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

Percentage Accurate: 98.3% → 98.3%

Time: 11.4s

Alternatives: 16

Speedup: 1.0×

• 48.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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: 98.3% 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: 98.3% 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 98.7%

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

Alternative 2: 42.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-84} \lor \neg \left(t\_1 \leq 5 \cdot 10^{-127}\right):\\ \;\;\;\;\frac{x \cdot \mathsf{fma}\left(\frac{0.5}{a} \cdot b - {a}^{-1}, b, {a}^{-1}\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;{a}^{-1} \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (* x (exp (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))) y)))
   (if (or (<= t_1 -1e-84) (not (<= t_1 5e-127)))
     (/ (* x (fma (- (* (/ 0.5 a) b) (pow a -1.0)) b (pow a -1.0))) y)
     (* (pow a -1.0) (/ x y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x * exp((((y * log(z)) + ((t - 1.0) * log(a))) - b))) / y;
	double tmp;
	if ((t_1 <= -1e-84) || !(t_1 <= 5e-127)) {
		tmp = (x * fma((((0.5 / a) * b) - pow(a, -1.0)), b, pow(a, -1.0))) / y;
	} else {
		tmp = pow(a, -1.0) * (x / y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x * exp(Float64(Float64(Float64(y * log(z)) + Float64(Float64(t - 1.0) * log(a))) - b))) / y)
	tmp = 0.0
	if ((t_1 <= -1e-84) || !(t_1 <= 5e-127))
		tmp = Float64(Float64(x * fma(Float64(Float64(Float64(0.5 / a) * b) - (a ^ -1.0)), b, (a ^ -1.0))) / y);
	else
		tmp = Float64((a ^ -1.0) * Float64(x / y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x * N[Exp[N[(N[(N[(y * N[Log[z], $MachinePrecision]), $MachinePrecision] + N[(N[(t - 1.0), $MachinePrecision] * N[Log[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -1e-84], N[Not[LessEqual[t$95$1, 5e-127]], $MachinePrecision]], N[(N[(x * N[(N[(N[(N[(0.5 / a), $MachinePrecision] * b), $MachinePrecision] - N[Power[a, -1.0], $MachinePrecision]), $MachinePrecision] * b + N[Power[a, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(N[Power[a, -1.0], $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (exp.f64 (-.f64 (+.f64 (*.f64 y (log.f64 z)) (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))) b))) y) < -1e-84 or 4.9999999999999997e-127 < (/.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 99.2%

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

   3. Taylor expanded in t around 0

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

      1. exp-diff N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. distribute-lft-neg-in N/A

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

      7. *-commutative N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg N/A

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

      10. exp-diff N/A

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

      11. rem-exp-log N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. exp-to-pow N/A

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

      15. lower-pow.f64 N/A

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

      16. lower-exp.f64 76.1

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

   5. Applied rewrites 76.1%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 58.3%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 56.5%

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

   if -1e-84 < (/.f64 (*.f64 x (exp.f64 (-.f64 (+.f64 (*.f64 y (log.f64 z)) (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))) b))) y) < 4.9999999999999997e-127

   1. Initial program 98.1%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. exp-sum N/A

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

      3. lower-*.f64 N/A

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

      4. exp-to-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. exp-to-pow N/A

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

      9. lower-pow.f64 73.8

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

   5. Applied rewrites 73.8%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 57.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 57.3

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

   10. Applied rewrites 57.3%

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

   11. Taylor expanded in y around 0

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

   13. Applied rewrites 26.7%

       \[\leadsto \frac{1}{a} \cdot \frac{x}{y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \leq -1 \cdot 10^{-84} \lor \neg \left(\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \leq 5 \cdot 10^{-127}\right):\\ \;\;\;\;\frac{x \cdot \mathsf{fma}\left(\frac{0.5}{a} \cdot b - {a}^{-1}, b, {a}^{-1}\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;{a}^{-1} \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t - 1\right) \cdot \log a\\ t_2 := \frac{{a}^{\left(t - 1\right)}}{y} \cdot x\\ \mathbf{if}\;t\_1 \leq -660:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -600:\\ \;\;\;\;e^{-b} \cdot \frac{x}{y}\\ \mathbf{elif}\;t\_1 \leq 690:\\ \;\;\;\;\frac{x}{a} \cdot \frac{{z}^{y}}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- t 1.0) (log a))) (t_2 (* (/ (pow a (- t 1.0)) y) x)))
   (if (<= t_1 -660.0)
     t_2
     (if (<= t_1 -600.0)
       (* (exp (- b)) (/ x y))
       (if (<= t_1 690.0) (* (/ x a) (/ (pow z y) 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 = (pow(a, (t - 1.0)) / y) * x;
	double tmp;
	if (t_1 <= -660.0) {
		tmp = t_2;
	} else if (t_1 <= -600.0) {
		tmp = exp(-b) * (x / y);
	} else if (t_1 <= 690.0) {
		tmp = (x / a) * (pow(z, y) / 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 = (t - 1.0d0) * log(a)
    t_2 = ((a ** (t - 1.0d0)) / y) * x
    if (t_1 <= (-660.0d0)) then
        tmp = t_2
    else if (t_1 <= (-600.0d0)) then
        tmp = exp(-b) * (x / y)
    else if (t_1 <= 690.0d0) then
        tmp = (x / a) * ((z ** y) / y)
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - 1.0) * Math.log(a);
	double t_2 = (Math.pow(a, (t - 1.0)) / y) * x;
	double tmp;
	if (t_1 <= -660.0) {
		tmp = t_2;
	} else if (t_1 <= -600.0) {
		tmp = Math.exp(-b) * (x / y);
	} else if (t_1 <= 690.0) {
		tmp = (x / a) * (Math.pow(z, y) / y);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (t - 1.0) * math.log(a)
	t_2 = (math.pow(a, (t - 1.0)) / y) * x
	tmp = 0
	if t_1 <= -660.0:
		tmp = t_2
	elif t_1 <= -600.0:
		tmp = math.exp(-b) * (x / y)
	elif t_1 <= 690.0:
		tmp = (x / a) * (math.pow(z, y) / y)
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - 1.0) * log(a))
	t_2 = Float64(Float64((a ^ Float64(t - 1.0)) / y) * x)
	tmp = 0.0
	if (t_1 <= -660.0)
		tmp = t_2;
	elseif (t_1 <= -600.0)
		tmp = Float64(exp(Float64(-b)) * Float64(x / y));
	elseif (t_1 <= 690.0)
		tmp = Float64(Float64(x / a) * Float64((z ^ y) / y));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (t - 1.0) * log(a);
	t_2 = ((a ^ (t - 1.0)) / y) * x;
	tmp = 0.0;
	if (t_1 <= -660.0)
		tmp = t_2;
	elseif (t_1 <= -600.0)
		tmp = exp(-b) * (x / y);
	elseif (t_1 <= 690.0)
		tmp = (x / a) * ((z ^ y) / y);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - 1.0), $MachinePrecision] * N[Log[a], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Power[a, N[(t - 1.0), $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[t$95$1, -660.0], t$95$2, If[LessEqual[t$95$1, -600.0], N[(N[Exp[(-b)], $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 690.0], N[(N[(x / a), $MachinePrecision] * N[(N[Power[z, y], $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)) < -660 or 690 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))

   1. Initial program 99.3%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. exp-sum N/A

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

      3. lower-*.f64 N/A

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

      4. exp-to-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. exp-to-pow N/A

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

      9. lower-pow.f64 74.8

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

   5. Applied rewrites 74.8%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 81.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 82.1

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

   10. Applied rewrites 82.1%

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

   if -660 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < -600

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

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

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

      2. metadata-eval N/A

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

      3. fp-cancel-sign-sub-inv N/A

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

      4. distribute-rgt-out N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. remove-double-neg N/A

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

      9. distribute-neg-in N/A

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

      10. mul-1-neg N/A

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

      11. lower-*.f64 N/A

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

      12. mul-1-neg N/A

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

      13. distribute-neg-in N/A

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

      14. metadata-eval N/A

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

      15. remove-double-neg N/A

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

      16. lower-+.f64 N/A

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

      17. lower-log.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 100.0

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

   8. Applied rewrites 100.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 100.0

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

   10. Applied rewrites 100.0%

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

   if -600 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < 690

   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. Add Preprocessing

   3. Taylor expanded in y around 0

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

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

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

      2. metadata-eval N/A

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

      3. fp-cancel-sign-sub-inv N/A

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

      4. distribute-rgt-out N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. remove-double-neg N/A

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

      9. distribute-neg-in N/A

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

      10. mul-1-neg N/A

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

      11. lower-*.f64 N/A

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

      12. mul-1-neg N/A

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

      13. distribute-neg-in N/A

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

      14. metadata-eval N/A

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

      15. remove-double-neg N/A

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

      16. lower-+.f64 N/A

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

      17. lower-log.f64 73.6

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

   5. Applied rewrites 73.6%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 79.6%

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

   9. Taylor expanded in t around 0

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

   11. Applied rewrites 79.0%

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

Alternative 4: 92.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -31000 \lor \neg \left(y \leq 1.18 \cdot 10^{+64}\right):\\ \;\;\;\;\frac{x \cdot e^{\mathsf{fma}\left(\log z, y, -\log a\right) - b}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot e^{\left(-1 + t\right) \cdot \log a - b}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -31000.0) (not (<= y 1.18e+64)))
   (/ (* x (exp (- (fma (log z) y (- (log a))) b))) y)
   (/ (* x (exp (- (* (+ -1.0 t) (log a)) b))) y)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -31000.0) || !(y <= 1.18e+64)) {
		tmp = (x * exp((fma(log(z), y, -log(a)) - b))) / y;
	} else {
		tmp = (x * exp((((-1.0 + t) * log(a)) - b))) / y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -31000.0) || !(y <= 1.18e+64))
		tmp = Float64(Float64(x * exp(Float64(fma(log(z), y, Float64(-log(a))) - b))) / y);
	else
		tmp = Float64(Float64(x * exp(Float64(Float64(Float64(-1.0 + t) * log(a)) - b))) / y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -31000.0], N[Not[LessEqual[y, 1.18e+64]], $MachinePrecision]], N[(N[(x * N[Exp[N[(N[(N[Log[z], $MachinePrecision] * y + (-N[Log[a], $MachinePrecision])), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(N[(x * N[Exp[N[(N[(N[(-1.0 + t), $MachinePrecision] * N[Log[a], $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -31000 or 1.18000000000000006e64 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. fp-cancel-sign-sub-inv N/A

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

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

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

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

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

      6. metadata-eval N/A

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

      7. fp-cancel-sub-sign-inv N/A

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

      8. *-commutative N/A

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

      9. mul-1-neg N/A

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

      10. *-rgt-identity N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-log.f64 94.3

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

   5. Applied rewrites 94.3%

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

   if -31000 < y < 1.18000000000000006e64

   1. Initial program 97.6%

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

   3. Taylor expanded in y around 0

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

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

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

      2. metadata-eval N/A

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

      3. fp-cancel-sign-sub-inv N/A

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

      4. distribute-rgt-out N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. remove-double-neg N/A

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

      9. distribute-neg-in N/A

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

      10. mul-1-neg N/A

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

      11. lower-*.f64 N/A

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

      12. mul-1-neg N/A

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

      13. distribute-neg-in N/A

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

      14. metadata-eval N/A

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

      15. remove-double-neg N/A

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

      16. lower-+.f64 N/A

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

      17. lower-log.f64 97.6

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

   5. Applied rewrites 97.6%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -31000 \lor \neg \left(y \leq 1.18 \cdot 10^{+64}\right):\\ \;\;\;\;\frac{x \cdot e^{\mathsf{fma}\left(\log z, y, -\log a\right) - b}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot e^{\left(-1 + t\right) \cdot \log a - b}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 51.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6.4 \cdot 10^{+32} \lor \neg \left(b \leq 1.1 \cdot 10^{+60}\right):\\ \;\;\;\;e^{-b} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \mathsf{fma}\left(\frac{0.5}{a} \cdot b - {a}^{-1}, b, {a}^{-1}\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= b -6.4e+32) (not (<= b 1.1e+60)))
   (* (exp (- b)) (/ x y))
   (/ (* x (fma (- (* (/ 0.5 a) b) (pow a -1.0)) b (pow a -1.0))) y)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((b <= -6.4e+32) || !(b <= 1.1e+60)) {
		tmp = exp(-b) * (x / y);
	} else {
		tmp = (x * fma((((0.5 / a) * b) - pow(a, -1.0)), b, pow(a, -1.0))) / y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((b <= -6.4e+32) || !(b <= 1.1e+60))
		tmp = Float64(exp(Float64(-b)) * Float64(x / y));
	else
		tmp = Float64(Float64(x * fma(Float64(Float64(Float64(0.5 / a) * b) - (a ^ -1.0)), b, (a ^ -1.0))) / y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[b, -6.4e+32], N[Not[LessEqual[b, 1.1e+60]], $MachinePrecision]], N[(N[Exp[(-b)], $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(N[(N[(N[(0.5 / a), $MachinePrecision] * b), $MachinePrecision] - N[Power[a, -1.0], $MachinePrecision]), $MachinePrecision] * b + N[Power[a, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -6.3999999999999998e32 or 1.09999999999999998e60 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

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

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

      2. metadata-eval N/A

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

      3. fp-cancel-sign-sub-inv N/A

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

      4. distribute-rgt-out N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. remove-double-neg N/A

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

      9. distribute-neg-in N/A

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

      10. mul-1-neg N/A

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

      11. lower-*.f64 N/A

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

      12. mul-1-neg N/A

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

      13. distribute-neg-in N/A

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

      14. metadata-eval N/A

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

      15. remove-double-neg N/A

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

      16. lower-+.f64 N/A

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

      17. lower-log.f64 93.7

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

   5. Applied rewrites 93.7%

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

   6. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 81.0

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

   8. Applied rewrites 81.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 72.7

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

   10. Applied rewrites 72.7%

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

   if -6.3999999999999998e32 < b < 1.09999999999999998e60

   1. Initial program 97.8%

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

   3. Taylor expanded in t around 0

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

      1. exp-diff N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. distribute-lft-neg-in N/A

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

      7. *-commutative N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg N/A

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

      10. exp-diff N/A

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

      11. rem-exp-log N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. exp-to-pow N/A

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

      15. lower-pow.f64 N/A

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

      16. lower-exp.f64 73.5

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

   5. Applied rewrites 73.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 42.1%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 40.8%

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

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.4 \cdot 10^{+32} \lor \neg \left(b \leq 1.1 \cdot 10^{+60}\right):\\ \;\;\;\;e^{-b} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \mathsf{fma}\left(\frac{0.5}{a} \cdot b - {a}^{-1}, b, {a}^{-1}\right)}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 40.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3 \cdot 10^{-232}:\\ \;\;\;\;\frac{x \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.16666666666666666, b, 0.5\right)}{a} \cdot b - {a}^{-1}, b, {a}^{-1}\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;{a}^{-1} \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b 3e-232)
   (/
    (*
     x
     (fma
      (- (* (/ (fma -0.16666666666666666 b 0.5) a) b) (pow a -1.0))
      b
      (pow a -1.0)))
    y)
   (* (pow a -1.0) (/ x y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= 3e-232) {
		tmp = (x * fma((((fma(-0.16666666666666666, b, 0.5) / a) * b) - pow(a, -1.0)), b, pow(a, -1.0))) / y;
	} else {
		tmp = pow(a, -1.0) * (x / y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= 3e-232)
		tmp = Float64(Float64(x * fma(Float64(Float64(Float64(fma(-0.16666666666666666, b, 0.5) / a) * b) - (a ^ -1.0)), b, (a ^ -1.0))) / y);
	else
		tmp = Float64((a ^ -1.0) * Float64(x / y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, 3e-232], N[(N[(x * N[(N[(N[(N[(N[(-0.16666666666666666 * b + 0.5), $MachinePrecision] / a), $MachinePrecision] * b), $MachinePrecision] - N[Power[a, -1.0], $MachinePrecision]), $MachinePrecision] * b + N[Power[a, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(N[Power[a, -1.0], $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.9999999999999999e-232

   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. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. exp-diff N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. distribute-lft-neg-in N/A

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

      7. *-commutative N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg N/A

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

      10. exp-diff N/A

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

      11. rem-exp-log N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. exp-to-pow N/A

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

      15. lower-pow.f64 N/A

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

      16. lower-exp.f64 77.5

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

   5. Applied rewrites 77.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 58.2%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 53.2%

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

   if 2.9999999999999999e-232 < b

   1. Initial program 98.7%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. exp-sum N/A

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

      3. lower-*.f64 N/A

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

      4. exp-to-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. exp-to-pow N/A

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

      9. lower-pow.f64 74.8

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

   5. Applied rewrites 74.8%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 62.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 58.0

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

   10. Applied rewrites 58.0%

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

   11. Taylor expanded in y around 0

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

   13. Applied rewrites 31.3%

       \[\leadsto \frac{1}{a} \cdot \frac{x}{y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3 \cdot 10^{-232}:\\ \;\;\;\;\frac{x \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.16666666666666666, b, 0.5\right)}{a} \cdot b - {a}^{-1}, b, {a}^{-1}\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;{a}^{-1} \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot \frac{{z}^{y}}{a}}{y}\\ \mathbf{if}\;y \leq -3.25 \cdot 10^{-8}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -2.5 \cdot 10^{-236}:\\ \;\;\;\;\frac{{a}^{\left(t - 1\right)}}{e^{b} \cdot y} \cdot x\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{+70}:\\ \;\;\;\;\frac{x \cdot e^{\log a \cdot t - b}}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (* x (/ (pow z y) a)) y)))
   (if (<= y -3.25e-8)
     t_1
     (if (<= y -2.5e-236)
       (* (/ (pow a (- t 1.0)) (* (exp b) y)) x)
       (if (<= y 9.5e+70) (/ (* x (exp (- (* (log a) t) 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 <= -3.25e-8) {
		tmp = t_1;
	} else if (y <= -2.5e-236) {
		tmp = (pow(a, (t - 1.0)) / (exp(b) * y)) * x;
	} else if (y <= 9.5e+70) {
		tmp = (x * exp(((log(a) * t) - 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 <= (-3.25d-8)) then
        tmp = t_1
    else if (y <= (-2.5d-236)) then
        tmp = ((a ** (t - 1.0d0)) / (exp(b) * y)) * x
    else if (y <= 9.5d+70) then
        tmp = (x * exp(((log(a) * t) - 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 <= -3.25e-8) {
		tmp = t_1;
	} else if (y <= -2.5e-236) {
		tmp = (Math.pow(a, (t - 1.0)) / (Math.exp(b) * y)) * x;
	} else if (y <= 9.5e+70) {
		tmp = (x * Math.exp(((Math.log(a) * t) - 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 <= -3.25e-8:
		tmp = t_1
	elif y <= -2.5e-236:
		tmp = (math.pow(a, (t - 1.0)) / (math.exp(b) * y)) * x
	elif y <= 9.5e+70:
		tmp = (x * math.exp(((math.log(a) * t) - 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 <= -3.25e-8)
		tmp = t_1;
	elseif (y <= -2.5e-236)
		tmp = Float64(Float64((a ^ Float64(t - 1.0)) / Float64(exp(b) * y)) * x);
	elseif (y <= 9.5e+70)
		tmp = Float64(Float64(x * exp(Float64(Float64(log(a) * t) - 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 <= -3.25e-8)
		tmp = t_1;
	elseif (y <= -2.5e-236)
		tmp = ((a ^ (t - 1.0)) / (exp(b) * y)) * x;
	elseif (y <= 9.5e+70)
		tmp = (x * exp(((log(a) * t) - 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, -3.25e-8], t$95$1, If[LessEqual[y, -2.5e-236], N[(N[(N[Power[a, N[(t - 1.0), $MachinePrecision]], $MachinePrecision] / N[(N[Exp[b], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[y, 9.5e+70], N[(N[(x * N[Exp[N[(N[(N[Log[a], $MachinePrecision] * t), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.24999999999999999e-8 or 9.5000000000000002e70 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. exp-sum N/A

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

      3. lower-*.f64 N/A

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

      4. exp-to-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. exp-to-pow N/A

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

      9. lower-pow.f64 80.4

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

   5. Applied rewrites 80.4%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 91.1%

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

   if -3.24999999999999999e-8 < y < -2.4999999999999999e-236

   1. Initial program 96.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 91.5%

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

   5. Taylor expanded in y around 0

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

      1. exp-to-pow N/A

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

      2. lower-pow.f64 N/A

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

      3. lower--.f64 91.5

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

   7. Applied rewrites 91.5%

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

   if -2.4999999999999999e-236 < y < 9.5000000000000002e70

   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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 84.3

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

   5. Applied rewrites 84.3%

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

Alternative 8: 80.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot e^{\log a \cdot t - b}}{y}\\ \mathbf{if}\;b \leq -900:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 3 \cdot 10^{-214}:\\ \;\;\;\;\frac{x \cdot \frac{{z}^{y}}{a}}{y}\\ \mathbf{elif}\;b \leq 1500000:\\ \;\;\;\;\frac{{a}^{\left(t - 1\right)}}{y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (* x (exp (- (* (log a) t) b))) y)))
   (if (<= b -900.0)
     t_1
     (if (<= b 3e-214)
       (/ (* x (/ (pow z y) a)) y)
       (if (<= b 1500000.0) (* (/ (pow a (- t 1.0)) y) x) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x * exp(((log(a) * t) - b))) / y;
	double tmp;
	if (b <= -900.0) {
		tmp = t_1;
	} else if (b <= 3e-214) {
		tmp = (x * (pow(z, y) / a)) / y;
	} else if (b <= 1500000.0) {
		tmp = (pow(a, (t - 1.0)) / y) * x;
	} 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(((log(a) * t) - b))) / y
    if (b <= (-900.0d0)) then
        tmp = t_1
    else if (b <= 3d-214) then
        tmp = (x * ((z ** y) / a)) / y
    else if (b <= 1500000.0d0) then
        tmp = ((a ** (t - 1.0d0)) / y) * x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x * Math.exp(((Math.log(a) * t) - b))) / y;
	double tmp;
	if (b <= -900.0) {
		tmp = t_1;
	} else if (b <= 3e-214) {
		tmp = (x * (Math.pow(z, y) / a)) / y;
	} else if (b <= 1500000.0) {
		tmp = (Math.pow(a, (t - 1.0)) / y) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (x * math.exp(((math.log(a) * t) - b))) / y
	tmp = 0
	if b <= -900.0:
		tmp = t_1
	elif b <= 3e-214:
		tmp = (x * (math.pow(z, y) / a)) / y
	elif b <= 1500000.0:
		tmp = (math.pow(a, (t - 1.0)) / y) * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x * exp(Float64(Float64(log(a) * t) - b))) / y)
	tmp = 0.0
	if (b <= -900.0)
		tmp = t_1;
	elseif (b <= 3e-214)
		tmp = Float64(Float64(x * Float64((z ^ y) / a)) / y);
	elseif (b <= 1500000.0)
		tmp = Float64(Float64((a ^ Float64(t - 1.0)) / y) * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (x * exp(((log(a) * t) - b))) / y;
	tmp = 0.0;
	if (b <= -900.0)
		tmp = t_1;
	elseif (b <= 3e-214)
		tmp = (x * ((z ^ y) / a)) / y;
	elseif (b <= 1500000.0)
		tmp = ((a ^ (t - 1.0)) / y) * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x * N[Exp[N[(N[(N[Log[a], $MachinePrecision] * t), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[b, -900.0], t$95$1, If[LessEqual[b, 3e-214], N[(N[(x * N[(N[Power[z, y], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[b, 1500000.0], N[(N[(N[Power[a, N[(t - 1.0), $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if b < -900 or 1.5e6 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 93.8

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

   5. Applied rewrites 93.8%

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

   if -900 < b < 2.99999999999999994e-214

   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. Add Preprocessing

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. exp-sum N/A

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

      3. lower-*.f64 N/A

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

      4. exp-to-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. exp-to-pow N/A

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

      9. lower-pow.f64 90.8

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

   5. Applied rewrites 90.8%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 81.5%

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

   if 2.99999999999999994e-214 < b < 1.5e6

   1. Initial program 96.8%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. exp-sum N/A

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

      3. lower-*.f64 N/A

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

      4. exp-to-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. exp-to-pow N/A

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

      9. lower-pow.f64 88.3

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

   5. Applied rewrites 88.3%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 79.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 82.9

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

   10. Applied rewrites 82.9%

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

Alternative 9: 88.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -58000 \lor \neg \left(y \leq 9.5 \cdot 10^{+70}\right):\\ \;\;\;\;\frac{x \cdot \frac{{z}^{y}}{a}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot e^{\left(-1 + t\right) \cdot \log a - b}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -58000.0) (not (<= y 9.5e+70)))
   (/ (* x (/ (pow z y) a)) y)
   (/ (* x (exp (- (* (+ -1.0 t) (log a)) b))) y)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -58000.0) || !(y <= 9.5e+70)) {
		tmp = (x * (pow(z, y) / a)) / y;
	} else {
		tmp = (x * exp((((-1.0 + t) * log(a)) - b))) / 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) :: tmp
    if ((y <= (-58000.0d0)) .or. (.not. (y <= 9.5d+70))) then
        tmp = (x * ((z ** y) / a)) / y
    else
        tmp = (x * exp(((((-1.0d0) + t) * log(a)) - b))) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -58000.0) || !(y <= 9.5e+70)) {
		tmp = (x * (Math.pow(z, y) / a)) / y;
	} else {
		tmp = (x * Math.exp((((-1.0 + t) * Math.log(a)) - b))) / y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -58000.0) or not (y <= 9.5e+70):
		tmp = (x * (math.pow(z, y) / a)) / y
	else:
		tmp = (x * math.exp((((-1.0 + t) * math.log(a)) - b))) / y
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -58000.0) || !(y <= 9.5e+70))
		tmp = Float64(Float64(x * Float64((z ^ y) / a)) / y);
	else
		tmp = Float64(Float64(x * exp(Float64(Float64(Float64(-1.0 + t) * log(a)) - b))) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -58000.0) || ~((y <= 9.5e+70)))
		tmp = (x * ((z ^ y) / a)) / y;
	else
		tmp = (x * exp((((-1.0 + t) * log(a)) - b))) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -58000.0], N[Not[LessEqual[y, 9.5e+70]], $MachinePrecision]], N[(N[(x * N[(N[Power[z, y], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(N[(x * N[Exp[N[(N[(N[(-1.0 + t), $MachinePrecision] * N[Log[a], $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -58000 or 9.5000000000000002e70 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. exp-sum N/A

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

      3. lower-*.f64 N/A

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

      4. exp-to-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. exp-to-pow N/A

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

      9. lower-pow.f64 79.9

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

   5. Applied rewrites 79.9%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 91.7%

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

   if -58000 < y < 9.5000000000000002e70

   1. Initial program 97.6%

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

   3. Taylor expanded in y around 0

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

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

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

      2. metadata-eval N/A

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

      3. fp-cancel-sign-sub-inv N/A

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

      4. distribute-rgt-out N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. remove-double-neg N/A

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

      9. distribute-neg-in N/A

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

      10. mul-1-neg N/A

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

      11. lower-*.f64 N/A

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

      12. mul-1-neg N/A

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

      13. distribute-neg-in N/A

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

      14. metadata-eval N/A

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

      15. remove-double-neg N/A

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

      16. lower-+.f64 N/A

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

      17. lower-log.f64 97.6

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

   5. Applied rewrites 97.6%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -58000 \lor \neg \left(y \leq 9.5 \cdot 10^{+70}\right):\\ \;\;\;\;\frac{x \cdot \frac{{z}^{y}}{a}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot e^{\left(-1 + t\right) \cdot \log a - b}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 86.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -940 \lor \neg \left(b \leq 5800000\right):\\ \;\;\;\;\frac{x \cdot e^{\log a \cdot t - b}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left({a}^{\left(t - 1\right)} \cdot {z}^{y}\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= b -940.0) (not (<= b 5800000.0)))
   (/ (* x (exp (- (* (log a) t) b))) y)
   (/ (* x (* (pow a (- t 1.0)) (pow z y))) y)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((b <= -940.0) || !(b <= 5800000.0)) {
		tmp = (x * exp(((log(a) * t) - b))) / y;
	} else {
		tmp = (x * (pow(a, (t - 1.0)) * pow(z, y))) / 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) :: tmp
    if ((b <= (-940.0d0)) .or. (.not. (b <= 5800000.0d0))) then
        tmp = (x * exp(((log(a) * t) - b))) / y
    else
        tmp = (x * ((a ** (t - 1.0d0)) * (z ** y))) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((b <= -940.0) || !(b <= 5800000.0)) {
		tmp = (x * Math.exp(((Math.log(a) * t) - b))) / y;
	} else {
		tmp = (x * (Math.pow(a, (t - 1.0)) * Math.pow(z, y))) / y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (b <= -940.0) or not (b <= 5800000.0):
		tmp = (x * math.exp(((math.log(a) * t) - b))) / y
	else:
		tmp = (x * (math.pow(a, (t - 1.0)) * math.pow(z, y))) / y
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((b <= -940.0) || !(b <= 5800000.0))
		tmp = Float64(Float64(x * exp(Float64(Float64(log(a) * t) - b))) / y);
	else
		tmp = Float64(Float64(x * Float64((a ^ Float64(t - 1.0)) * (z ^ y))) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((b <= -940.0) || ~((b <= 5800000.0)))
		tmp = (x * exp(((log(a) * t) - b))) / y;
	else
		tmp = (x * ((a ^ (t - 1.0)) * (z ^ y))) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[b, -940.0], N[Not[LessEqual[b, 5800000.0]], $MachinePrecision]], N[(N[(x * N[Exp[N[(N[(N[Log[a], $MachinePrecision] * t), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(N[(x * N[(N[Power[a, N[(t - 1.0), $MachinePrecision]], $MachinePrecision] * N[Power[z, y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -940 or 5.8e6 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 93.8

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

   5. Applied rewrites 93.8%

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

   if -940 < b < 5.8e6

   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. Add Preprocessing

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. exp-sum N/A

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

      3. lower-*.f64 N/A

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

      4. exp-to-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. exp-to-pow N/A

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

      9. lower-pow.f64 89.8

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

   5. Applied rewrites 89.8%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -940 \lor \neg \left(b \leq 5800000\right):\\ \;\;\;\;\frac{x \cdot e^{\log a \cdot t - b}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left({a}^{\left(t - 1\right)} \cdot {z}^{y}\right)}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 85.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -900 \lor \neg \left(b \leq 4000000\right):\\ \;\;\;\;\frac{x \cdot e^{\log a \cdot t - b}}{y}\\ \mathbf{else}:\\ \;\;\;\;\left({a}^{\left(t - 1\right)} \cdot x\right) \cdot \frac{{z}^{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= b -900.0) (not (<= b 4000000.0)))
   (/ (* x (exp (- (* (log a) t) b))) y)
   (* (* (pow a (- t 1.0)) x) (/ (pow z y) y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((b <= -900.0) || !(b <= 4000000.0)) {
		tmp = (x * exp(((log(a) * t) - b))) / y;
	} else {
		tmp = (pow(a, (t - 1.0)) * x) * (pow(z, y) / 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) :: tmp
    if ((b <= (-900.0d0)) .or. (.not. (b <= 4000000.0d0))) then
        tmp = (x * exp(((log(a) * t) - b))) / y
    else
        tmp = ((a ** (t - 1.0d0)) * x) * ((z ** y) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((b <= -900.0) || !(b <= 4000000.0)) {
		tmp = (x * Math.exp(((Math.log(a) * t) - b))) / y;
	} else {
		tmp = (Math.pow(a, (t - 1.0)) * x) * (Math.pow(z, y) / y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (b <= -900.0) or not (b <= 4000000.0):
		tmp = (x * math.exp(((math.log(a) * t) - b))) / y
	else:
		tmp = (math.pow(a, (t - 1.0)) * x) * (math.pow(z, y) / y)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((b <= -900.0) || !(b <= 4000000.0))
		tmp = Float64(Float64(x * exp(Float64(Float64(log(a) * t) - b))) / y);
	else
		tmp = Float64(Float64((a ^ Float64(t - 1.0)) * x) * Float64((z ^ y) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((b <= -900.0) || ~((b <= 4000000.0)))
		tmp = (x * exp(((log(a) * t) - b))) / y;
	else
		tmp = ((a ^ (t - 1.0)) * x) * ((z ^ y) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[b, -900.0], N[Not[LessEqual[b, 4000000.0]], $MachinePrecision]], N[(N[(x * N[Exp[N[(N[(N[Log[a], $MachinePrecision] * t), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(N[(N[Power[a, N[(t - 1.0), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision] * N[(N[Power[z, y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -900 or 4e6 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 93.8

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

   5. Applied rewrites 93.8%

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

   if -900 < b < 4e6

   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. Add Preprocessing

   3. Taylor expanded in y around 0

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

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

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

      2. metadata-eval N/A

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

      3. fp-cancel-sign-sub-inv N/A

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

      4. distribute-rgt-out N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. remove-double-neg N/A

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

      9. distribute-neg-in N/A

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

      10. mul-1-neg N/A

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

      11. lower-*.f64 N/A

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

      12. mul-1-neg N/A

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

      13. distribute-neg-in N/A

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

      14. metadata-eval N/A

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

      15. remove-double-neg N/A

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

      16. lower-+.f64 N/A

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

      17. lower-log.f64 72.2

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

   5. Applied rewrites 72.2%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 89.0%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -900 \lor \neg \left(b \leq 4000000\right):\\ \;\;\;\;\frac{x \cdot e^{\log a \cdot t - b}}{y}\\ \mathbf{else}:\\ \;\;\;\;\left({a}^{\left(t - 1\right)} \cdot x\right) \cdot \frac{{z}^{y}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 75.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -40000 \lor \neg \left(y \leq 1.18 \cdot 10^{+64}\right):\\ \;\;\;\;\frac{x \cdot \frac{{z}^{y}}{a}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{{a}^{\left(t - 1\right)}}{y} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -40000.0) (not (<= y 1.18e+64)))
   (/ (* x (/ (pow z y) a)) y)
   (* (/ (pow a (- t 1.0)) y) x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -40000.0) || !(y <= 1.18e+64)) {
		tmp = (x * (pow(z, y) / a)) / y;
	} else {
		tmp = (pow(a, (t - 1.0)) / y) * x;
	}
	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) :: tmp
    if ((y <= (-40000.0d0)) .or. (.not. (y <= 1.18d+64))) then
        tmp = (x * ((z ** y) / a)) / y
    else
        tmp = ((a ** (t - 1.0d0)) / y) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -40000.0) || !(y <= 1.18e+64)) {
		tmp = (x * (Math.pow(z, y) / a)) / y;
	} else {
		tmp = (Math.pow(a, (t - 1.0)) / y) * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -40000.0) or not (y <= 1.18e+64):
		tmp = (x * (math.pow(z, y) / a)) / y
	else:
		tmp = (math.pow(a, (t - 1.0)) / y) * x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -40000.0) || !(y <= 1.18e+64))
		tmp = Float64(Float64(x * Float64((z ^ y) / a)) / y);
	else
		tmp = Float64(Float64((a ^ Float64(t - 1.0)) / y) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -40000.0) || ~((y <= 1.18e+64)))
		tmp = (x * ((z ^ y) / a)) / y;
	else
		tmp = ((a ^ (t - 1.0)) / y) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -40000.0], N[Not[LessEqual[y, 1.18e+64]], $MachinePrecision]], N[(N[(x * N[(N[Power[z, y], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(N[(N[Power[a, N[(t - 1.0), $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4e4 or 1.18000000000000006e64 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. exp-sum N/A

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

      3. lower-*.f64 N/A

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

      4. exp-to-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. exp-to-pow N/A

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

      9. lower-pow.f64 79.2

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

   5. Applied rewrites 79.2%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 91.0%

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

   if -4e4 < y < 1.18000000000000006e64

   1. Initial program 97.6%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. exp-sum N/A

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

      3. lower-*.f64 N/A

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

      4. exp-to-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. exp-to-pow N/A

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

      9. lower-pow.f64 72.5

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

   5. Applied rewrites 72.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 76.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 77.2

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

   10. Applied rewrites 77.2%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -40000 \lor \neg \left(y \leq 1.18 \cdot 10^{+64}\right):\\ \;\;\;\;\frac{x \cdot \frac{{z}^{y}}{a}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{{a}^{\left(t - 1\right)}}{y} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 73.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.4 \cdot 10^{+83} \lor \neg \left(b \leq 1.1 \cdot 10^{+60}\right):\\ \;\;\;\;\frac{e^{-b}}{y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{{a}^{\left(t - 1\right)}}{y} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= b -3.4e+83) (not (<= b 1.1e+60)))
   (* (/ (exp (- b)) y) x)
   (* (/ (pow a (- t 1.0)) y) x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((b <= -3.4e+83) || !(b <= 1.1e+60)) {
		tmp = (exp(-b) / y) * x;
	} else {
		tmp = (pow(a, (t - 1.0)) / y) * x;
	}
	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) :: tmp
    if ((b <= (-3.4d+83)) .or. (.not. (b <= 1.1d+60))) then
        tmp = (exp(-b) / y) * x
    else
        tmp = ((a ** (t - 1.0d0)) / y) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((b <= -3.4e+83) || !(b <= 1.1e+60)) {
		tmp = (Math.exp(-b) / y) * x;
	} else {
		tmp = (Math.pow(a, (t - 1.0)) / y) * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (b <= -3.4e+83) or not (b <= 1.1e+60):
		tmp = (math.exp(-b) / y) * x
	else:
		tmp = (math.pow(a, (t - 1.0)) / y) * x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((b <= -3.4e+83) || !(b <= 1.1e+60))
		tmp = Float64(Float64(exp(Float64(-b)) / y) * x);
	else
		tmp = Float64(Float64((a ^ Float64(t - 1.0)) / y) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((b <= -3.4e+83) || ~((b <= 1.1e+60)))
		tmp = (exp(-b) / y) * x;
	else
		tmp = ((a ^ (t - 1.0)) / y) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[b, -3.4e+83], N[Not[LessEqual[b, 1.1e+60]], $MachinePrecision]], N[(N[(N[Exp[(-b)], $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision], N[(N[(N[Power[a, N[(t - 1.0), $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -3.3999999999999998e83 or 1.09999999999999998e60 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

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

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

      2. metadata-eval N/A

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

      3. fp-cancel-sign-sub-inv N/A

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

      4. distribute-rgt-out N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. remove-double-neg N/A

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

      9. distribute-neg-in N/A

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

      10. mul-1-neg N/A

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

      11. lower-*.f64 N/A

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

      12. mul-1-neg N/A

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

      13. distribute-neg-in N/A

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

      14. metadata-eval N/A

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

      15. remove-double-neg N/A

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

      16. lower-+.f64 N/A

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

      17. lower-log.f64 93.0

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

   5. Applied rewrites 93.0%

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

   6. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 84.1

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

   8. Applied rewrites 84.1%

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

      1. lift-/.f64 N/A

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

      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. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 84.1

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

   10. Applied rewrites 84.1%

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

   if -3.3999999999999998e83 < b < 1.09999999999999998e60

   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. Add Preprocessing

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. exp-sum N/A

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

      3. lower-*.f64 N/A

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

      4. exp-to-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. exp-to-pow N/A

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

      9. lower-pow.f64 87.8

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

   5. Applied rewrites 87.8%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 74.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 74.8

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

   10. Applied rewrites 74.8%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.4 \cdot 10^{+83} \lor \neg \left(b \leq 1.1 \cdot 10^{+60}\right):\\ \;\;\;\;\frac{e^{-b}}{y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{{a}^{\left(t - 1\right)}}{y} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 57.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6.4 \cdot 10^{+32} \lor \neg \left(b \leq 22500000\right):\\ \;\;\;\;\frac{e^{-b}}{y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;{a}^{-1} \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= b -6.4e+32) (not (<= b 22500000.0)))
   (* (/ (exp (- b)) y) x)
   (* (pow a -1.0) (/ x y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((b <= -6.4e+32) || !(b <= 22500000.0)) {
		tmp = (exp(-b) / y) * x;
	} else {
		tmp = pow(a, -1.0) * (x / 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) :: tmp
    if ((b <= (-6.4d+32)) .or. (.not. (b <= 22500000.0d0))) then
        tmp = (exp(-b) / y) * x
    else
        tmp = (a ** (-1.0d0)) * (x / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((b <= -6.4e+32) || !(b <= 22500000.0)) {
		tmp = (Math.exp(-b) / y) * x;
	} else {
		tmp = Math.pow(a, -1.0) * (x / y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (b <= -6.4e+32) or not (b <= 22500000.0):
		tmp = (math.exp(-b) / y) * x
	else:
		tmp = math.pow(a, -1.0) * (x / y)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((b <= -6.4e+32) || !(b <= 22500000.0))
		tmp = Float64(Float64(exp(Float64(-b)) / y) * x);
	else
		tmp = Float64((a ^ -1.0) * Float64(x / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((b <= -6.4e+32) || ~((b <= 22500000.0)))
		tmp = (exp(-b) / y) * x;
	else
		tmp = (a ^ -1.0) * (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[b, -6.4e+32], N[Not[LessEqual[b, 22500000.0]], $MachinePrecision]], N[(N[(N[Exp[(-b)], $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision], N[(N[Power[a, -1.0], $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -6.3999999999999998e32 or 2.25e7 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

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

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

      2. metadata-eval N/A

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

      3. fp-cancel-sign-sub-inv N/A

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

      4. distribute-rgt-out N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. remove-double-neg N/A

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

      9. distribute-neg-in N/A

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

      10. mul-1-neg N/A

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

      11. lower-*.f64 N/A

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

      12. mul-1-neg N/A

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

      13. distribute-neg-in N/A

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

      14. metadata-eval N/A

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

      15. remove-double-neg N/A

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

      16. lower-+.f64 N/A

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

      17. lower-log.f64 93.5

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

   5. Applied rewrites 93.5%

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

   6. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 78.8

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

   8. Applied rewrites 78.8%

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

      1. lift-/.f64 N/A

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

      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. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 78.8

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

   10. Applied rewrites 78.8%

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

   if -6.3999999999999998e32 < b < 2.25e7

   1. Initial program 97.6%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. exp-sum N/A

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

      3. lower-*.f64 N/A

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

      4. exp-to-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. exp-to-pow N/A

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

      9. lower-pow.f64 89.4

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

   5. Applied rewrites 89.4%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 77.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 71.9

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

   10. Applied rewrites 71.9%

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

   11. Taylor expanded in y around 0

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

   13. Applied rewrites 40.7%

       \[\leadsto \frac{1}{a} \cdot \frac{x}{y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.4 \cdot 10^{+32} \lor \neg \left(b \leq 22500000\right):\\ \;\;\;\;\frac{e^{-b}}{y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;{a}^{-1} \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 34.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.5 \cdot 10^{-236}:\\ \;\;\;\;\frac{x \cdot \frac{\mathsf{fma}\left(-1, b, 1\right)}{a}}{y}\\ \mathbf{else}:\\ \;\;\;\;{a}^{-1} \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b 5.5e-236)
   (/ (* x (/ (fma -1.0 b 1.0) a)) y)
   (* (pow a -1.0) (/ x y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= 5.5e-236) {
		tmp = (x * (fma(-1.0, b, 1.0) / a)) / y;
	} else {
		tmp = pow(a, -1.0) * (x / y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= 5.5e-236)
		tmp = Float64(Float64(x * Float64(fma(-1.0, b, 1.0) / a)) / y);
	else
		tmp = Float64((a ^ -1.0) * Float64(x / y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, 5.5e-236], N[(N[(x * N[(N[(-1.0 * b + 1.0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(N[Power[a, -1.0], $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.49999999999999959e-236

   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. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. exp-diff N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. distribute-lft-neg-in N/A

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

      7. *-commutative N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg N/A

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

      10. exp-diff N/A

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

      11. rem-exp-log N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. exp-to-pow N/A

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

      15. lower-pow.f64 N/A

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

      16. lower-exp.f64 77.8

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

   5. Applied rewrites 77.8%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 58.3%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 38.1%

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

   if 5.49999999999999959e-236 < b

   1. Initial program 98.6%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. exp-sum N/A

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

      3. lower-*.f64 N/A

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

      4. exp-to-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. exp-to-pow N/A

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

      9. lower-pow.f64 75.0

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

   5. Applied rewrites 75.0%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 62.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 57.9

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

   10. Applied rewrites 57.9%

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

   11. Taylor expanded in y around 0

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

   13. Applied rewrites 31.4%

       \[\leadsto \frac{1}{a} \cdot \frac{x}{y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.5 \cdot 10^{-236}:\\ \;\;\;\;\frac{x \cdot \frac{\mathsf{fma}\left(-1, b, 1\right)}{a}}{y}\\ \mathbf{else}:\\ \;\;\;\;{a}^{-1} \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 30.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ {a}^{-1} \cdot \frac{x}{y} \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double t, double a, double b) {
	return pow(a, -1.0) * (x / 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 = (a ** (-1.0d0)) * (x / y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in b around 0

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

   1. +-commutative N/A

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

   2. exp-sum N/A

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

   3. lower-*.f64 N/A

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

   4. exp-to-pow N/A

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

   5. lower-pow.f64 N/A

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

   6. lower--.f64 N/A

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

   7. *-commutative N/A

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

   8. exp-to-pow N/A

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

   9. lower-pow.f64 75.6

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

5. Applied rewrites 75.6%

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

6. Taylor expanded in t around 0

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

8. Applied rewrites 65.4%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 61.2

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

10. Applied rewrites 61.2%

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

11. Taylor expanded in y around 0

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

13. Applied rewrites 31.6%

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

14. Final simplification 31.6%

    \[\leadsto {a}^{-1} \cdot \frac{x}{y} \]
15. Add Preprocessing

Developer Target 1: 71.6% 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 2024352 
(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))