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

Percentage Accurate: 98.5% → 98.5%

Time: 11.6s

Alternatives: 16

Speedup: 1.0×

• 46.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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 98.3%

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

Alternative 2: 75.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t - 1\right) \cdot \log a\\ \mathbf{if}\;t\_1 \leq -670:\\ \;\;\;\;\frac{\frac{{a}^{t}}{\mathsf{fma}\left(b, a, a\right)} \cdot x}{y}\\ \mathbf{elif}\;t\_1 \leq 540:\\ \;\;\;\;\frac{x \cdot \frac{{z}^{y}}{a}}{y}\\ \mathbf{elif}\;t\_1 \leq 1000:\\ \;\;\;\;\frac{x \cdot {\left(e^{b} \cdot a\right)}^{-1}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot {a}^{\left(t - 1\right)}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- t 1.0) (log a))))
   (if (<= t_1 -670.0)
     (/ (* (/ (pow a t) (fma b a a)) x) y)
     (if (<= t_1 540.0)
       (/ (* x (/ (pow z y) a)) y)
       (if (<= t_1 1000.0)
         (/ (* x (pow (* (exp b) a) -1.0)) y)
         (/ (* x (pow a (- t 1.0))) y))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - 1.0) * log(a);
	double tmp;
	if (t_1 <= -670.0) {
		tmp = ((pow(a, t) / fma(b, a, a)) * x) / y;
	} else if (t_1 <= 540.0) {
		tmp = (x * (pow(z, y) / a)) / y;
	} else if (t_1 <= 1000.0) {
		tmp = (x * pow((exp(b) * a), -1.0)) / y;
	} else {
		tmp = (x * pow(a, (t - 1.0))) / y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - 1.0) * log(a))
	tmp = 0.0
	if (t_1 <= -670.0)
		tmp = Float64(Float64(Float64((a ^ t) / fma(b, a, a)) * x) / y);
	elseif (t_1 <= 540.0)
		tmp = Float64(Float64(x * Float64((z ^ y) / a)) / y);
	elseif (t_1 <= 1000.0)
		tmp = Float64(Float64(x * (Float64(exp(b) * a) ^ -1.0)) / y);
	else
		tmp = Float64(Float64(x * (a ^ Float64(t - 1.0))) / y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - 1.0), $MachinePrecision] * N[Log[a], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -670.0], N[(N[(N[(N[Power[a, t], $MachinePrecision] / N[(b * a + a), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[t$95$1, 540.0], N[(N[(x * N[(N[Power[z, y], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[t$95$1, 1000.0], N[(N[(x * N[Power[N[(N[Exp[b], $MachinePrecision] * a), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(N[(x * N[Power[a, N[(t - 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < -670

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. exp-diff N/A

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

      2. lower-/.f64 N/A

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

      3. exp-to-pow N/A

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

      4. lower-pow.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-exp.f64 83.4

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

   5. Applied rewrites 83.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 83.4

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

   7. Applied rewrites 83.4%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 92.5%

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

   if -670 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < 540

   1. Initial program 96.5%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   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 66.5

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

   5. Applied rewrites 66.5%

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

   6. Taylor expanded in b around 0

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

      1. exp-sum N/A

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

      2. *-commutative N/A

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

      3. exp-to-pow N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. exp-to-pow N/A

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

      7. lower-pow.f64 N/A

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

      8. lower--.f64 81.3

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

   8. Applied rewrites 81.3%

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

   9. Taylor expanded in t around 0

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

   11. Applied rewrites 80.1%

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

   if 540 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < 1e3

   1. Initial program 98.0%

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

   3. Taylor expanded in y around 0

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

      1. exp-diff N/A

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

      2. lower-/.f64 N/A

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

      3. exp-to-pow N/A

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

      4. lower-pow.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-exp.f64 93.2

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

   5. Applied rewrites 93.2%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 89.2%

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

   if 1e3 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot \left(t - 1\right)} - b}}{y} \]
   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 0

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

      1. exp-sum N/A

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

      2. *-commutative N/A

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

      3. exp-to-pow N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. exp-to-pow N/A

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

      7. lower-pow.f64 N/A

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

      8. lower--.f64 69.0

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

   8. Applied rewrites 69.0%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 87.1%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(t - 1\right) \cdot \log a \leq -670:\\ \;\;\;\;\frac{\frac{{a}^{t}}{\mathsf{fma}\left(b, a, a\right)} \cdot x}{y}\\ \mathbf{elif}\;\left(t - 1\right) \cdot \log a \leq 540:\\ \;\;\;\;\frac{x \cdot \frac{{z}^{y}}{a}}{y}\\ \mathbf{elif}\;\left(t - 1\right) \cdot \log a \leq 1000:\\ \;\;\;\;\frac{x \cdot {\left(e^{b} \cdot a\right)}^{-1}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot {a}^{\left(t - 1\right)}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t - 1\right) \cdot \log a\\ t_2 := \frac{x \cdot {a}^{\left(t - 1\right)}}{y}\\ \mathbf{if}\;t\_1 \leq -1000000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 540:\\ \;\;\;\;\frac{x \cdot \frac{{z}^{y}}{a}}{y}\\ \mathbf{elif}\;t\_1 \leq 1000:\\ \;\;\;\;\frac{x \cdot {\left(e^{b} \cdot a\right)}^{-1}}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- t 1.0) (log a))) (t_2 (/ (* x (pow a (- t 1.0))) y)))
   (if (<= t_1 -1000000000.0)
     t_2
     (if (<= t_1 540.0)
       (/ (* x (/ (pow z y) a)) y)
       (if (<= t_1 1000.0) (/ (* x (pow (* (exp b) a) -1.0)) y) t_2)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - 1.0) * log(a);
	double t_2 = (x * pow(a, (t - 1.0))) / y;
	double tmp;
	if (t_1 <= -1000000000.0) {
		tmp = t_2;
	} else if (t_1 <= 540.0) {
		tmp = (x * (pow(z, y) / a)) / y;
	} else if (t_1 <= 1000.0) {
		tmp = (x * pow((exp(b) * a), -1.0)) / 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 = (x * (a ** (t - 1.0d0))) / y
    if (t_1 <= (-1000000000.0d0)) then
        tmp = t_2
    else if (t_1 <= 540.0d0) then
        tmp = (x * ((z ** y) / a)) / y
    else if (t_1 <= 1000.0d0) then
        tmp = (x * ((exp(b) * a) ** (-1.0d0))) / y
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - 1.0) * Math.log(a);
	double t_2 = (x * Math.pow(a, (t - 1.0))) / y;
	double tmp;
	if (t_1 <= -1000000000.0) {
		tmp = t_2;
	} else if (t_1 <= 540.0) {
		tmp = (x * (Math.pow(z, y) / a)) / y;
	} else if (t_1 <= 1000.0) {
		tmp = (x * Math.pow((Math.exp(b) * a), -1.0)) / y;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (t - 1.0) * math.log(a)
	t_2 = (x * math.pow(a, (t - 1.0))) / y
	tmp = 0
	if t_1 <= -1000000000.0:
		tmp = t_2
	elif t_1 <= 540.0:
		tmp = (x * (math.pow(z, y) / a)) / y
	elif t_1 <= 1000.0:
		tmp = (x * math.pow((math.exp(b) * a), -1.0)) / y
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - 1.0) * log(a))
	t_2 = Float64(Float64(x * (a ^ Float64(t - 1.0))) / y)
	tmp = 0.0
	if (t_1 <= -1000000000.0)
		tmp = t_2;
	elseif (t_1 <= 540.0)
		tmp = Float64(Float64(x * Float64((z ^ y) / a)) / y);
	elseif (t_1 <= 1000.0)
		tmp = Float64(Float64(x * (Float64(exp(b) * a) ^ -1.0)) / y);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (t - 1.0) * log(a);
	t_2 = (x * (a ^ (t - 1.0))) / y;
	tmp = 0.0;
	if (t_1 <= -1000000000.0)
		tmp = t_2;
	elseif (t_1 <= 540.0)
		tmp = (x * ((z ^ y) / a)) / y;
	elseif (t_1 <= 1000.0)
		tmp = (x * ((exp(b) * a) ^ -1.0)) / y;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - 1.0), $MachinePrecision] * N[Log[a], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * N[Power[a, N[(t - 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$1, -1000000000.0], t$95$2, If[LessEqual[t$95$1, 540.0], N[(N[(x * N[(N[Power[z, y], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[t$95$1, 1000.0], N[(N[(x * N[Power[N[(N[Exp[b], $MachinePrecision] * a), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < -1e9 or 1e3 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot \left(t - 1\right)} - b}}{y} \]
   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 98.4

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

   5. Applied rewrites 98.4%

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

   6. Taylor expanded in b around 0

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

      1. exp-sum N/A

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

      2. *-commutative N/A

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

      3. exp-to-pow N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. exp-to-pow N/A

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

      7. lower-pow.f64 N/A

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

      8. lower--.f64 74.1

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

   8. Applied rewrites 74.1%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 89.6%

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

   if -1e9 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < 540

   1. Initial program 96.5%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   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 67.6

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

   5. Applied rewrites 67.6%

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

   6. Taylor expanded in b around 0

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

      1. exp-sum N/A

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

      2. *-commutative N/A

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

      3. exp-to-pow N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. exp-to-pow N/A

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

      7. lower-pow.f64 N/A

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

      8. lower--.f64 81.2

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

   8. Applied rewrites 81.2%

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

   9. Taylor expanded in t around 0

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

   11. Applied rewrites 80.0%

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

   if 540 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < 1e3

   1. Initial program 98.0%

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

   3. Taylor expanded in y around 0

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

      1. exp-diff N/A

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

      2. lower-/.f64 N/A

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

      3. exp-to-pow N/A

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

      4. lower-pow.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-exp.f64 93.2

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

   5. Applied rewrites 93.2%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 89.2%

      \[\leadsto \frac{x \cdot \frac{1}{\color{blue}{e^{b} \cdot a}}}{y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(t - 1\right) \cdot \log a \leq -1000000000:\\ \;\;\;\;\frac{x \cdot {a}^{\left(t - 1\right)}}{y}\\ \mathbf{elif}\;\left(t - 1\right) \cdot \log a \leq 540:\\ \;\;\;\;\frac{x \cdot \frac{{z}^{y}}{a}}{y}\\ \mathbf{elif}\;\left(t - 1\right) \cdot \log a \leq 1000:\\ \;\;\;\;\frac{x \cdot {\left(e^{b} \cdot a\right)}^{-1}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot {a}^{\left(t - 1\right)}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t - 1\right) \cdot \log a\\ t_2 := \frac{x \cdot {a}^{\left(t - 1\right)}}{y}\\ \mathbf{if}\;t\_1 \leq -1000000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 540:\\ \;\;\;\;\frac{x \cdot \frac{{z}^{y}}{a}}{y}\\ \mathbf{elif}\;t\_1 \leq 1000:\\ \;\;\;\;\frac{x \cdot \frac{e^{-b}}{a}}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- t 1.0) (log a))) (t_2 (/ (* x (pow a (- t 1.0))) y)))
   (if (<= t_1 -1000000000.0)
     t_2
     (if (<= t_1 540.0)
       (/ (* x (/ (pow z y) a)) y)
       (if (<= t_1 1000.0) (/ (* x (/ (exp (- b)) a)) y) t_2)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - 1.0) * log(a);
	double t_2 = (x * pow(a, (t - 1.0))) / y;
	double tmp;
	if (t_1 <= -1000000000.0) {
		tmp = t_2;
	} else if (t_1 <= 540.0) {
		tmp = (x * (pow(z, y) / a)) / y;
	} else if (t_1 <= 1000.0) {
		tmp = (x * (exp(-b) / a)) / 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 = (x * (a ** (t - 1.0d0))) / y
    if (t_1 <= (-1000000000.0d0)) then
        tmp = t_2
    else if (t_1 <= 540.0d0) then
        tmp = (x * ((z ** y) / a)) / y
    else if (t_1 <= 1000.0d0) then
        tmp = (x * (exp(-b) / a)) / y
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - 1.0) * Math.log(a);
	double t_2 = (x * Math.pow(a, (t - 1.0))) / y;
	double tmp;
	if (t_1 <= -1000000000.0) {
		tmp = t_2;
	} else if (t_1 <= 540.0) {
		tmp = (x * (Math.pow(z, y) / a)) / y;
	} else if (t_1 <= 1000.0) {
		tmp = (x * (Math.exp(-b) / a)) / y;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (t - 1.0) * math.log(a)
	t_2 = (x * math.pow(a, (t - 1.0))) / y
	tmp = 0
	if t_1 <= -1000000000.0:
		tmp = t_2
	elif t_1 <= 540.0:
		tmp = (x * (math.pow(z, y) / a)) / y
	elif t_1 <= 1000.0:
		tmp = (x * (math.exp(-b) / a)) / y
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - 1.0) * log(a))
	t_2 = Float64(Float64(x * (a ^ Float64(t - 1.0))) / y)
	tmp = 0.0
	if (t_1 <= -1000000000.0)
		tmp = t_2;
	elseif (t_1 <= 540.0)
		tmp = Float64(Float64(x * Float64((z ^ y) / a)) / y);
	elseif (t_1 <= 1000.0)
		tmp = Float64(Float64(x * Float64(exp(Float64(-b)) / a)) / y);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (t - 1.0) * log(a);
	t_2 = (x * (a ^ (t - 1.0))) / y;
	tmp = 0.0;
	if (t_1 <= -1000000000.0)
		tmp = t_2;
	elseif (t_1 <= 540.0)
		tmp = (x * ((z ^ y) / a)) / y;
	elseif (t_1 <= 1000.0)
		tmp = (x * (exp(-b) / a)) / y;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - 1.0), $MachinePrecision] * N[Log[a], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * N[Power[a, N[(t - 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$1, -1000000000.0], t$95$2, If[LessEqual[t$95$1, 540.0], N[(N[(x * N[(N[Power[z, y], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[t$95$1, 1000.0], N[(N[(x * N[(N[Exp[(-b)], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < -1e9 or 1e3 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot \left(t - 1\right)} - b}}{y} \]
   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 98.4

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

   5. Applied rewrites 98.4%

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

   6. Taylor expanded in b around 0

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

      1. exp-sum N/A

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

      2. *-commutative N/A

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

      3. exp-to-pow N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. exp-to-pow N/A

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

      7. lower-pow.f64 N/A

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

      8. lower--.f64 74.1

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

   8. Applied rewrites 74.1%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 89.6%

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

   if -1e9 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < 540

   1. Initial program 96.5%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   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 67.6

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

   5. Applied rewrites 67.6%

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

   6. Taylor expanded in b around 0

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

      1. exp-sum N/A

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

      2. *-commutative N/A

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

      3. exp-to-pow N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. exp-to-pow N/A

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

      7. lower-pow.f64 N/A

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

      8. lower--.f64 81.2

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

   8. Applied rewrites 81.2%

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

   9. Taylor expanded in t around 0

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

   11. Applied rewrites 80.0%

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

   if 540 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < 1e3

   1. Initial program 98.0%

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

   3. Taylor expanded in y around 0

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

      1. exp-diff N/A

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

      2. lower-/.f64 N/A

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

      3. exp-to-pow N/A

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

      4. lower-pow.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-exp.f64 93.2

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

   5. Applied rewrites 93.2%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 89.2%

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

   9. Taylor expanded in t around 0

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

   11. Applied rewrites 89.2%

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

Alternative 5: 86.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t - 1\right) \cdot \log a\\ \mathbf{if}\;t\_1 \leq -600 \lor \neg \left(t\_1 \leq 480\right):\\ \;\;\;\;\frac{x \cdot e^{\left(-1 + t\right) \cdot \log a - b}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{e^{b} \cdot a}{{z}^{y}} \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- t 1.0) (log a))))
   (if (or (<= t_1 -600.0) (not (<= t_1 480.0)))
     (/ (* x (exp (- (* (+ -1.0 t) (log a)) b))) y)
     (/ x (* (/ (* (exp b) a) (pow z y)) y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - 1.0) * log(a);
	double tmp;
	if ((t_1 <= -600.0) || !(t_1 <= 480.0)) {
		tmp = (x * exp((((-1.0 + t) * log(a)) - b))) / y;
	} else {
		tmp = x / (((exp(b) * a) / 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) :: t_1
    real(8) :: tmp
    t_1 = (t - 1.0d0) * log(a)
    if ((t_1 <= (-600.0d0)) .or. (.not. (t_1 <= 480.0d0))) then
        tmp = (x * exp(((((-1.0d0) + t) * log(a)) - b))) / y
    else
        tmp = x / (((exp(b) * a) / (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 t_1 = (t - 1.0) * Math.log(a);
	double tmp;
	if ((t_1 <= -600.0) || !(t_1 <= 480.0)) {
		tmp = (x * Math.exp((((-1.0 + t) * Math.log(a)) - b))) / y;
	} else {
		tmp = x / (((Math.exp(b) * a) / Math.pow(z, y)) * y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (t - 1.0) * math.log(a)
	tmp = 0
	if (t_1 <= -600.0) or not (t_1 <= 480.0):
		tmp = (x * math.exp((((-1.0 + t) * math.log(a)) - b))) / y
	else:
		tmp = x / (((math.exp(b) * a) / math.pow(z, y)) * y)
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - 1.0) * log(a))
	tmp = 0.0
	if ((t_1 <= -600.0) || !(t_1 <= 480.0))
		tmp = Float64(Float64(x * exp(Float64(Float64(Float64(-1.0 + t) * log(a)) - b))) / y);
	else
		tmp = Float64(x / Float64(Float64(Float64(exp(b) * a) / (z ^ y)) * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (t - 1.0) * log(a);
	tmp = 0.0;
	if ((t_1 <= -600.0) || ~((t_1 <= 480.0)))
		tmp = (x * exp((((-1.0 + t) * log(a)) - b))) / y;
	else
		tmp = x / (((exp(b) * a) / (z ^ y)) * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - 1.0), $MachinePrecision] * N[Log[a], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -600.0], N[Not[LessEqual[t$95$1, 480.0]], $MachinePrecision]], N[(N[(x * N[Exp[N[(N[(N[(-1.0 + t), $MachinePrecision] * N[Log[a], $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(x / N[(N[(N[(N[Exp[b], $MachinePrecision] * a), $MachinePrecision] / N[Power[z, y], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < -600 or 480 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))

   1. Initial program 99.4%

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

   3. Taylor expanded in y around 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 95.0

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

   5. Applied rewrites 95.0%

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

   if -600 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < 480

   1. Initial program 96.5%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   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. *-commutative N/A

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

      4. associate-/l* N/A

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

   4. Applied rewrites 88.2%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-pow.f64 87.2

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

   7. Applied rewrites 87.2%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(t - 1\right) \cdot \log a \leq -600 \lor \neg \left(\left(t - 1\right) \cdot \log a \leq 480\right):\\ \;\;\;\;\frac{x \cdot e^{\left(-1 + t\right) \cdot \log a - b}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{e^{b} \cdot a}{{z}^{y}} \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t - 1\right) \cdot \log a\\ \mathbf{if}\;t\_1 \leq -600 \lor \neg \left(t\_1 \leq 480\right):\\ \;\;\;\;\frac{x \cdot e^{\left(-1 + t\right) \cdot \log a - b}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{z}^{y}}{a}}{e^{b} \cdot y} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- t 1.0) (log a))))
   (if (or (<= t_1 -600.0) (not (<= t_1 480.0)))
     (/ (* x (exp (- (* (+ -1.0 t) (log a)) b))) y)
     (* (/ (/ (pow z y) a) (* (exp b) y)) x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - 1.0) * log(a);
	double tmp;
	if ((t_1 <= -600.0) || !(t_1 <= 480.0)) {
		tmp = (x * exp((((-1.0 + t) * log(a)) - b))) / y;
	} else {
		tmp = ((pow(z, y) / a) / (exp(b) * 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) :: t_1
    real(8) :: tmp
    t_1 = (t - 1.0d0) * log(a)
    if ((t_1 <= (-600.0d0)) .or. (.not. (t_1 <= 480.0d0))) then
        tmp = (x * exp(((((-1.0d0) + t) * log(a)) - b))) / y
    else
        tmp = (((z ** y) / a) / (exp(b) * 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 t_1 = (t - 1.0) * Math.log(a);
	double tmp;
	if ((t_1 <= -600.0) || !(t_1 <= 480.0)) {
		tmp = (x * Math.exp((((-1.0 + t) * Math.log(a)) - b))) / y;
	} else {
		tmp = ((Math.pow(z, y) / a) / (Math.exp(b) * y)) * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (t - 1.0) * math.log(a)
	tmp = 0
	if (t_1 <= -600.0) or not (t_1 <= 480.0):
		tmp = (x * math.exp((((-1.0 + t) * math.log(a)) - b))) / y
	else:
		tmp = ((math.pow(z, y) / a) / (math.exp(b) * y)) * x
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - 1.0) * log(a))
	tmp = 0.0
	if ((t_1 <= -600.0) || !(t_1 <= 480.0))
		tmp = Float64(Float64(x * exp(Float64(Float64(Float64(-1.0 + t) * log(a)) - b))) / y);
	else
		tmp = Float64(Float64(Float64((z ^ y) / a) / Float64(exp(b) * y)) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (t - 1.0) * log(a);
	tmp = 0.0;
	if ((t_1 <= -600.0) || ~((t_1 <= 480.0)))
		tmp = (x * exp((((-1.0 + t) * log(a)) - b))) / y;
	else
		tmp = (((z ^ y) / a) / (exp(b) * y)) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - 1.0), $MachinePrecision] * N[Log[a], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -600.0], N[Not[LessEqual[t$95$1, 480.0]], $MachinePrecision]], N[(N[(x * N[Exp[N[(N[(N[(-1.0 + t), $MachinePrecision] * N[Log[a], $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(N[(N[(N[Power[z, y], $MachinePrecision] / a), $MachinePrecision] / N[(N[Exp[b], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < -600 or 480 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))

   1. Initial program 99.4%

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

   3. Taylor expanded in y around 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 95.0

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

   5. Applied rewrites 95.0%

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

   if -600 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < 480

   1. Initial program 96.5%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   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 66.2

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

   5. Applied rewrites 66.2%

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

   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 46.1

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

   8. Applied rewrites 46.1%

      \[\leadsto \frac{x \cdot e^{\color{blue}{-b}}}{y} \]
   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 46.1

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

   10. Applied rewrites 46.1%

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

   11. Taylor expanded in t around 0

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

       1. exp-diff N/A

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

       2. associate-/l/ N/A

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

       3. +-commutative N/A

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

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

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

       5. metadata-eval N/A

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

       6. *-lft-identity N/A

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

       7. div-exp N/A

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

       8. *-commutative N/A

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

       9. exp-to-pow N/A

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

       10. rem-exp-log N/A

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

       11. lower-/.f64 N/A

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

       12. lower-/.f64 N/A

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

       13. lower-pow.f64 N/A

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

       14. lower-*.f64 N/A

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

       15. lower-exp.f64 86.7

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

   13. Applied rewrites 86.7%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(t - 1\right) \cdot \log a \leq -600 \lor \neg \left(\left(t - 1\right) \cdot \log a \leq 480\right):\\ \;\;\;\;\frac{x \cdot e^{\left(-1 + t\right) \cdot \log a - b}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{z}^{y}}{a}}{e^{b} \cdot y} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t - 1\right) \cdot \log a\\ \mathbf{if}\;t\_1 \leq -480 \lor \neg \left(t\_1 \leq 540\right):\\ \;\;\;\;\frac{x \cdot e^{\left(-1 + t\right) \cdot \log a - b}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left({a}^{\left(t - 1\right)} \cdot {z}^{y}\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- t 1.0) (log a))))
   (if (or (<= t_1 -480.0) (not (<= t_1 540.0)))
     (/ (* x (exp (- (* (+ -1.0 t) (log a)) 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 t_1 = (t - 1.0) * log(a);
	double tmp;
	if ((t_1 <= -480.0) || !(t_1 <= 540.0)) {
		tmp = (x * exp((((-1.0 + t) * log(a)) - 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) :: t_1
    real(8) :: tmp
    t_1 = (t - 1.0d0) * log(a)
    if ((t_1 <= (-480.0d0)) .or. (.not. (t_1 <= 540.0d0))) then
        tmp = (x * exp(((((-1.0d0) + t) * log(a)) - 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 t_1 = (t - 1.0) * Math.log(a);
	double tmp;
	if ((t_1 <= -480.0) || !(t_1 <= 540.0)) {
		tmp = (x * Math.exp((((-1.0 + t) * Math.log(a)) - 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):
	t_1 = (t - 1.0) * math.log(a)
	tmp = 0
	if (t_1 <= -480.0) or not (t_1 <= 540.0):
		tmp = (x * math.exp((((-1.0 + t) * math.log(a)) - 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)
	t_1 = Float64(Float64(t - 1.0) * log(a))
	tmp = 0.0
	if ((t_1 <= -480.0) || !(t_1 <= 540.0))
		tmp = Float64(Float64(x * exp(Float64(Float64(Float64(-1.0 + t) * log(a)) - 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)
	t_1 = (t - 1.0) * log(a);
	tmp = 0.0;
	if ((t_1 <= -480.0) || ~((t_1 <= 540.0)))
		tmp = (x * exp((((-1.0 + t) * log(a)) - 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_] := Block[{t$95$1 = N[(N[(t - 1.0), $MachinePrecision] * N[Log[a], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -480.0], N[Not[LessEqual[t$95$1, 540.0]], $MachinePrecision]], N[(N[(x * N[Exp[N[(N[(N[(-1.0 + t), $MachinePrecision] * N[Log[a], $MachinePrecision]), $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 (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < -480 or 540 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))

   1. Initial program 99.5%

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

   3. Taylor expanded in 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 96.4

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

   5. Applied rewrites 96.4%

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

   if -480 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < 540

   1. Initial program 96.4%

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

   3. Taylor expanded in 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 82.7

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

   5. Applied rewrites 82.7%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(t - 1\right) \cdot \log a \leq -480 \lor \neg \left(\left(t - 1\right) \cdot \log a \leq 540\right):\\ \;\;\;\;\frac{x \cdot e^{\left(-1 + t\right) \cdot \log a - b}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left({a}^{\left(t - 1\right)} \cdot {z}^{y}\right)}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t - 1\right) \cdot \log a\\ \mathbf{if}\;t\_1 \leq -1000000000 \lor \neg \left(t\_1 \leq 590\right):\\ \;\;\;\;\frac{e^{\log a \cdot t - b}}{y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{{z}^{y}}{a}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- t 1.0) (log a))))
   (if (or (<= t_1 -1000000000.0) (not (<= t_1 590.0)))
     (* (/ (exp (- (* (log a) t) b)) y) x)
     (/ (* x (/ (pow z y) a)) y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - 1.0) * log(a);
	double tmp;
	if ((t_1 <= -1000000000.0) || !(t_1 <= 590.0)) {
		tmp = (exp(((log(a) * t) - b)) / y) * x;
	} else {
		tmp = (x * (pow(z, y) / a)) / y;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (t - 1.0d0) * log(a)
    if ((t_1 <= (-1000000000.0d0)) .or. (.not. (t_1 <= 590.0d0))) then
        tmp = (exp(((log(a) * t) - b)) / y) * x
    else
        tmp = (x * ((z ** y) / a)) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - 1.0) * Math.log(a);
	double tmp;
	if ((t_1 <= -1000000000.0) || !(t_1 <= 590.0)) {
		tmp = (Math.exp(((Math.log(a) * t) - b)) / y) * x;
	} else {
		tmp = (x * (Math.pow(z, y) / a)) / y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (t - 1.0) * math.log(a)
	tmp = 0
	if (t_1 <= -1000000000.0) or not (t_1 <= 590.0):
		tmp = (math.exp(((math.log(a) * t) - b)) / y) * x
	else:
		tmp = (x * (math.pow(z, y) / a)) / y
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - 1.0) * log(a))
	tmp = 0.0
	if ((t_1 <= -1000000000.0) || !(t_1 <= 590.0))
		tmp = Float64(Float64(exp(Float64(Float64(log(a) * t) - b)) / y) * x);
	else
		tmp = Float64(Float64(x * Float64((z ^ y) / a)) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (t - 1.0) * log(a);
	tmp = 0.0;
	if ((t_1 <= -1000000000.0) || ~((t_1 <= 590.0)))
		tmp = (exp(((log(a) * t) - b)) / y) * x;
	else
		tmp = (x * ((z ^ y) / a)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - 1.0), $MachinePrecision] * N[Log[a], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -1000000000.0], N[Not[LessEqual[t$95$1, 590.0]], $MachinePrecision]], N[(N[(N[Exp[N[(N[(N[Log[a], $MachinePrecision] * t), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision], N[(N[(x * N[(N[Power[z, y], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < -1e9 or 590 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 97.1

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

   5. Applied rewrites 97.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 97.1

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

   7. Applied rewrites 97.1%

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

   if -1e9 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < 590

   1. Initial program 96.4%

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

   3. Taylor expanded in 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 68.1

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

   5. Applied rewrites 68.1%

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

   6. Taylor expanded in b around 0

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

      1. exp-sum N/A

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

      2. *-commutative N/A

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

      3. exp-to-pow N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. exp-to-pow N/A

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

      7. lower-pow.f64 N/A

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

      8. lower--.f64 80.8

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

   8. Applied rewrites 80.8%

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

   9. Taylor expanded in t around 0

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

   11. Applied rewrites 79.1%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(t - 1\right) \cdot \log a \leq -1000000000 \lor \neg \left(\left(t - 1\right) \cdot \log a \leq 590\right):\\ \;\;\;\;\frac{e^{\log a \cdot t - b}}{y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{{z}^{y}}{a}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t - 1\right) \cdot \log a\\ \mathbf{if}\;t\_1 \leq -1000000000 \lor \neg \left(t\_1 \leq 600\right):\\ \;\;\;\;\frac{x \cdot {a}^{\left(t - 1\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{z}^{y}}{a}}{y} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- t 1.0) (log a))))
   (if (or (<= t_1 -1000000000.0) (not (<= t_1 600.0)))
     (/ (* x (pow a (- t 1.0))) y)
     (* (/ (/ (pow z y) a) y) x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - 1.0) * log(a);
	double tmp;
	if ((t_1 <= -1000000000.0) || !(t_1 <= 600.0)) {
		tmp = (x * pow(a, (t - 1.0))) / y;
	} else {
		tmp = ((pow(z, y) / a) / 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) :: t_1
    real(8) :: tmp
    t_1 = (t - 1.0d0) * log(a)
    if ((t_1 <= (-1000000000.0d0)) .or. (.not. (t_1 <= 600.0d0))) then
        tmp = (x * (a ** (t - 1.0d0))) / y
    else
        tmp = (((z ** y) / a) / 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 t_1 = (t - 1.0) * Math.log(a);
	double tmp;
	if ((t_1 <= -1000000000.0) || !(t_1 <= 600.0)) {
		tmp = (x * Math.pow(a, (t - 1.0))) / y;
	} else {
		tmp = ((Math.pow(z, y) / a) / y) * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (t - 1.0) * math.log(a)
	tmp = 0
	if (t_1 <= -1000000000.0) or not (t_1 <= 600.0):
		tmp = (x * math.pow(a, (t - 1.0))) / y
	else:
		tmp = ((math.pow(z, y) / a) / y) * x
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - 1.0) * log(a))
	tmp = 0.0
	if ((t_1 <= -1000000000.0) || !(t_1 <= 600.0))
		tmp = Float64(Float64(x * (a ^ Float64(t - 1.0))) / y);
	else
		tmp = Float64(Float64(Float64((z ^ y) / a) / y) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (t - 1.0) * log(a);
	tmp = 0.0;
	if ((t_1 <= -1000000000.0) || ~((t_1 <= 600.0)))
		tmp = (x * (a ^ (t - 1.0))) / y;
	else
		tmp = (((z ^ y) / a) / y) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - 1.0), $MachinePrecision] * N[Log[a], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -1000000000.0], N[Not[LessEqual[t$95$1, 600.0]], $MachinePrecision]], N[(N[(x * N[Power[a, N[(t - 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(N[(N[(N[Power[z, y], $MachinePrecision] / a), $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < -1e9 or 600 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot \left(t - 1\right)} - b}}{y} \]
   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 98.5

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

   5. Applied rewrites 98.5%

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

   6. Taylor expanded in b around 0

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

      1. exp-sum N/A

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

      2. *-commutative N/A

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

      3. exp-to-pow N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. exp-to-pow N/A

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

      7. lower-pow.f64 N/A

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

      8. lower--.f64 71.9

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

   8. Applied rewrites 71.9%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 87.2%

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

   if -1e9 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < 600

   1. Initial program 96.5%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   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 68.6

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

   5. Applied rewrites 68.6%

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

   6. Taylor expanded in b around 0

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

      1. exp-sum N/A

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

      2. *-commutative N/A

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

      3. exp-to-pow N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. exp-to-pow N/A

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

      7. lower-pow.f64 N/A

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

      8. lower--.f64 79.6

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

   8. Applied rewrites 79.6%

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

   9. Taylor expanded in t around 0

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

   11. Applied rewrites 77.9%

       \[\leadsto \frac{x \cdot \frac{{z}^{y}}{\color{blue}{a}}}{y} \]
   12. 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. associate-/l* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 77.8

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

   13. Applied rewrites 77.8%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(t - 1\right) \cdot \log a \leq -1000000000 \lor \neg \left(\left(t - 1\right) \cdot \log a \leq 600\right):\\ \;\;\;\;\frac{x \cdot {a}^{\left(t - 1\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{z}^{y}}{a}}{y} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 93.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t - 1 \leq -1.000005:\\ \;\;\;\;\frac{x \cdot e^{\left(-1 + t\right) \cdot \log a - b}}{y}\\ \mathbf{elif}\;t - 1 \leq -0.05:\\ \;\;\;\;\frac{x \cdot e^{\mathsf{fma}\left(\log z, y, -\log a\right) - b}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log a \cdot t - b}}{y} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (- t 1.0) -1.000005)
   (/ (* x (exp (- (* (+ -1.0 t) (log a)) b))) y)
   (if (<= (- t 1.0) -0.05)
     (/ (* x (exp (- (fma (log z) y (- (log a))) b))) y)
     (* (/ (exp (- (* (log a) t) b)) y) x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t - 1.0) <= -1.000005) {
		tmp = (x * exp((((-1.0 + t) * log(a)) - b))) / y;
	} else if ((t - 1.0) <= -0.05) {
		tmp = (x * exp((fma(log(z), y, -log(a)) - b))) / y;
	} else {
		tmp = (exp(((log(a) * t) - b)) / y) * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(t - 1.0) <= -1.000005)
		tmp = Float64(Float64(x * exp(Float64(Float64(Float64(-1.0 + t) * log(a)) - b))) / y);
	elseif (Float64(t - 1.0) <= -0.05)
		tmp = Float64(Float64(x * exp(Float64(fma(log(z), y, Float64(-log(a))) - b))) / y);
	else
		tmp = Float64(Float64(exp(Float64(Float64(log(a) * t) - b)) / y) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(t - 1.0), $MachinePrecision], -1.000005], N[(N[(x * N[Exp[N[(N[(N[(-1.0 + t), $MachinePrecision] * N[Log[a], $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[N[(t - 1.0), $MachinePrecision], -0.05], 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[(N[Exp[N[(N[(N[Log[a], $MachinePrecision] * t), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 t #s(literal 1 binary64)) < -1.00000500000000003

   1. Initial program 99.9%

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

   3. Taylor expanded in 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 97.2

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

   5. Applied rewrites 97.2%

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

   if -1.00000500000000003 < (-.f64 t #s(literal 1 binary64)) < -0.050000000000000003

   1. Initial program 96.7%

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

   3. Taylor expanded in t around 0

      \[\leadsto \frac{x \cdot e^{\color{blue}{\left(-1 \cdot \log a + y \cdot \log z\right)} - b}}{y} \]
   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 95.5

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

   5. Applied rewrites 95.5%

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

   if -0.050000000000000003 < (-.f64 t #s(literal 1 binary64))

   1. Initial program 100.0%

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

   3. Taylor expanded in 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 100.0

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

   5. Applied rewrites 100.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 100.0

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

   7. Applied rewrites 100.0%

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

Alternative 11: 86.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.1 \cdot 10^{+30} \lor \neg \left(b \leq 3.6 \cdot 10^{+56}\right):\\ \;\;\;\;\frac{e^{\log a \cdot t - b}}{y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left({a}^{\left(t - 1\right)} \cdot {z}^{y}\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= b -4.1e+30) (not (<= b 3.6e+56)))
   (* (/ (exp (- (* (log a) t) b)) y) x)
   (/ (* 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 <= -4.1e+30) || !(b <= 3.6e+56)) {
		tmp = (exp(((log(a) * t) - b)) / y) * x;
	} 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 <= (-4.1d+30)) .or. (.not. (b <= 3.6d+56))) then
        tmp = (exp(((log(a) * t) - b)) / y) * x
    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 <= -4.1e+30) || !(b <= 3.6e+56)) {
		tmp = (Math.exp(((Math.log(a) * t) - b)) / y) * x;
	} 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 <= -4.1e+30) or not (b <= 3.6e+56):
		tmp = (math.exp(((math.log(a) * t) - b)) / y) * x
	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 <= -4.1e+30) || !(b <= 3.6e+56))
		tmp = Float64(Float64(exp(Float64(Float64(log(a) * t) - b)) / y) * x);
	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 <= -4.1e+30) || ~((b <= 3.6e+56)))
		tmp = (exp(((log(a) * t) - b)) / y) * x;
	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, -4.1e+30], N[Not[LessEqual[b, 3.6e+56]], $MachinePrecision]], N[(N[(N[Exp[N[(N[(N[Log[a], $MachinePrecision] * t), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision] * x), $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 < -4.10000000000000005e30 or 3.59999999999999998e56 < 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 89.3

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

   5. Applied rewrites 89.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 89.3

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

   7. Applied rewrites 89.3%

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

   if -4.10000000000000005e30 < b < 3.59999999999999998e56

   1. Initial program 97.2%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. 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 90.9

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

   5. Applied rewrites 90.9%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.1 \cdot 10^{+30} \lor \neg \left(b \leq 3.6 \cdot 10^{+56}\right):\\ \;\;\;\;\frac{e^{\log a \cdot t - b}}{y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left({a}^{\left(t - 1\right)} \cdot {z}^{y}\right)}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 86.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.1 \cdot 10^{+30} \lor \neg \left(b \leq 3.6 \cdot 10^{+56}\right):\\ \;\;\;\;\frac{e^{\log a \cdot t - b}}{y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{{a}^{\left(t - 1\right)} \cdot {z}^{y}}{y} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= b -4.1e+30) (not (<= b 3.6e+56)))
   (* (/ (exp (- (* (log a) t) b)) y) x)
   (* (/ (* (pow a (- t 1.0)) (pow z y)) y) x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((b <= -4.1e+30) || !(b <= 3.6e+56)) {
		tmp = (exp(((log(a) * t) - b)) / y) * x;
	} else {
		tmp = ((pow(a, (t - 1.0)) * pow(z, y)) / 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 <= (-4.1d+30)) .or. (.not. (b <= 3.6d+56))) then
        tmp = (exp(((log(a) * t) - b)) / y) * x
    else
        tmp = (((a ** (t - 1.0d0)) * (z ** y)) / 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 <= -4.1e+30) || !(b <= 3.6e+56)) {
		tmp = (Math.exp(((Math.log(a) * t) - b)) / y) * x;
	} else {
		tmp = ((Math.pow(a, (t - 1.0)) * Math.pow(z, y)) / y) * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (b <= -4.1e+30) or not (b <= 3.6e+56):
		tmp = (math.exp(((math.log(a) * t) - b)) / y) * x
	else:
		tmp = ((math.pow(a, (t - 1.0)) * math.pow(z, y)) / y) * x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((b <= -4.1e+30) || !(b <= 3.6e+56))
		tmp = Float64(Float64(exp(Float64(Float64(log(a) * t) - b)) / y) * x);
	else
		tmp = Float64(Float64(Float64((a ^ Float64(t - 1.0)) * (z ^ y)) / y) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((b <= -4.1e+30) || ~((b <= 3.6e+56)))
		tmp = (exp(((log(a) * t) - b)) / y) * x;
	else
		tmp = (((a ^ (t - 1.0)) * (z ^ y)) / y) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[b, -4.1e+30], N[Not[LessEqual[b, 3.6e+56]], $MachinePrecision]], N[(N[(N[Exp[N[(N[(N[Log[a], $MachinePrecision] * t), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision], N[(N[(N[(N[Power[a, N[(t - 1.0), $MachinePrecision]], $MachinePrecision] * N[Power[z, y], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.10000000000000005e30 or 3.59999999999999998e56 < 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 89.3

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

   5. Applied rewrites 89.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 89.3

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

   7. Applied rewrites 89.3%

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

   if -4.10000000000000005e30 < b < 3.59999999999999998e56

   1. Initial program 97.2%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. 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 80.4

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

   5. Applied rewrites 80.4%

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

   6. Taylor expanded in b around 0

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

      1. exp-sum N/A

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

      2. *-commutative N/A

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

      3. exp-to-pow N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. exp-to-pow N/A

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

      7. lower-pow.f64 N/A

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

      8. lower--.f64 90.9

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

   8. Applied rewrites 90.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 89.5

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

   10. Applied rewrites 89.5%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.1 \cdot 10^{+30} \lor \neg \left(b \leq 3.6 \cdot 10^{+56}\right):\\ \;\;\;\;\frac{e^{\log a \cdot t - b}}{y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{{a}^{\left(t - 1\right)} \cdot {z}^{y}}{y} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 75.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -9 \cdot 10^{-6} \lor \neg \left(t \leq 1.3 \cdot 10^{-15}\right):\\ \;\;\;\;\frac{\frac{{a}^{t}}{a} \cdot x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{{z}^{y}}{a}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= t -9e-6) (not (<= t 1.3e-15)))
   (/ (* (/ (pow a t) a) x) y)
   (/ (* x (/ (pow z y) a)) y)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -9e-6) || !(t <= 1.3e-15)) {
		tmp = ((pow(a, t) / a) * x) / y;
	} else {
		tmp = (x * (pow(z, y) / a)) / 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 ((t <= (-9d-6)) .or. (.not. (t <= 1.3d-15))) then
        tmp = (((a ** t) / a) * x) / y
    else
        tmp = (x * ((z ** y) / a)) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -9e-6) || !(t <= 1.3e-15)) {
		tmp = ((Math.pow(a, t) / a) * x) / y;
	} else {
		tmp = (x * (Math.pow(z, y) / a)) / y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (t <= -9e-6) or not (t <= 1.3e-15):
		tmp = ((math.pow(a, t) / a) * x) / y
	else:
		tmp = (x * (math.pow(z, y) / a)) / y
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((t <= -9e-6) || !(t <= 1.3e-15))
		tmp = Float64(Float64(Float64((a ^ t) / a) * x) / y);
	else
		tmp = Float64(Float64(x * Float64((z ^ y) / a)) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((t <= -9e-6) || ~((t <= 1.3e-15)))
		tmp = (((a ^ t) / a) * x) / y;
	else
		tmp = (x * ((z ^ y) / a)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[t, -9e-6], N[Not[LessEqual[t, 1.3e-15]], $MachinePrecision]], N[(N[(N[(N[Power[a, t], $MachinePrecision] / a), $MachinePrecision] * x), $MachinePrecision] / y), $MachinePrecision], N[(N[(x * N[(N[Power[z, y], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -9.00000000000000023e-6 or 1.30000000000000002e-15 < t

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0

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

      1. exp-diff N/A

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

      2. lower-/.f64 N/A

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

      3. exp-to-pow N/A

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

      4. lower-pow.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-exp.f64 78.8

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

   5. Applied rewrites 78.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 78.8

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

   7. Applied rewrites 79.0%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 88.0%

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

   if -9.00000000000000023e-6 < t < 1.30000000000000002e-15

   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 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 70.0

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

   5. Applied rewrites 70.0%

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

   6. Taylor expanded in b around 0

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

      1. exp-sum N/A

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

      2. *-commutative N/A

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

      3. exp-to-pow N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. exp-to-pow N/A

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

      7. lower-pow.f64 N/A

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

      8. lower--.f64 77.2

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

   8. Applied rewrites 77.2%

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

   9. Taylor expanded in t around 0

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

   11. Applied rewrites 77.2%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -9 \cdot 10^{-6} \lor \neg \left(t \leq 1.3 \cdot 10^{-15}\right):\\ \;\;\;\;\frac{\frac{{a}^{t}}{a} \cdot x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{{z}^{y}}{a}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 75.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -9 \cdot 10^{-6} \lor \neg \left(t \leq 1.3 \cdot 10^{-15}\right):\\ \;\;\;\;\frac{x \cdot {a}^{\left(t - 1\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{{z}^{y}}{a}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= t -9e-6) (not (<= t 1.3e-15)))
   (/ (* x (pow a (- t 1.0))) y)
   (/ (* x (/ (pow z y) a)) y)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -9e-6) || !(t <= 1.3e-15)) {
		tmp = (x * pow(a, (t - 1.0))) / y;
	} else {
		tmp = (x * (pow(z, y) / a)) / 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 ((t <= (-9d-6)) .or. (.not. (t <= 1.3d-15))) then
        tmp = (x * (a ** (t - 1.0d0))) / y
    else
        tmp = (x * ((z ** y) / a)) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -9e-6) || !(t <= 1.3e-15)) {
		tmp = (x * Math.pow(a, (t - 1.0))) / y;
	} else {
		tmp = (x * (Math.pow(z, y) / a)) / y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (t <= -9e-6) or not (t <= 1.3e-15):
		tmp = (x * math.pow(a, (t - 1.0))) / y
	else:
		tmp = (x * (math.pow(z, y) / a)) / y
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((t <= -9e-6) || !(t <= 1.3e-15))
		tmp = Float64(Float64(x * (a ^ Float64(t - 1.0))) / y);
	else
		tmp = Float64(Float64(x * Float64((z ^ y) / a)) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((t <= -9e-6) || ~((t <= 1.3e-15)))
		tmp = (x * (a ^ (t - 1.0))) / y;
	else
		tmp = (x * ((z ^ y) / a)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[t, -9e-6], N[Not[LessEqual[t, 1.3e-15]], $MachinePrecision]], N[(N[(x * N[Power[a, N[(t - 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(N[(x * N[(N[Power[z, y], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -9.00000000000000023e-6 or 1.30000000000000002e-15 < t

   1. Initial program 99.7%

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

   3. Taylor expanded in 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 96.7

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

   5. Applied rewrites 96.7%

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

   6. Taylor expanded in b around 0

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

      1. exp-sum N/A

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

      2. *-commutative N/A

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

      3. exp-to-pow N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. exp-to-pow N/A

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

      7. lower-pow.f64 N/A

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

      8. lower--.f64 74.2

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

   8. Applied rewrites 74.2%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 87.8%

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

   if -9.00000000000000023e-6 < t < 1.30000000000000002e-15

   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 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 70.0

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

   5. Applied rewrites 70.0%

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

   6. Taylor expanded in b around 0

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

      1. exp-sum N/A

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

      2. *-commutative N/A

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

      3. exp-to-pow N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. exp-to-pow N/A

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

      7. lower-pow.f64 N/A

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

      8. lower--.f64 77.2

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

   8. Applied rewrites 77.2%

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

   9. Taylor expanded in t around 0

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

   11. Applied rewrites 77.2%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -9 \cdot 10^{-6} \lor \neg \left(t \leq 1.3 \cdot 10^{-15}\right):\\ \;\;\;\;\frac{x \cdot {a}^{\left(t - 1\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{{z}^{y}}{a}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 74.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -9 \cdot 10^{+92} \lor \neg \left(b \leq 8.4 \cdot 10^{+27}\right):\\ \;\;\;\;\frac{e^{-b}}{y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot {a}^{\left(t - 1\right)}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= b -9e+92) (not (<= b 8.4e+27)))
   (* (/ (exp (- b)) y) x)
   (/ (* x (pow a (- t 1.0))) y)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((b <= -9e+92) || !(b <= 8.4e+27)) {
		tmp = (exp(-b) / y) * x;
	} else {
		tmp = (x * pow(a, (t - 1.0))) / y;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b <= (-9d+92)) .or. (.not. (b <= 8.4d+27))) then
        tmp = (exp(-b) / y) * x
    else
        tmp = (x * (a ** (t - 1.0d0))) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((b <= -9e+92) || !(b <= 8.4e+27)) {
		tmp = (Math.exp(-b) / y) * x;
	} else {
		tmp = (x * Math.pow(a, (t - 1.0))) / y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (b <= -9e+92) or not (b <= 8.4e+27):
		tmp = (math.exp(-b) / y) * x
	else:
		tmp = (x * math.pow(a, (t - 1.0))) / y
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((b <= -9e+92) || !(b <= 8.4e+27))
		tmp = Float64(Float64(exp(Float64(-b)) / y) * x);
	else
		tmp = Float64(Float64(x * (a ^ Float64(t - 1.0))) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((b <= -9e+92) || ~((b <= 8.4e+27)))
		tmp = (exp(-b) / y) * x;
	else
		tmp = (x * (a ^ (t - 1.0))) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[b, -9e+92], N[Not[LessEqual[b, 8.4e+27]], $MachinePrecision]], N[(N[(N[Exp[(-b)], $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision], N[(N[(x * N[Power[a, N[(t - 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -8.9999999999999998e92 or 8.39999999999999978e27 < 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 90.3

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

   5. Applied rewrites 90.3%

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

   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 82.7

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

   8. Applied rewrites 82.7%

      \[\leadsto \frac{x \cdot e^{\color{blue}{-b}}}{y} \]
   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 82.7

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

   10. Applied rewrites 82.7%

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

   if -8.9999999999999998e92 < b < 8.39999999999999978e27

   1. Initial program 97.3%

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

   3. Taylor expanded in y around 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 80.4

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

   5. Applied rewrites 80.4%

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

   6. Taylor expanded in b around 0

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

      1. exp-sum N/A

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

      2. *-commutative N/A

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

      3. exp-to-pow N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. exp-to-pow N/A

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

      7. lower-pow.f64 N/A

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

      8. lower--.f64 90.2

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

   8. Applied rewrites 90.2%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 80.8%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9 \cdot 10^{+92} \lor \neg \left(b \leq 8.4 \cdot 10^{+27}\right):\\ \;\;\;\;\frac{e^{-b}}{y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot {a}^{\left(t - 1\right)}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 48.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \frac{e^{-b}}{y} \cdot x \end{array} \]

FPCore

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

C

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

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 = (exp(-b) / y) * x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.3%

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

3. Taylor expanded in 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 83.9

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

5. Applied rewrites 83.9%

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

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 43.8

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

8. Applied rewrites 43.8%

   \[\leadsto \frac{x \cdot e^{\color{blue}{-b}}}{y} \]
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 43.8

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

10. Applied rewrites 43.8%

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

Developer Target 1: 71.8% 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 2024364 
(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))