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

Percentage Accurate: 98.2% → 98.9%

Time: 13.9s

Alternatives: 17

Speedup: 1.0×

• 46.6% 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: 98.2% 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: 98.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left({a}^{\left(\frac{1 - t}{-2}\right)}\right)}^{\left(\frac{y \cdot \log z - b}{\left(t - 1\right) \cdot \log a} + 1\right)}\\ \frac{t\_1 \cdot t\_1}{y} \cdot x \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1
         (pow
          (pow a (/ (- 1.0 t) -2.0))
          (+ (/ (- (* y (log z)) b) (* (- t 1.0) (log a))) 1.0))))
   (* (/ (* t_1 t_1) y) x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = pow(pow(a, ((1.0 - t) / -2.0)), ((((y * log(z)) - b) / ((t - 1.0) * log(a))) + 1.0));
	return ((t_1 * t_1) / 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
    real(8) :: t_1
    t_1 = (a ** ((1.0d0 - t) / (-2.0d0))) ** ((((y * log(z)) - b) / ((t - 1.0d0) * log(a))) + 1.0d0)
    code = ((t_1 * t_1) / y) * x
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = Math.pow(Math.pow(a, ((1.0 - t) / -2.0)), ((((y * Math.log(z)) - b) / ((t - 1.0) * Math.log(a))) + 1.0));
	return ((t_1 * t_1) / y) * x;
}

Python

def code(x, y, z, t, a, b):
	t_1 = math.pow(math.pow(a, ((1.0 - t) / -2.0)), ((((y * math.log(z)) - b) / ((t - 1.0) * math.log(a))) + 1.0))
	return ((t_1 * t_1) / y) * x

Julia

function code(x, y, z, t, a, b)
	t_1 = (a ^ Float64(Float64(1.0 - t) / -2.0)) ^ Float64(Float64(Float64(Float64(y * log(z)) - b) / Float64(Float64(t - 1.0) * log(a))) + 1.0)
	return Float64(Float64(Float64(t_1 * t_1) / y) * x)
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	t_1 = (a ^ ((1.0 - t) / -2.0)) ^ ((((y * log(z)) - b) / ((t - 1.0) * log(a))) + 1.0);
	tmp = ((t_1 * t_1) / y) * x;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[Power[N[Power[a, N[(N[(1.0 - t), $MachinePrecision] / -2.0), $MachinePrecision]], $MachinePrecision], N[(N[(N[(N[(y * N[Log[z], $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] / N[(N[(t - 1.0), $MachinePrecision] * N[Log[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(t$95$1 * t$95$1), $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Initial program 98.2%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

3. Applied rewrites 98.5%

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

   1. lift-exp.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. sum-to-mult N/A

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

   5. *-commutative N/A

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

   6. pow-exp N/A

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

   7. lift-*.f64 N/A

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

   8. lift-log.f64 N/A

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

   9. pow-to-exp N/A

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

   10. sqr-pow N/A

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

   11. unpow-prod-down N/A

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

5. Applied rewrites 98.9%

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

Alternative 2: 98.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \frac{{\left({a}^{\left(-0.5 \cdot \left(1 - t\right)\right)}\right)}^{\left(\mathsf{fma}\left(\frac{\log z \cdot y - b}{\log a \cdot \left(t - 1\right)}, 2, 2\right)\right)}}{y} \cdot x \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (*
  (/
   (pow
    (pow a (* -0.5 (- 1.0 t)))
    (fma (/ (- (* (log z) y) b) (* (log a) (- t 1.0))) 2.0 2.0))
   y)
  x))

C

double code(double x, double y, double z, double t, double a, double b) {
	return (pow(pow(a, (-0.5 * (1.0 - t))), fma((((log(z) * y) - b) / (log(a) * (t - 1.0))), 2.0, 2.0)) / y) * x;
}

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(((a ^ Float64(-0.5 * Float64(1.0 - t))) ^ fma(Float64(Float64(Float64(log(z) * y) - b) / Float64(log(a) * Float64(t - 1.0))), 2.0, 2.0)) / y) * x)
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(N[Power[N[Power[a, N[(-0.5 * N[(1.0 - t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(N[(N[(N[Log[z], $MachinePrecision] * y), $MachinePrecision] - b), $MachinePrecision] / N[(N[Log[a], $MachinePrecision] * N[(t - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0 + 2.0), $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision]

Derivation

1. Initial program 98.2%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

3. Applied rewrites 98.5%

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

   1. lift-exp.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. sum-to-mult N/A

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

   5. *-commutative N/A

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

   6. pow-exp N/A

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

   7. lift-*.f64 N/A

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

   8. lift-log.f64 N/A

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

   9. pow-to-exp N/A

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

   10. sqr-pow N/A

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

   11. unpow-prod-down N/A

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

5. Applied rewrites 98.9%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-pow.f64 N/A

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

   4. pow-sqr N/A

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

   5. lower-pow.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. div-flip N/A

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

   8. associate-/r/ N/A

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

   9. lower-*.f64 N/A

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

   10. metadata-eval N/A

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

7. Applied rewrites 98.9%

   \[\leadsto \frac{\color{blue}{{\left({a}^{\left(-0.5 \cdot \left(1 - t\right)\right)}\right)}^{\left(\mathsf{fma}\left(\frac{\log z \cdot y - b}{\log a \cdot \left(t - 1\right)}, 2, 2\right)\right)}}}{y} \cdot x \]
8. Add Preprocessing

Alternative 3: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.2%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

3. Applied rewrites 98.5%

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

   1. lift-exp.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. sum-to-mult N/A

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

   5. *-commutative N/A

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

   6. pow-exp N/A

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

   7. lift-*.f64 N/A

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

   8. lift-log.f64 N/A

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

   9. pow-to-exp N/A

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

   10. sqr-pow N/A

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

   11. unpow-prod-down N/A

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

5. Applied rewrites 98.9%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. div-flip N/A

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

   5. mult-flip-rev N/A

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

7. Applied rewrites 98.5%

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

Alternative 4: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.2%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

3. Applied rewrites 98.5%

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

   1. lift-exp.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. sum-to-mult N/A

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

   5. *-commutative N/A

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

   6. pow-exp N/A

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

   7. lift-*.f64 N/A

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

   8. lift-log.f64 N/A

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

   9. pow-to-exp N/A

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

   10. sqr-pow N/A

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

   11. unpow-prod-down N/A

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

5. Applied rewrites 98.9%

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

7. Applied rewrites 98.5%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. associate-/l/ N/A

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

   6. mult-flip-rev N/A

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

   7. lower-/.f64 N/A

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

   8. lower-*.f64 98.5

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 98.5

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

9. Applied rewrites 98.5%

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

Alternative 5: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.2%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

3. Applied rewrites 98.5%

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

Alternative 6: 93.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t - 1\right) \cdot \log a\\ \mathbf{if}\;t\_1 \leq -595:\\ \;\;\;\;e^{\mathsf{fma}\left(\log a, t - 1, \log z \cdot y - b\right)} \cdot \frac{x}{y}\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+65}:\\ \;\;\;\;\frac{x \cdot e^{\mathsf{fma}\left(-1, \log a, y \cdot \log z\right) - b}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \sqrt{e^{\left(t\_1 - b\right) \cdot 2}}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- t 1.0) (log a))))
   (if (<= t_1 -595.0)
     (* (exp (fma (log a) (- t 1.0) (- (* (log z) y) b))) (/ x y))
     (if (<= t_1 4e+65)
       (/ (* x (exp (- (fma -1.0 (log a) (* y (log z))) b))) y)
       (/ (* x (sqrt (exp (* (- t_1 b) 2.0)))) 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 <= -595.0) {
		tmp = exp(fma(log(a), (t - 1.0), ((log(z) * y) - b))) * (x / y);
	} else if (t_1 <= 4e+65) {
		tmp = (x * exp((fma(-1.0, log(a), (y * log(z))) - b))) / y;
	} else {
		tmp = (x * sqrt(exp(((t_1 - b) * 2.0)))) / 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 <= -595.0)
		tmp = Float64(exp(fma(log(a), Float64(t - 1.0), Float64(Float64(log(z) * y) - b))) * Float64(x / y));
	elseif (t_1 <= 4e+65)
		tmp = Float64(Float64(x * exp(Float64(fma(-1.0, log(a), Float64(y * log(z))) - b))) / y);
	else
		tmp = Float64(Float64(x * sqrt(exp(Float64(Float64(t_1 - b) * 2.0)))) / 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, -595.0], N[(N[Exp[N[(N[Log[a], $MachinePrecision] * N[(t - 1.0), $MachinePrecision] + N[(N[(N[Log[z], $MachinePrecision] * y), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e+65], N[(N[(x * N[Exp[N[(N[(-1.0 * N[Log[a], $MachinePrecision] + N[(y * N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(N[(x * N[Sqrt[N[Exp[N[(N[(t$95$1 - b), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < -595

   1. Initial program 98.2%

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

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 88.1%

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

   if -595 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < 4e65

   1. Initial program 98.2%

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

   2. Taylor expanded in t around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-log.f64 79.2

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

   4. Applied rewrites 79.2%

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

   if 4e65 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))

   1. Initial program 98.2%

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

   2. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. lower--.f64 79.7

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

   4. Applied rewrites 79.7%

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

      1. lift-exp.f64 N/A

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

      2. exp-fabs N/A

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

      3. lift-exp.f64 N/A

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

      4. rem-sqrt-square-rev N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-exp.f64 N/A

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

   6. Applied rewrites 78.6%

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

Alternative 7: 92.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t - 1\right) \cdot \log a\\ t_2 := t\_1 - b\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+27}:\\ \;\;\;\;\frac{e^{t\_2}}{y} \cdot x\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+65}:\\ \;\;\;\;\frac{e^{\mathsf{fma}\left(\log a, -1, \log z \cdot y - b\right)}}{y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \sqrt{e^{t\_2 \cdot 2}}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- t 1.0) (log a))) (t_2 (- t_1 b)))
   (if (<= t_1 -2e+27)
     (* (/ (exp t_2) y) x)
     (if (<= t_1 4e+65)
       (* (/ (exp (fma (log a) -1.0 (- (* (log z) y) b))) y) x)
       (/ (* x (sqrt (exp (* t_2 2.0)))) 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 t_2 = t_1 - b;
	double tmp;
	if (t_1 <= -2e+27) {
		tmp = (exp(t_2) / y) * x;
	} else if (t_1 <= 4e+65) {
		tmp = (exp(fma(log(a), -1.0, ((log(z) * y) - b))) / y) * x;
	} else {
		tmp = (x * sqrt(exp((t_2 * 2.0)))) / y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - 1.0) * log(a))
	t_2 = Float64(t_1 - b)
	tmp = 0.0
	if (t_1 <= -2e+27)
		tmp = Float64(Float64(exp(t_2) / y) * x);
	elseif (t_1 <= 4e+65)
		tmp = Float64(Float64(exp(fma(log(a), -1.0, Float64(Float64(log(z) * y) - b))) / y) * x);
	else
		tmp = Float64(Float64(x * sqrt(exp(Float64(t_2 * 2.0)))) / 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]}, Block[{t$95$2 = N[(t$95$1 - b), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+27], N[(N[(N[Exp[t$95$2], $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$1, 4e+65], N[(N[(N[Exp[N[(N[Log[a], $MachinePrecision] * -1.0 + N[(N[(N[Log[z], $MachinePrecision] * y), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision], N[(N[(x * N[Sqrt[N[Exp[N[(t$95$2 * 2.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < -2e27

   1. Initial program 98.2%

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

   2. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. lower--.f64 79.7

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

   4. Applied rewrites 79.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   6. Applied rewrites 80.0%

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

   if -2e27 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < 4e65

   1. Initial program 98.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 98.5%

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

   4. Taylor expanded in t around 0

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

   6. Applied rewrites 79.7%

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

   if 4e65 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))

   1. Initial program 98.2%

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

   2. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. lower--.f64 79.7

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

   4. Applied rewrites 79.7%

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

      1. lift-exp.f64 N/A

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

      2. exp-fabs N/A

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

      3. lift-exp.f64 N/A

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

      4. rem-sqrt-square-rev N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-exp.f64 N/A

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

   6. Applied rewrites 78.6%

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

Alternative 8: 90.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t - 1\right) \cdot \log a\\ t_2 := t\_1 - b\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+27}:\\ \;\;\;\;\frac{e^{t\_2}}{y} \cdot x\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+65}:\\ \;\;\;\;\frac{x}{y \cdot e^{\left(b + \log a\right) - y \cdot \log z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \sqrt{e^{t\_2 \cdot 2}}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- t 1.0) (log a))) (t_2 (- t_1 b)))
   (if (<= t_1 -2e+27)
     (* (/ (exp t_2) y) x)
     (if (<= t_1 4e+65)
       (/ x (* y (exp (- (+ b (log a)) (* y (log z))))))
       (/ (* x (sqrt (exp (* t_2 2.0)))) 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 t_2 = t_1 - b;
	double tmp;
	if (t_1 <= -2e+27) {
		tmp = (exp(t_2) / y) * x;
	} else if (t_1 <= 4e+65) {
		tmp = x / (y * exp(((b + log(a)) - (y * log(z)))));
	} else {
		tmp = (x * sqrt(exp((t_2 * 2.0)))) / y;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (t - 1.0d0) * log(a)
    t_2 = t_1 - b
    if (t_1 <= (-2d+27)) then
        tmp = (exp(t_2) / y) * x
    else if (t_1 <= 4d+65) then
        tmp = x / (y * exp(((b + log(a)) - (y * log(z)))))
    else
        tmp = (x * sqrt(exp((t_2 * 2.0d0)))) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - 1.0) * Math.log(a);
	double t_2 = t_1 - b;
	double tmp;
	if (t_1 <= -2e+27) {
		tmp = (Math.exp(t_2) / y) * x;
	} else if (t_1 <= 4e+65) {
		tmp = x / (y * Math.exp(((b + Math.log(a)) - (y * Math.log(z)))));
	} else {
		tmp = (x * Math.sqrt(Math.exp((t_2 * 2.0)))) / y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (t - 1.0) * math.log(a)
	t_2 = t_1 - b
	tmp = 0
	if t_1 <= -2e+27:
		tmp = (math.exp(t_2) / y) * x
	elif t_1 <= 4e+65:
		tmp = x / (y * math.exp(((b + math.log(a)) - (y * math.log(z)))))
	else:
		tmp = (x * math.sqrt(math.exp((t_2 * 2.0)))) / y
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - 1.0) * log(a))
	t_2 = Float64(t_1 - b)
	tmp = 0.0
	if (t_1 <= -2e+27)
		tmp = Float64(Float64(exp(t_2) / y) * x);
	elseif (t_1 <= 4e+65)
		tmp = Float64(x / Float64(y * exp(Float64(Float64(b + log(a)) - Float64(y * log(z))))));
	else
		tmp = Float64(Float64(x * sqrt(exp(Float64(t_2 * 2.0)))) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (t - 1.0) * log(a);
	t_2 = t_1 - b;
	tmp = 0.0;
	if (t_1 <= -2e+27)
		tmp = (exp(t_2) / y) * x;
	elseif (t_1 <= 4e+65)
		tmp = x / (y * exp(((b + log(a)) - (y * log(z)))));
	else
		tmp = (x * sqrt(exp((t_2 * 2.0)))) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - 1.0), $MachinePrecision] * N[Log[a], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - b), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+27], N[(N[(N[Exp[t$95$2], $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$1, 4e+65], N[(x / N[(y * N[Exp[N[(N[(b + N[Log[a], $MachinePrecision]), $MachinePrecision] - N[(y * N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[Sqrt[N[Exp[N[(t$95$2 * 2.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < -2e27

   1. Initial program 98.2%

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

   2. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. lower--.f64 79.7

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

   4. Applied rewrites 79.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   6. Applied rewrites 80.0%

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

   if -2e27 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < 4e65

   1. Initial program 98.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 98.5%

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

      1. lift-exp.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. sum-to-mult N/A

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

      5. *-commutative N/A

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

      6. pow-exp N/A

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

      7. lift-*.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. pow-to-exp N/A

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

      10. sqr-pow N/A

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

      11. unpow-prod-down N/A

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

   5. Applied rewrites 98.9%

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

   7. Applied rewrites 98.5%

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

   8. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-log.f64 79.7

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

   10. Applied rewrites 79.7%

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

   if 4e65 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))

   1. Initial program 98.2%

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

   2. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. lower--.f64 79.7

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

   4. Applied rewrites 79.7%

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

      1. lift-exp.f64 N/A

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

      2. exp-fabs N/A

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

      3. lift-exp.f64 N/A

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

      4. rem-sqrt-square-rev N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-exp.f64 N/A

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

   6. Applied rewrites 78.6%

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

Alternative 9: 89.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot \sqrt{e^{2 \cdot \left(y \cdot \log z\right)}}}{y}\\ \mathbf{if}\;y \leq -1.1 \cdot 10^{+111}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{+123}:\\ \;\;\;\;\frac{e^{\left(t - 1\right) \cdot \log a - b}}{y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (* x (sqrt (exp (* 2.0 (* y (log z)))))) y)))
   (if (<= y -1.1e+111)
     t_1
     (if (<= y 2.8e+123) (* (/ (exp (- (* (- t 1.0) (log a)) b)) y) x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x * sqrt(exp((2.0 * (y * log(z)))))) / y;
	double tmp;
	if (y <= -1.1e+111) {
		tmp = t_1;
	} else if (y <= 2.8e+123) {
		tmp = (exp((((t - 1.0) * log(a)) - b)) / y) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x * sqrt(exp((2.0d0 * (y * log(z)))))) / y
    if (y <= (-1.1d+111)) then
        tmp = t_1
    else if (y <= 2.8d+123) then
        tmp = (exp((((t - 1.0d0) * log(a)) - b)) / y) * x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x * Math.sqrt(Math.exp((2.0 * (y * Math.log(z)))))) / y;
	double tmp;
	if (y <= -1.1e+111) {
		tmp = t_1;
	} else if (y <= 2.8e+123) {
		tmp = (Math.exp((((t - 1.0) * Math.log(a)) - b)) / y) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (x * math.sqrt(math.exp((2.0 * (y * math.log(z)))))) / y
	tmp = 0
	if y <= -1.1e+111:
		tmp = t_1
	elif y <= 2.8e+123:
		tmp = (math.exp((((t - 1.0) * math.log(a)) - b)) / y) * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x * sqrt(exp(Float64(2.0 * Float64(y * log(z)))))) / y)
	tmp = 0.0
	if (y <= -1.1e+111)
		tmp = t_1;
	elseif (y <= 2.8e+123)
		tmp = Float64(Float64(exp(Float64(Float64(Float64(t - 1.0) * log(a)) - b)) / y) * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (x * sqrt(exp((2.0 * (y * log(z)))))) / y;
	tmp = 0.0;
	if (y <= -1.1e+111)
		tmp = t_1;
	elseif (y <= 2.8e+123)
		tmp = (exp((((t - 1.0) * log(a)) - b)) / y) * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x * N[Sqrt[N[Exp[N[(2.0 * N[(y * N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[y, -1.1e+111], t$95$1, If[LessEqual[y, 2.8e+123], N[(N[(N[Exp[N[(N[(N[(t - 1.0), $MachinePrecision] * N[Log[a], $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.09999999999999999e111 or 2.80000000000000011e123 < y

   1. Initial program 98.2%

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

   2. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. lower--.f64 79.7

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

   4. Applied rewrites 79.7%

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

      1. lift-exp.f64 N/A

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

      2. exp-fabs N/A

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

      3. lift-exp.f64 N/A

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

      4. rem-sqrt-square-rev N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-exp.f64 N/A

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

   6. Applied rewrites 78.6%

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

   7. Taylor expanded in y around inf

      \[\leadsto \frac{x \cdot \sqrt{e^{\color{blue}{2 \cdot \left(y \cdot \log z\right)}}}}{y} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{x \cdot \sqrt{e^{2 \cdot \color{blue}{\left(y \cdot \log z\right)}}}}{y} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{x \cdot \sqrt{e^{2 \cdot \left(y \cdot \color{blue}{\log z}\right)}}}{y} \]

      3. lower-log.f64 48.1

         \[\leadsto \frac{x \cdot \sqrt{e^{2 \cdot \left(y \cdot \log z\right)}}}{y} \]

   9. Applied rewrites 48.1%

      \[\leadsto \frac{x \cdot \sqrt{e^{\color{blue}{2 \cdot \left(y \cdot \log z\right)}}}}{y} \]

   if -1.09999999999999999e111 < y < 2.80000000000000011e123

   1. Initial program 98.2%

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

   2. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. lower--.f64 79.7

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

   4. Applied rewrites 79.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   6. Applied rewrites 80.0%

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

Alternative 10: 84.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot \sqrt{e^{2 \cdot \left(y \cdot \log z\right)}}}{y}\\ \mathbf{if}\;y \leq -3.7 \cdot 10^{+79}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{+123}:\\ \;\;\;\;e^{\left(t - 1\right) \cdot \log a - b} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (* x (sqrt (exp (* 2.0 (* y (log z)))))) y)))
   (if (<= y -3.7e+79)
     t_1
     (if (<= y 2.8e+123) (* (exp (- (* (- t 1.0) (log a)) b)) (/ x y)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x * sqrt(exp((2.0 * (y * log(z)))))) / y;
	double tmp;
	if (y <= -3.7e+79) {
		tmp = t_1;
	} else if (y <= 2.8e+123) {
		tmp = exp((((t - 1.0) * log(a)) - b)) * (x / y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x * sqrt(exp((2.0d0 * (y * log(z)))))) / y
    if (y <= (-3.7d+79)) then
        tmp = t_1
    else if (y <= 2.8d+123) then
        tmp = exp((((t - 1.0d0) * log(a)) - b)) * (x / y)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x * Math.sqrt(Math.exp((2.0 * (y * Math.log(z)))))) / y;
	double tmp;
	if (y <= -3.7e+79) {
		tmp = t_1;
	} else if (y <= 2.8e+123) {
		tmp = Math.exp((((t - 1.0) * Math.log(a)) - b)) * (x / y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (x * math.sqrt(math.exp((2.0 * (y * math.log(z)))))) / y
	tmp = 0
	if y <= -3.7e+79:
		tmp = t_1
	elif y <= 2.8e+123:
		tmp = math.exp((((t - 1.0) * math.log(a)) - b)) * (x / y)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x * sqrt(exp(Float64(2.0 * Float64(y * log(z)))))) / y)
	tmp = 0.0
	if (y <= -3.7e+79)
		tmp = t_1;
	elseif (y <= 2.8e+123)
		tmp = Float64(exp(Float64(Float64(Float64(t - 1.0) * log(a)) - b)) * Float64(x / y));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (x * sqrt(exp((2.0 * (y * log(z)))))) / y;
	tmp = 0.0;
	if (y <= -3.7e+79)
		tmp = t_1;
	elseif (y <= 2.8e+123)
		tmp = exp((((t - 1.0) * log(a)) - b)) * (x / y);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x * N[Sqrt[N[Exp[N[(2.0 * N[(y * N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[y, -3.7e+79], t$95$1, If[LessEqual[y, 2.8e+123], N[(N[Exp[N[(N[(N[(t - 1.0), $MachinePrecision] * N[Log[a], $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.70000000000000009e79 or 2.80000000000000011e123 < y

   1. Initial program 98.2%

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

   2. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. lower--.f64 79.7

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

   4. Applied rewrites 79.7%

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

      1. lift-exp.f64 N/A

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

      2. exp-fabs N/A

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

      3. lift-exp.f64 N/A

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

      4. rem-sqrt-square-rev N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-exp.f64 N/A

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

   6. Applied rewrites 78.6%

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

   7. Taylor expanded in y around inf

      \[\leadsto \frac{x \cdot \sqrt{e^{\color{blue}{2 \cdot \left(y \cdot \log z\right)}}}}{y} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{x \cdot \sqrt{e^{2 \cdot \color{blue}{\left(y \cdot \log z\right)}}}}{y} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{x \cdot \sqrt{e^{2 \cdot \left(y \cdot \color{blue}{\log z}\right)}}}{y} \]

      3. lower-log.f64 48.1

         \[\leadsto \frac{x \cdot \sqrt{e^{2 \cdot \left(y \cdot \log z\right)}}}{y} \]

   9. Applied rewrites 48.1%

      \[\leadsto \frac{x \cdot \sqrt{e^{\color{blue}{2 \cdot \left(y \cdot \log z\right)}}}}{y} \]

   if -3.70000000000000009e79 < y < 2.80000000000000011e123

   1. Initial program 98.2%

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

   2. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. lower--.f64 79.7

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

   4. Applied rewrites 79.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. mult-flip-rev N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-*.f64 72.7

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

   6. Applied rewrites 72.7%

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

Alternative 11: 74.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{e^{t \cdot \log a}}{y} \cdot x\\ \mathbf{if}\;t \leq -76:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 3.7 \cdot 10^{+54}:\\ \;\;\;\;\frac{x \cdot e^{\log a \cdot -1 - b}}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (/ (exp (* t (log a))) y) x)))
   (if (<= t -76.0)
     t_1
     (if (<= t 3.7e+54) (/ (* x (exp (- (* (log a) -1.0) b))) y) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (exp((t * log(a))) / y) * x;
	double tmp;
	if (t <= -76.0) {
		tmp = t_1;
	} else if (t <= 3.7e+54) {
		tmp = (x * exp(((log(a) * -1.0) - b))) / y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (exp((t * log(a))) / y) * x
    if (t <= (-76.0d0)) then
        tmp = t_1
    else if (t <= 3.7d+54) then
        tmp = (x * exp(((log(a) * (-1.0d0)) - b))) / y
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (Math.exp((t * Math.log(a))) / y) * x;
	double tmp;
	if (t <= -76.0) {
		tmp = t_1;
	} else if (t <= 3.7e+54) {
		tmp = (x * Math.exp(((Math.log(a) * -1.0) - b))) / y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (math.exp((t * math.log(a))) / y) * x
	tmp = 0
	if t <= -76.0:
		tmp = t_1
	elif t <= 3.7e+54:
		tmp = (x * math.exp(((math.log(a) * -1.0) - b))) / y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(exp(Float64(t * log(a))) / y) * x)
	tmp = 0.0
	if (t <= -76.0)
		tmp = t_1;
	elseif (t <= 3.7e+54)
		tmp = Float64(Float64(x * exp(Float64(Float64(log(a) * -1.0) - b))) / y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (exp((t * log(a))) / y) * x;
	tmp = 0.0;
	if (t <= -76.0)
		tmp = t_1;
	elseif (t <= 3.7e+54)
		tmp = (x * exp(((log(a) * -1.0) - b))) / y;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[Exp[N[(t * N[Log[a], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[t, -76.0], t$95$1, If[LessEqual[t, 3.7e+54], N[(N[(x * N[Exp[N[(N[(N[Log[a], $MachinePrecision] * -1.0), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -76 or 3.7000000000000002e54 < t

   1. Initial program 98.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 98.5%

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

   4. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 48.4

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

   6. Applied rewrites 48.4%

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

   if -76 < t < 3.7000000000000002e54

   1. Initial program 98.2%

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

   2. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. lower--.f64 79.7

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

   4. Applied rewrites 79.7%

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

   5. Taylor expanded in t around 0

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

   7. Applied rewrites 57.5%

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

Alternative 12: 73.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t - 1\right) \cdot \log a\\ t_2 := \frac{e^{t \cdot \log a}}{y} \cdot x\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+27}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+51}:\\ \;\;\;\;\frac{e^{-1 \cdot \log a - b}}{y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- t 1.0) (log a))) (t_2 (* (/ (exp (* t (log a))) y) x)))
   (if (<= t_1 -2e+27)
     t_2
     (if (<= t_1 5e+51) (* (/ (exp (- (* -1.0 (log a)) b)) y) x) 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 = (exp((t * log(a))) / y) * x;
	double tmp;
	if (t_1 <= -2e+27) {
		tmp = t_2;
	} else if (t_1 <= 5e+51) {
		tmp = (exp(((-1.0 * log(a)) - b)) / y) * x;
	} 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 = (exp((t * log(a))) / y) * x
    if (t_1 <= (-2d+27)) then
        tmp = t_2
    else if (t_1 <= 5d+51) then
        tmp = (exp((((-1.0d0) * log(a)) - b)) / y) * x
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - 1.0) * Math.log(a);
	double t_2 = (Math.exp((t * Math.log(a))) / y) * x;
	double tmp;
	if (t_1 <= -2e+27) {
		tmp = t_2;
	} else if (t_1 <= 5e+51) {
		tmp = (Math.exp(((-1.0 * Math.log(a)) - b)) / y) * x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (t - 1.0) * math.log(a)
	t_2 = (math.exp((t * math.log(a))) / y) * x
	tmp = 0
	if t_1 <= -2e+27:
		tmp = t_2
	elif t_1 <= 5e+51:
		tmp = (math.exp(((-1.0 * math.log(a)) - b)) / y) * x
	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(exp(Float64(t * log(a))) / y) * x)
	tmp = 0.0
	if (t_1 <= -2e+27)
		tmp = t_2;
	elseif (t_1 <= 5e+51)
		tmp = Float64(Float64(exp(Float64(Float64(-1.0 * log(a)) - b)) / y) * x);
	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 = (exp((t * log(a))) / y) * x;
	tmp = 0.0;
	if (t_1 <= -2e+27)
		tmp = t_2;
	elseif (t_1 <= 5e+51)
		tmp = (exp(((-1.0 * log(a)) - b)) / y) * x;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - 1.0), $MachinePrecision] * N[Log[a], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Exp[N[(t * N[Log[a], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+27], t$95$2, If[LessEqual[t$95$1, 5e+51], N[(N[(N[Exp[N[(N[(-1.0 * N[Log[a], $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < -2e27 or 5e51 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))

   1. Initial program 98.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 98.5%

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

   4. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 48.4

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

   6. Applied rewrites 48.4%

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

   if -2e27 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < 5e51

   1. Initial program 98.2%

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

   2. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. lower--.f64 79.7

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

   4. Applied rewrites 79.7%

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

   5. Taylor expanded in t around 0

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

   7. Applied rewrites 57.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   9. Applied rewrites 58.1%

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

Alternative 13: 71.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t - 1\right) \cdot \log a\\ t_2 := \frac{e^{t \cdot \log a}}{y} \cdot x\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+27}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+51}:\\ \;\;\;\;e^{-1 \cdot \log a - b} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- t 1.0) (log a))) (t_2 (* (/ (exp (* t (log a))) y) x)))
   (if (<= t_1 -2e+27)
     t_2
     (if (<= t_1 5e+51) (* (exp (- (* -1.0 (log a)) b)) (/ x 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 = (exp((t * log(a))) / y) * x;
	double tmp;
	if (t_1 <= -2e+27) {
		tmp = t_2;
	} else if (t_1 <= 5e+51) {
		tmp = exp(((-1.0 * log(a)) - b)) * (x / 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 = (exp((t * log(a))) / y) * x
    if (t_1 <= (-2d+27)) then
        tmp = t_2
    else if (t_1 <= 5d+51) then
        tmp = exp((((-1.0d0) * log(a)) - b)) * (x / y)
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - 1.0) * Math.log(a);
	double t_2 = (Math.exp((t * Math.log(a))) / y) * x;
	double tmp;
	if (t_1 <= -2e+27) {
		tmp = t_2;
	} else if (t_1 <= 5e+51) {
		tmp = Math.exp(((-1.0 * Math.log(a)) - b)) * (x / y);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (t - 1.0) * math.log(a)
	t_2 = (math.exp((t * math.log(a))) / y) * x
	tmp = 0
	if t_1 <= -2e+27:
		tmp = t_2
	elif t_1 <= 5e+51:
		tmp = math.exp(((-1.0 * math.log(a)) - b)) * (x / 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(exp(Float64(t * log(a))) / y) * x)
	tmp = 0.0
	if (t_1 <= -2e+27)
		tmp = t_2;
	elseif (t_1 <= 5e+51)
		tmp = Float64(exp(Float64(Float64(-1.0 * log(a)) - b)) * Float64(x / 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 = (exp((t * log(a))) / y) * x;
	tmp = 0.0;
	if (t_1 <= -2e+27)
		tmp = t_2;
	elseif (t_1 <= 5e+51)
		tmp = exp(((-1.0 * log(a)) - b)) * (x / y);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - 1.0), $MachinePrecision] * N[Log[a], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Exp[N[(t * N[Log[a], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+27], t$95$2, If[LessEqual[t$95$1, 5e+51], N[(N[Exp[N[(N[(-1.0 * N[Log[a], $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < -2e27 or 5e51 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))

   1. Initial program 98.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 98.5%

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

   4. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 48.4

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

   6. Applied rewrites 48.4%

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

   if -2e27 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < 5e51

   1. Initial program 98.2%

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

   2. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. lower--.f64 79.7

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

   4. Applied rewrites 79.7%

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

   5. Taylor expanded in t around 0

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

   7. Applied rewrites 57.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. lower-*.f64 53.6

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

   9. Applied rewrites 53.6%

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

Alternative 14: 65.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t - 1\right) \cdot \log a\\ t_2 := \frac{e^{t \cdot \log a}}{y} \cdot x\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+27}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+51}:\\ \;\;\;\;\frac{e^{-b}}{y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- t 1.0) (log a))) (t_2 (* (/ (exp (* t (log a))) y) x)))
   (if (<= t_1 -2e+27) t_2 (if (<= t_1 5e+51) (* (/ (exp (- b)) y) x) 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 = (exp((t * log(a))) / y) * x;
	double tmp;
	if (t_1 <= -2e+27) {
		tmp = t_2;
	} else if (t_1 <= 5e+51) {
		tmp = (exp(-b) / y) * x;
	} 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 = (exp((t * log(a))) / y) * x
    if (t_1 <= (-2d+27)) then
        tmp = t_2
    else if (t_1 <= 5d+51) then
        tmp = (exp(-b) / y) * x
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - 1.0) * Math.log(a);
	double t_2 = (Math.exp((t * Math.log(a))) / y) * x;
	double tmp;
	if (t_1 <= -2e+27) {
		tmp = t_2;
	} else if (t_1 <= 5e+51) {
		tmp = (Math.exp(-b) / y) * x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (t - 1.0) * math.log(a)
	t_2 = (math.exp((t * math.log(a))) / y) * x
	tmp = 0
	if t_1 <= -2e+27:
		tmp = t_2
	elif t_1 <= 5e+51:
		tmp = (math.exp(-b) / y) * x
	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(exp(Float64(t * log(a))) / y) * x)
	tmp = 0.0
	if (t_1 <= -2e+27)
		tmp = t_2;
	elseif (t_1 <= 5e+51)
		tmp = Float64(Float64(exp(Float64(-b)) / y) * x);
	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 = (exp((t * log(a))) / y) * x;
	tmp = 0.0;
	if (t_1 <= -2e+27)
		tmp = t_2;
	elseif (t_1 <= 5e+51)
		tmp = (exp(-b) / y) * x;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - 1.0), $MachinePrecision] * N[Log[a], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Exp[N[(t * N[Log[a], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+27], t$95$2, If[LessEqual[t$95$1, 5e+51], N[(N[(N[Exp[(-b)], $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < -2e27 or 5e51 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))

   1. Initial program 98.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 98.5%

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

   4. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 48.4

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

   6. Applied rewrites 48.4%

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

   if -2e27 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < 5e51

   1. Initial program 98.2%

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

   2. Taylor expanded in b around inf

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

      1. lower-*.f64 47.1

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

   4. Applied rewrites 47.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 47.1

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

      7. lift-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 47.1

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

   6. Applied rewrites 47.1%

      \[\leadsto \color{blue}{\frac{e^{-b}}{y} \cdot x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 62.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{e^{-b}}{y} \cdot x\\ \mathbf{if}\;b \leq -11500000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 290000000:\\ \;\;\;\;e^{t \cdot \log a} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (/ (exp (- b)) y) x)))
   (if (<= b -11500000000000.0)
     t_1
     (if (<= b 290000000.0) (* (exp (* t (log a))) (/ x y)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (exp(-b) / y) * x;
	double tmp;
	if (b <= -11500000000000.0) {
		tmp = t_1;
	} else if (b <= 290000000.0) {
		tmp = exp((t * log(a))) * (x / y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (exp(-b) / y) * x
    if (b <= (-11500000000000.0d0)) then
        tmp = t_1
    else if (b <= 290000000.0d0) then
        tmp = exp((t * log(a))) * (x / y)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (Math.exp(-b) / y) * x;
	double tmp;
	if (b <= -11500000000000.0) {
		tmp = t_1;
	} else if (b <= 290000000.0) {
		tmp = Math.exp((t * Math.log(a))) * (x / y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (math.exp(-b) / y) * x
	tmp = 0
	if b <= -11500000000000.0:
		tmp = t_1
	elif b <= 290000000.0:
		tmp = math.exp((t * math.log(a))) * (x / y)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(exp(Float64(-b)) / y) * x)
	tmp = 0.0
	if (b <= -11500000000000.0)
		tmp = t_1;
	elseif (b <= 290000000.0)
		tmp = Float64(exp(Float64(t * log(a))) * Float64(x / y));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (exp(-b) / y) * x;
	tmp = 0.0;
	if (b <= -11500000000000.0)
		tmp = t_1;
	elseif (b <= 290000000.0)
		tmp = exp((t * log(a))) * (x / y);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[Exp[(-b)], $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[b, -11500000000000.0], t$95$1, If[LessEqual[b, 290000000.0], N[(N[Exp[N[(t * N[Log[a], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if b < -1.15e13 or 2.9e8 < b

   1. Initial program 98.2%

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

   2. Taylor expanded in b around inf

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

      1. lower-*.f64 47.1

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

   4. Applied rewrites 47.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 47.1

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

      7. lift-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 47.1

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

   6. Applied rewrites 47.1%

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

   if -1.15e13 < b < 2.9e8

   1. Initial program 98.2%

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

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 88.1%

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

   4. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 44.1

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

   6. Applied rewrites 44.1%

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

Alternative 16: 47.1% accurate, 2.3× 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 98.2%

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

2. Taylor expanded in b around inf

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

   1. lower-*.f64 47.1

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

4. Applied rewrites 47.1%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 47.1

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

   7. lift-*.f64 N/A

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

   8. mul-1-neg N/A

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

   9. lower-neg.f64 47.1

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

6. Applied rewrites 47.1%

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

Alternative 17: 42.9% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.2%

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

2. Taylor expanded in b around inf

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

   1. lower-*.f64 47.1

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

4. Applied rewrites 47.1%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. mult-flip-rev N/A

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

   6. lift-/.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lower-*.f64 42.9

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

   9. lift-*.f64 N/A

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

   10. mul-1-neg N/A

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

   11. lower-neg.f64 42.9

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

   12. lower-neg.f64 N/A

       \[\leadsto e^{-b} \cdot \mathsf{Rewrite=>}\left(lift-*.f64, \left(x \cdot \frac{1}{y}\right)\right) \]

   13. lower-neg.f64 N/A

       \[\leadsto e^{-b} \cdot \left(x \cdot \mathsf{Rewrite=>}\left(lift-/.f64, \left(\frac{1}{y}\right)\right)\right) \]

   14. lower-neg.f64 N/A

       \[\leadsto e^{-b} \cdot \mathsf{Rewrite=>}\left(mult-flip-rev, \left(\frac{x}{y}\right)\right) \]

   15. lower-neg.f64 N/A

       \[\leadsto e^{-b} \cdot \mathsf{Rewrite<=}\left(lift-/.f64, \left(\frac{x}{y}\right)\right) \]

6. Applied rewrites 42.9%

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

Reproduce

herbie shell --seed 2025142 
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:incompleteBetaWorker from math-functions-0.1.5.2, A"
  :precision binary64
  (/ (* x (exp (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))) y))