp = fma(((-b * (t_1 * -0.5)) - t_1), b, t_1);
	} else {
		tmp = x / ((fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 1.0) * y) * a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x / Float64(a * y))
	tmp = 0.0
	if (b <= -0.78)
		tmp = fma(Float64(Float64(Float64(-b) * Float64(t_1 * -0.5)) - t_1), b, t_1);
	else
		tmp = Float64(x / Float64(Float64(fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 1.0) * y) * a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x / N[(a * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -0.78], N[(N[(N[((-b) * N[(t$95$1 * -0.5), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision] * b + t$95$1), $MachinePrecision], N[(x / N[(N[(N[(N[(N[(0.16666666666666666 * b + 0.5), $MachinePrecision] * b + 1.0), $MachinePrecision] * b + 1.0), $MachinePrecision] * y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < -0.78000000000000003

   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. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. div-exp N/A

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

      5. pow-to-exp N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-exp.f64 67.4

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

   4. Applied rewrites 67.4%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-exp.f64 78.4

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

   7. Applied rewrites 78.4%

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

   8. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   10. Applied rewrites 50.8%

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

   if -0.78000000000000003 < b

   1. Initial program 97.9%

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

   2. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. div-exp N/A

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

      5. pow-to-exp N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-exp.f64 70.6

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

   4. Applied rewrites 70.6%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-exp.f64 52.9

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

   7. Applied rewrites 52.9%

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

   8. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 49.0

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

   10. Applied rewrites 49.0%

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

Alternative 14: 46.4% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{a \cdot y}\\ \mathbf{if}\;b \leq -0.78:\\ \;\;\;\;\mathsf{fma}\left(b \cdot t\_1, -1, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 1\right) \cdot y\right) \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (* a y))))
   (if (<= b -0.78)
     (fma (* b t_1) -1.0 t_1)
     (/
      x
      (* (* (fma (fma (fma 0.16666666666666666 b 0.5) b 1.0) b 1.0) y) a)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / (a * y);
	double tmp;
	if (b <= -0.78) {
		tmp = fma((b * t_1), -1.0, t_1);
	} else {
		tmp = x / ((fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 1.0) * y) * a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x / Float64(a * y))
	tmp = 0.0
	if (b <= -0.78)
		tmp = fma(Float64(b * t_1), -1.0, t_1);
	else
		tmp = Float64(x / Float64(Float64(fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 1.0) * y) * a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x / N[(a * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -0.78], N[(N[(b * t$95$1), $MachinePrecision] * -1.0 + t$95$1), $MachinePrecision], N[(x / N[(N[(N[(N[(N[(0.16666666666666666 * b + 0.5), $MachinePrecision] * b + 1.0), $MachinePrecision] * b + 1.0), $MachinePrecision] * y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < -0.78000000000000003

   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. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. div-exp N/A

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

      5. pow-to-exp N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-exp.f64 67.4

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

   4. Applied rewrites 67.4%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-exp.f64 78.4

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

   7. Applied rewrites 78.4%

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

   8. Taylor expanded in b around 0

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

      1. *-commutative N/A

         \[\leadsto \frac{b \cdot x}{a \cdot y} \cdot -1 + \frac{x}{a \cdot y} \]

      2. lower-fma.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 38.4

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

   10. Applied rewrites 38.4%

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

   if -0.78000000000000003 < b

   1. Initial program 97.9%

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

   2. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. div-exp N/A

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

      5. pow-to-exp N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-exp.f64 70.6

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

   4. Applied rewrites 70.6%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-exp.f64 52.9

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

   7. Applied rewrites 52.9%

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

   8. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 49.0

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

   10. Applied rewrites 49.0%

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

Alternative 15: 35.8% accurate, 9.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.65 \cdot 10^{+20}:\\ \;\;\;\;\frac{x \cdot \frac{1}{a}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(b, y, y\right) \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a 1.65e+20) (/ (* x (/ 1.0 a)) y) (/ x (* (fma b y y) a))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= 1.65e+20) {
		tmp = (x * (1.0 / a)) / y;
	} else {
		tmp = x / (fma(b, y, y) * a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= 1.65e+20)
		tmp = Float64(Float64(x * Float64(1.0 / a)) / y);
	else
		tmp = Float64(x / Float64(fma(b, y, y) * a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, 1.65e+20], N[(N[(x * N[(1.0 / a), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(x / N[(N[(b * y + y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.65e20

   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. Taylor expanded in b around 0

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

      1. exp-sum N/A

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

      2. *-commutative N/A

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

      3. pow-to-exp N/A

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

      4. pow-to-exp N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift--.f64 72.0

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

   4. Applied rewrites 72.0%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lift-pow.f64 61.5

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

   7. Applied rewrites 61.5%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 32.2%

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

   if 1.65e20 < a

   1. Initial program 97.2%

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

   2. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. div-exp N/A

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

      5. pow-to-exp N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-exp.f64 71.8

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

   4. Applied rewrites 71.8%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-exp.f64 61.7

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

   7. Applied rewrites 61.7%

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

   8. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 39.9

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

   10. Applied rewrites 39.9%

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

Alternative 16: 36.6% accurate, 9.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.78:\\ \;\;\;\;x \cdot \frac{\frac{1}{a}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(b, y, y\right) \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -0.78) (* x (/ (/ 1.0 a) y)) (/ x (* (fma b y y) a))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -0.78) {
		tmp = x * ((1.0 / a) / y);
	} else {
		tmp = x / (fma(b, y, y) * a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -0.78)
		tmp = Float64(x * Float64(Float64(1.0 / a) / y));
	else
		tmp = Float64(x / Float64(fma(b, y, y) * a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -0.78], N[(x * N[(N[(1.0 / a), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(b * y + y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -0.78000000000000003

   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. Taylor expanded in b around 0

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

      1. exp-sum N/A

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

      2. *-commutative N/A

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

      3. pow-to-exp N/A

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

      4. pow-to-exp N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift--.f64 57.4

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

   4. Applied rewrites 57.4%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lift-pow.f64 50.0

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

   7. Applied rewrites 50.0%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 26.5%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-/l* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-/.f64 27.9

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

   12. Applied rewrites 27.9%

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

   if -0.78000000000000003 < b

   1. Initial program 97.9%

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

   2. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. div-exp N/A

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

      5. pow-to-exp N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-exp.f64 70.6

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

   4. Applied rewrites 70.6%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-exp.f64 52.9

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

   7. Applied rewrites 52.9%

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

   8. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 39.4

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

   10. Applied rewrites 39.4%

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

Alternative 17: 36.5% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.78:\\ \;\;\;\;\frac{x}{y \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(b, y, y\right) \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -0.78) (/ x (* y a)) (/ x (* (fma b y y) a))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -0.78) {
		tmp = x / (y * a);
	} else {
		tmp = x / (fma(b, y, y) * a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -0.78)
		tmp = Float64(x / Float64(y * a));
	else
		tmp = Float64(x / Float64(fma(b, y, y) * a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -0.78], N[(x / N[(y * a), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(b * y + y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -0.78000000000000003

   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. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. div-exp N/A

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

      5. pow-to-exp N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-exp.f64 67.4

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

   4. Applied rewrites 67.4%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-exp.f64 78.4

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

   7. Applied rewrites 78.4%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 27.4%

       \[\leadsto \frac{x}{y \cdot a} \]

   if -0.78000000000000003 < b

   1. Initial program 97.9%

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

   2. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. div-exp N/A

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

      5. pow-to-exp N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-exp.f64 70.6

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

   4. Applied rewrites 70.6%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-exp.f64 52.9

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

   7. Applied rewrites 52.9%

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

   8. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 39.4

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

   10. Applied rewrites 39.4%

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

Alternative 18: 31.8% accurate, 19.8× speedup

Math

\[\begin{array}{l} \\ \frac{x}{y \cdot a} \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = x / (y * a)
end function

Java

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

Python

def code(x, y, z, t, a, b):
	return x / (y * a)

Julia

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

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = x / (y * a);
end

Wolfram

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

Derivation

1. Initial program 98.4%

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

2. Taylor expanded in y around 0

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

   1. associate-/l* N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. div-exp N/A

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

   5. pow-to-exp N/A

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

   6. lower-/.f64 N/A

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

   7. lower-pow.f64 N/A

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

   8. lift--.f64 N/A

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

   9. lower-exp.f64 69.8

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

4. Applied rewrites 69.8%

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

5. Taylor expanded in t around 0

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lift-exp.f64 59.2

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

7. Applied rewrites 59.2%

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

8. Taylor expanded in b around 0

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

10. Applied rewrites 31.8%

    \[\leadsto \frac{x}{y \cdot a} \]
11. Add Preprocessing

Developer Target 1: 72.9% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    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 2025095 
(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))