Numeric.Signal.Multichannel:$cput from hsignal-0.2.7.1

Percentage Accurate: 96.8% → 94.5%

Time: 3.7s

Alternatives: 15

Speedup: N/A×

Specification

Math

\[\begin{array}{l} \\ \frac{x - y}{z - y} \cdot t \end{array} \]

FPCore

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

C

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

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

Java

public static double code(double x, double y, double z, double t) {
	return ((x - y) / (z - y)) * t;
}

Python

def code(x, y, z, t):
	return ((x - y) / (z - y)) * t

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]

Initial Program: 96.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x - y}{z - y} \cdot t \end{array} \]

FPCore

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

C

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

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

Java

public static double code(double x, double y, double z, double t) {
	return ((x - y) / (z - y)) * t;
}

Python

def code(x, y, z, t):
	return ((x - y) / (z - y)) * t

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]

Alternative 1: 94.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.52 \cdot 10^{+174}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-t}{z} + t \cdot \frac{x}{z \cdot z}, y, \frac{t \cdot x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{z - y} - \frac{y}{z - y}\right) \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -1.52e+174)
   (fma (+ (/ (- t) z) (* t (/ x (* z z)))) y (/ (* t x) z))
   (* (- (/ x (- z y)) (/ y (- z y))) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.52e+174) {
		tmp = fma(((-t / z) + (t * (x / (z * z)))), y, ((t * x) / z));
	} else {
		tmp = ((x / (z - y)) - (y / (z - y))) * t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -1.52e+174)
		tmp = fma(Float64(Float64(Float64(-t) / z) + Float64(t * Float64(x / Float64(z * z)))), y, Float64(Float64(t * x) / z));
	else
		tmp = Float64(Float64(Float64(x / Float64(z - y)) - Float64(y / Float64(z - y))) * t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -1.52e+174], N[(N[(N[((-t) / z), $MachinePrecision] + N[(t * N[(x / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y + N[(N[(t * x), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision] - N[(y / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.52000000000000004e174

   1. Initial program 75.2%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(-1 \cdot \frac{t}{z} - -1 \cdot \frac{t \cdot x}{{z}^{2}}, \color{blue}{y}, \frac{t \cdot x}{z}\right) \]

      3. distribute-lft-out-- N/A

         \[\leadsto \mathsf{fma}\left(-1 \cdot \left(\frac{t}{z} - \frac{t \cdot x}{{z}^{2}}\right), y, \frac{t \cdot x}{z}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-1 \cdot \left(\frac{t}{z} - \frac{t \cdot x}{{z}^{2}}\right), y, \frac{t \cdot x}{z}\right) \]

      5. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(-1 \cdot \left(\frac{t}{z} - \frac{t \cdot x}{{z}^{2}}\right), y, \frac{t \cdot x}{z}\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(-1 \cdot \left(\frac{t}{z} - \frac{t \cdot x}{{z}^{2}}\right), y, \frac{t \cdot x}{z}\right) \]

      7. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(-1 \cdot \left(\frac{t}{z} - t \cdot \frac{x}{{z}^{2}}\right), y, \frac{t \cdot x}{z}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-1 \cdot \left(\frac{t}{z} - t \cdot \frac{x}{{z}^{2}}\right), y, \frac{t \cdot x}{z}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(-1 \cdot \left(\frac{t}{z} - t \cdot \frac{x}{{z}^{2}}\right), y, \frac{t \cdot x}{z}\right) \]

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 96.6

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

   5. Applied rewrites 96.6%

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

   if -1.52000000000000004e174 < z

   1. Initial program 99.0%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{x - y}}{z - y} \cdot t \]

      2. lift--.f64 N/A

         \[\leadsto \frac{x - y}{\color{blue}{z - y}} \cdot t \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x - y}{z - y}} \cdot t \]

      4. div-sub N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 99.0

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

   4. Applied rewrites 99.0%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.52 \cdot 10^{+174}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-t}{z} + t \cdot \frac{x}{z \cdot z}, y, \frac{t \cdot x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{z - y} - \frac{y}{z - y}\right) \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y}{z - y}\\ t_2 := \frac{x}{z - y} \cdot t\\ \mathbf{if}\;t\_1 \leq -200:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0.5:\\ \;\;\;\;\frac{x - y}{z} \cdot t\\ \mathbf{elif}\;t\_1 \leq 5000000:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{-x}{y}, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x y) (- z y))) (t_2 (* (/ x (- z y)) t)))
   (if (<= t_1 -200.0)
     t_2
     (if (<= t_1 0.5)
       (* (/ (- x y) z) t)
       (if (<= t_1 5000000.0) (fma t (/ (- x) y) t) t_2)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - y) / (z - y);
	double t_2 = (x / (z - y)) * t;
	double tmp;
	if (t_1 <= -200.0) {
		tmp = t_2;
	} else if (t_1 <= 0.5) {
		tmp = ((x - y) / z) * t;
	} else if (t_1 <= 5000000.0) {
		tmp = fma(t, (-x / y), t);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - y) / Float64(z - y))
	t_2 = Float64(Float64(x / Float64(z - y)) * t)
	tmp = 0.0
	if (t_1 <= -200.0)
		tmp = t_2;
	elseif (t_1 <= 0.5)
		tmp = Float64(Float64(Float64(x - y) / z) * t);
	elseif (t_1 <= 5000000.0)
		tmp = fma(t, Float64(Float64(-x) / y), t);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$1, -200.0], t$95$2, If[LessEqual[t$95$1, 0.5], N[(N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[t$95$1, 5000000.0], N[(t * N[((-x) / y), $MachinePrecision] + t), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -200 or 5e6 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 97.5%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 95.5%

      \[\leadsto \frac{\color{blue}{x}}{z - y} \cdot t \]

   if -200 < (/.f64 (-.f64 x y) (-.f64 z y)) < 0.5

   1. Initial program 94.0%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 90.9%

      \[\leadsto \frac{x - y}{\color{blue}{z}} \cdot t \]

   if 0.5 < (/.f64 (-.f64 x y) (-.f64 z y)) < 5e6

   1. Initial program 99.9%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{x - y}}{z - y} \cdot t \]

      2. lift--.f64 N/A

         \[\leadsto \frac{x - y}{\color{blue}{z - y}} \cdot t \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x - y}{z - y}} \cdot t \]

      4. div-sub N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 4.7%

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

   8. Taylor expanded in y around inf

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

      1. frac-sub N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

         \[\leadsto t \cdot \frac{z + -1 \cdot x}{y} + t \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{\frac{z + -1 \cdot x}{y}}, t\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(t, \frac{z + -1 \cdot x}{\color{blue}{y}}, t\right) \]

      7. +-commutative N/A

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

      8. lower-fma.f64 99.7

         \[\leadsto \mathsf{fma}\left(t, \frac{\mathsf{fma}\left(-1, x, z\right)}{y}, t\right) \]

   10. Applied rewrites 99.7%

       \[\leadsto \color{blue}{\mathsf{fma}\left(t, \frac{\mathsf{fma}\left(-1, x, z\right)}{y}, t\right)} \]

   11. Taylor expanded in x around inf

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

       1. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(t, \frac{\mathsf{neg}\left(x\right)}{y}, t\right) \]

       2. lift-neg.f64 99.3

          \[\leadsto \mathsf{fma}\left(t, \frac{-x}{y}, t\right) \]

   13. Applied rewrites 99.3%

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

Alternative 3: 93.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y}{z - y}\\ t_2 := \frac{x}{z - y} \cdot t\\ \mathbf{if}\;t\_1 \leq -2:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0.5:\\ \;\;\;\;\frac{\left(x - y\right) \cdot t}{z}\\ \mathbf{elif}\;t\_1 \leq 5000000:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{-x}{y}, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x y) (- z y))) (t_2 (* (/ x (- z y)) t)))
   (if (<= t_1 -2.0)
     t_2
     (if (<= t_1 0.5)
       (/ (* (- x y) t) z)
       (if (<= t_1 5000000.0) (fma t (/ (- x) y) t) t_2)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - y) / (z - y);
	double t_2 = (x / (z - y)) * t;
	double tmp;
	if (t_1 <= -2.0) {
		tmp = t_2;
	} else if (t_1 <= 0.5) {
		tmp = ((x - y) * t) / z;
	} else if (t_1 <= 5000000.0) {
		tmp = fma(t, (-x / y), t);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - y) / Float64(z - y))
	t_2 = Float64(Float64(x / Float64(z - y)) * t)
	tmp = 0.0
	if (t_1 <= -2.0)
		tmp = t_2;
	elseif (t_1 <= 0.5)
		tmp = Float64(Float64(Float64(x - y) * t) / z);
	elseif (t_1 <= 5000000.0)
		tmp = fma(t, Float64(Float64(-x) / y), t);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$1, -2.0], t$95$2, If[LessEqual[t$95$1, 0.5], N[(N[(N[(x - y), $MachinePrecision] * t), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$1, 5000000.0], N[(t * N[((-x) / y), $MachinePrecision] + t), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -2 or 5e6 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 97.5%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 95.5%

      \[\leadsto \frac{\color{blue}{x}}{z - y} \cdot t \]

   if -2 < (/.f64 (-.f64 x y) (-.f64 z y)) < 0.5

   1. Initial program 93.9%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift--.f64 86.7

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

   5. Applied rewrites 86.7%

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

   if 0.5 < (/.f64 (-.f64 x y) (-.f64 z y)) < 5e6

   1. Initial program 99.9%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{x - y}}{z - y} \cdot t \]

      2. lift--.f64 N/A

         \[\leadsto \frac{x - y}{\color{blue}{z - y}} \cdot t \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x - y}{z - y}} \cdot t \]

      4. div-sub N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 4.7%

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

   8. Taylor expanded in y around inf

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

      1. frac-sub N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

         \[\leadsto t \cdot \frac{z + -1 \cdot x}{y} + t \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{\frac{z + -1 \cdot x}{y}}, t\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(t, \frac{z + -1 \cdot x}{\color{blue}{y}}, t\right) \]

      7. +-commutative N/A

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

      8. lower-fma.f64 99.7

         \[\leadsto \mathsf{fma}\left(t, \frac{\mathsf{fma}\left(-1, x, z\right)}{y}, t\right) \]

   10. Applied rewrites 99.7%

       \[\leadsto \color{blue}{\mathsf{fma}\left(t, \frac{\mathsf{fma}\left(-1, x, z\right)}{y}, t\right)} \]

   11. Taylor expanded in x around inf

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

       1. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(t, \frac{\mathsf{neg}\left(x\right)}{y}, t\right) \]

       2. lift-neg.f64 99.3

          \[\leadsto \mathsf{fma}\left(t, \frac{-x}{y}, t\right) \]

   13. Applied rewrites 99.3%

       \[\leadsto \mathsf{fma}\left(t, \frac{-x}{y}, t\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq -2:\\ \;\;\;\;\frac{x}{z - y} \cdot t\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 0.5:\\ \;\;\;\;\frac{\left(x - y\right) \cdot t}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 5000000:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{-x}{y}, t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z - y} \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y}{z - y}\\ t_2 := \frac{t \cdot x}{z - y}\\ \mathbf{if}\;t\_1 \leq -200:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0.5:\\ \;\;\;\;\frac{\left(x - y\right) \cdot t}{z}\\ \mathbf{elif}\;t\_1 \leq 5000000:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{-x}{y}, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x y) (- z y))) (t_2 (/ (* t x) (- z y))))
   (if (<= t_1 -200.0)
     t_2
     (if (<= t_1 0.5)
       (/ (* (- x y) t) z)
       (if (<= t_1 5000000.0) (fma t (/ (- x) y) t) t_2)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - y) / (z - y);
	double t_2 = (t * x) / (z - y);
	double tmp;
	if (t_1 <= -200.0) {
		tmp = t_2;
	} else if (t_1 <= 0.5) {
		tmp = ((x - y) * t) / z;
	} else if (t_1 <= 5000000.0) {
		tmp = fma(t, (-x / y), t);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - y) / Float64(z - y))
	t_2 = Float64(Float64(t * x) / Float64(z - y))
	tmp = 0.0
	if (t_1 <= -200.0)
		tmp = t_2;
	elseif (t_1 <= 0.5)
		tmp = Float64(Float64(Float64(x - y) * t) / z);
	elseif (t_1 <= 5000000.0)
		tmp = fma(t, Float64(Float64(-x) / y), t);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t * x), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -200.0], t$95$2, If[LessEqual[t$95$1, 0.5], N[(N[(N[(x - y), $MachinePrecision] * t), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$1, 5000000.0], N[(t * N[((-x) / y), $MachinePrecision] + t), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -200 or 5e6 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 97.5%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{x - y}}{z - y} \cdot t \]

      2. lift--.f64 N/A

         \[\leadsto \frac{x - y}{\color{blue}{z - y}} \cdot t \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x - y}{z - y}} \cdot t \]

      4. div-sub N/A

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

      5. frac-sub N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift--.f64 67.1

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

   4. Applied rewrites 67.1%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{t \cdot x}{\color{blue}{z - y}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{t \cdot x}{\color{blue}{z} - y} \]

      3. lift--.f64 92.0

         \[\leadsto \frac{t \cdot x}{z - \color{blue}{y}} \]

   7. Applied rewrites 92.0%

      \[\leadsto \color{blue}{\frac{t \cdot x}{z - y}} \]

   if -200 < (/.f64 (-.f64 x y) (-.f64 z y)) < 0.5

   1. Initial program 94.0%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift--.f64 85.8

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

   5. Applied rewrites 85.8%

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

   if 0.5 < (/.f64 (-.f64 x y) (-.f64 z y)) < 5e6

   1. Initial program 99.9%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{x - y}}{z - y} \cdot t \]

      2. lift--.f64 N/A

         \[\leadsto \frac{x - y}{\color{blue}{z - y}} \cdot t \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x - y}{z - y}} \cdot t \]

      4. div-sub N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 4.7%

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

   8. Taylor expanded in y around inf

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

      1. frac-sub N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

         \[\leadsto t \cdot \frac{z + -1 \cdot x}{y} + t \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{\frac{z + -1 \cdot x}{y}}, t\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(t, \frac{z + -1 \cdot x}{\color{blue}{y}}, t\right) \]

      7. +-commutative N/A

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

      8. lower-fma.f64 99.7

         \[\leadsto \mathsf{fma}\left(t, \frac{\mathsf{fma}\left(-1, x, z\right)}{y}, t\right) \]

   10. Applied rewrites 99.7%

       \[\leadsto \color{blue}{\mathsf{fma}\left(t, \frac{\mathsf{fma}\left(-1, x, z\right)}{y}, t\right)} \]

   11. Taylor expanded in x around inf

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

       1. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(t, \frac{\mathsf{neg}\left(x\right)}{y}, t\right) \]

       2. lift-neg.f64 99.3

          \[\leadsto \mathsf{fma}\left(t, \frac{-x}{y}, t\right) \]

   13. Applied rewrites 99.3%

       \[\leadsto \mathsf{fma}\left(t, \frac{-x}{y}, t\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq -200:\\ \;\;\;\;\frac{t \cdot x}{z - y}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 0.5:\\ \;\;\;\;\frac{\left(x - y\right) \cdot t}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 5000000:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{-x}{y}, t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot x}{z - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y}{z - y}\\ t_2 := \frac{t \cdot x}{z - y}\\ \mathbf{if}\;t\_1 \leq -200:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0.5:\\ \;\;\;\;\frac{\left(x - y\right) \cdot t}{z}\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{z}{y}, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x y) (- z y))) (t_2 (/ (* t x) (- z y))))
   (if (<= t_1 -200.0)
     t_2
     (if (<= t_1 0.5)
       (/ (* (- x y) t) z)
       (if (<= t_1 2.0) (fma t (/ z y) t) t_2)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - y) / (z - y);
	double t_2 = (t * x) / (z - y);
	double tmp;
	if (t_1 <= -200.0) {
		tmp = t_2;
	} else if (t_1 <= 0.5) {
		tmp = ((x - y) * t) / z;
	} else if (t_1 <= 2.0) {
		tmp = fma(t, (z / y), t);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - y) / Float64(z - y))
	t_2 = Float64(Float64(t * x) / Float64(z - y))
	tmp = 0.0
	if (t_1 <= -200.0)
		tmp = t_2;
	elseif (t_1 <= 0.5)
		tmp = Float64(Float64(Float64(x - y) * t) / z);
	elseif (t_1 <= 2.0)
		tmp = fma(t, Float64(z / y), t);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t * x), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -200.0], t$95$2, If[LessEqual[t$95$1, 0.5], N[(N[(N[(x - y), $MachinePrecision] * t), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(t * N[(z / y), $MachinePrecision] + t), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -200 or 2 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 97.6%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{x - y}}{z - y} \cdot t \]

      2. lift--.f64 N/A

         \[\leadsto \frac{x - y}{\color{blue}{z - y}} \cdot t \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x - y}{z - y}} \cdot t \]

      4. div-sub N/A

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

      5. frac-sub N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift--.f64 64.7

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

   4. Applied rewrites 64.7%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{t \cdot x}{\color{blue}{z - y}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{t \cdot x}{\color{blue}{z} - y} \]

      3. lift--.f64 90.8

         \[\leadsto \frac{t \cdot x}{z - \color{blue}{y}} \]

   7. Applied rewrites 90.8%

      \[\leadsto \color{blue}{\frac{t \cdot x}{z - y}} \]

   if -200 < (/.f64 (-.f64 x y) (-.f64 z y)) < 0.5

   1. Initial program 94.0%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift--.f64 85.8

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

   5. Applied rewrites 85.8%

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

   if 0.5 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 99.9%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{x - y}}{z - y} \cdot t \]

      2. lift--.f64 N/A

         \[\leadsto \frac{x - y}{\color{blue}{z - y}} \cdot t \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x - y}{z - y}} \cdot t \]

      4. div-sub N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 3.6%

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

   8. Taylor expanded in y around inf

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

      1. frac-sub N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

         \[\leadsto t \cdot \frac{z + -1 \cdot x}{y} + t \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{\frac{z + -1 \cdot x}{y}}, t\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(t, \frac{z + -1 \cdot x}{\color{blue}{y}}, t\right) \]

      7. +-commutative N/A

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

      8. lower-fma.f64 99.7

         \[\leadsto \mathsf{fma}\left(t, \frac{\mathsf{fma}\left(-1, x, z\right)}{y}, t\right) \]

   10. Applied rewrites 99.7%

       \[\leadsto \color{blue}{\mathsf{fma}\left(t, \frac{\mathsf{fma}\left(-1, x, z\right)}{y}, t\right)} \]

   11. Taylor expanded in x around 0

       \[\leadsto \mathsf{fma}\left(t, \frac{z}{y}, t\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 97.6%

       \[\leadsto \mathsf{fma}\left(t, \frac{z}{y}, t\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq -200:\\ \;\;\;\;\frac{t \cdot x}{z - y}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 0.5:\\ \;\;\;\;\frac{\left(x - y\right) \cdot t}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 2:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{z}{y}, t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot x}{z - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 70.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y}{z - y}\\ t_2 := \frac{x}{z} \cdot t\\ \mathbf{if}\;t\_1 \leq 4 \cdot 10^{-246}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{-13}:\\ \;\;\;\;\frac{-y}{z} \cdot t\\ \mathbf{elif}\;t\_1 \leq 5:\\ \;\;\;\;t\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x y) (- z y))) (t_2 (* (/ x z) t)))
   (if (<= t_1 4e-246)
     t_2
     (if (<= t_1 1e-13) (* (/ (- y) z) t) (if (<= t_1 5.0) t t_2)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - y) / (z - y);
	double t_2 = (x / z) * t;
	double tmp;
	if (t_1 <= 4e-246) {
		tmp = t_2;
	} else if (t_1 <= 1e-13) {
		tmp = (-y / z) * t;
	} else if (t_1 <= 5.0) {
		tmp = t;
	} 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)
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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (x - y) / (z - y)
    t_2 = (x / z) * t
    if (t_1 <= 4d-246) then
        tmp = t_2
    else if (t_1 <= 1d-13) then
        tmp = (-y / z) * t
    else if (t_1 <= 5.0d0) then
        tmp = t
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x - y) / (z - y);
	double t_2 = (x / z) * t;
	double tmp;
	if (t_1 <= 4e-246) {
		tmp = t_2;
	} else if (t_1 <= 1e-13) {
		tmp = (-y / z) * t;
	} else if (t_1 <= 5.0) {
		tmp = t;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x - y) / (z - y)
	t_2 = (x / z) * t
	tmp = 0
	if t_1 <= 4e-246:
		tmp = t_2
	elif t_1 <= 1e-13:
		tmp = (-y / z) * t
	elif t_1 <= 5.0:
		tmp = t
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - y) / Float64(z - y))
	t_2 = Float64(Float64(x / z) * t)
	tmp = 0.0
	if (t_1 <= 4e-246)
		tmp = t_2;
	elseif (t_1 <= 1e-13)
		tmp = Float64(Float64(Float64(-y) / z) * t);
	elseif (t_1 <= 5.0)
		tmp = t;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x - y) / (z - y);
	t_2 = (x / z) * t;
	tmp = 0.0;
	if (t_1 <= 4e-246)
		tmp = t_2;
	elseif (t_1 <= 1e-13)
		tmp = (-y / z) * t;
	elseif (t_1 <= 5.0)
		tmp = t;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x / z), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$1, 4e-246], t$95$2, If[LessEqual[t$95$1, 1e-13], N[(N[((-y) / z), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[t$95$1, 5.0], t, t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 3.99999999999999982e-246 or 5 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 94.6%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x}{z}} \cdot t \]
   4. Step-by-step derivation

      1. lower-/.f64 58.0

         \[\leadsto \frac{x}{\color{blue}{z}} \cdot t \]

   5. Applied rewrites 58.0%

      \[\leadsto \color{blue}{\frac{x}{z}} \cdot t \]

   if 3.99999999999999982e-246 < (/.f64 (-.f64 x y) (-.f64 z y)) < 1e-13

   1. Initial program 99.8%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 43.6%

      \[\leadsto \frac{\color{blue}{x}}{z - y} \cdot t \]

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 43.6%

      \[\leadsto \frac{x}{\color{blue}{z}} \cdot t \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{-1 \cdot y}}{z} \cdot t \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \frac{\mathsf{neg}\left(y\right)}{z} \cdot t \]

       2. lower-neg.f64 69.4

          \[\leadsto \frac{-y}{z} \cdot t \]

   11. Applied rewrites 69.4%

       \[\leadsto \frac{\color{blue}{-y}}{z} \cdot t \]

   if 1e-13 < (/.f64 (-.f64 x y) (-.f64 z y)) < 5

   1. Initial program 99.9%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{t} \]
   4. Step-by-step derivation

   5. Applied rewrites 92.5%

      \[\leadsto \color{blue}{t} \]
3. Recombined 3 regimes into one program.

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq 4 \cdot 10^{-246}:\\ \;\;\;\;\frac{x}{z} \cdot t\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 10^{-13}:\\ \;\;\;\;\frac{-y}{z} \cdot t\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 5:\\ \;\;\;\;t\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y}{z - y}\\ t_2 := \frac{x}{z} \cdot t\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-113}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{-38}:\\ \;\;\;\;\frac{\left(-t\right) \cdot y}{z}\\ \mathbf{elif}\;t\_1 \leq 5:\\ \;\;\;\;t\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x y) (- z y))) (t_2 (* (/ x z) t)))
   (if (<= t_1 -2e-113)
     t_2
     (if (<= t_1 1e-38) (/ (* (- t) y) z) (if (<= t_1 5.0) t t_2)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - y) / (z - y);
	double t_2 = (x / z) * t;
	double tmp;
	if (t_1 <= -2e-113) {
		tmp = t_2;
	} else if (t_1 <= 1e-38) {
		tmp = (-t * y) / z;
	} else if (t_1 <= 5.0) {
		tmp = t;
	} 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)
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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (x - y) / (z - y)
    t_2 = (x / z) * t
    if (t_1 <= (-2d-113)) then
        tmp = t_2
    else if (t_1 <= 1d-38) then
        tmp = (-t * y) / z
    else if (t_1 <= 5.0d0) then
        tmp = t
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x - y) / (z - y);
	double t_2 = (x / z) * t;
	double tmp;
	if (t_1 <= -2e-113) {
		tmp = t_2;
	} else if (t_1 <= 1e-38) {
		tmp = (-t * y) / z;
	} else if (t_1 <= 5.0) {
		tmp = t;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x - y) / (z - y)
	t_2 = (x / z) * t
	tmp = 0
	if t_1 <= -2e-113:
		tmp = t_2
	elif t_1 <= 1e-38:
		tmp = (-t * y) / z
	elif t_1 <= 5.0:
		tmp = t
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - y) / Float64(z - y))
	t_2 = Float64(Float64(x / z) * t)
	tmp = 0.0
	if (t_1 <= -2e-113)
		tmp = t_2;
	elseif (t_1 <= 1e-38)
		tmp = Float64(Float64(Float64(-t) * y) / z);
	elseif (t_1 <= 5.0)
		tmp = t;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x - y) / (z - y);
	t_2 = (x / z) * t;
	tmp = 0.0;
	if (t_1 <= -2e-113)
		tmp = t_2;
	elseif (t_1 <= 1e-38)
		tmp = (-t * y) / z;
	elseif (t_1 <= 5.0)
		tmp = t;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x / z), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-113], t$95$2, If[LessEqual[t$95$1, 1e-38], N[(N[((-t) * y), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$1, 5.0], t, t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -1.99999999999999996e-113 or 5 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 97.9%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x}{z}} \cdot t \]
   4. Step-by-step derivation

      1. lower-/.f64 54.8

         \[\leadsto \frac{x}{\color{blue}{z}} \cdot t \]

   5. Applied rewrites 54.8%

      \[\leadsto \color{blue}{\frac{x}{z}} \cdot t \]

   if -1.99999999999999996e-113 < (/.f64 (-.f64 x y) (-.f64 z y)) < 9.9999999999999996e-39

   1. Initial program 91.7%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{x - y}}{z - y} \cdot t \]

      2. lift--.f64 N/A

         \[\leadsto \frac{x - y}{\color{blue}{z - y}} \cdot t \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x - y}{z - y}} \cdot t \]

      4. div-sub N/A

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

      5. frac-sub N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift--.f64 60.6

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

   4. Applied rewrites 60.6%

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

   5. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lift--.f64 95.3

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

   7. Applied rewrites 95.3%

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

   8. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(t\right)\right) \cdot y}{z} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(t\right)\right) \cdot y}{z} \]

      4. lower-neg.f64 69.7

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

   10. Applied rewrites 69.7%

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

   if 9.9999999999999996e-39 < (/.f64 (-.f64 x y) (-.f64 z y)) < 5

   1. Initial program 99.9%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{t} \]
   4. Step-by-step derivation

   5. Applied rewrites 89.7%

      \[\leadsto \color{blue}{t} \]
3. Recombined 3 regimes into one program.

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq -2 \cdot 10^{-113}:\\ \;\;\;\;\frac{x}{z} \cdot t\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 10^{-38}:\\ \;\;\;\;\frac{\left(-t\right) \cdot y}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 5:\\ \;\;\;\;t\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 95.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y}{z - y}\\ \mathbf{if}\;t\_1 \leq -200 \lor \neg \left(t\_1 \leq 0.5\right):\\ \;\;\;\;\left(\frac{x}{z - y} - -1\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{z} \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x y) (- z y))))
   (if (or (<= t_1 -200.0) (not (<= t_1 0.5)))
     (* (- (/ x (- z y)) -1.0) t)
     (* (/ (- x y) z) t))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - y) / (z - y);
	double tmp;
	if ((t_1 <= -200.0) || !(t_1 <= 0.5)) {
		tmp = ((x / (z - y)) - -1.0) * t;
	} else {
		tmp = ((x - y) / z) * t;
	}
	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)
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) :: t_1
    real(8) :: tmp
    t_1 = (x - y) / (z - y)
    if ((t_1 <= (-200.0d0)) .or. (.not. (t_1 <= 0.5d0))) then
        tmp = ((x / (z - y)) - (-1.0d0)) * t
    else
        tmp = ((x - y) / z) * t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x - y) / (z - y);
	double tmp;
	if ((t_1 <= -200.0) || !(t_1 <= 0.5)) {
		tmp = ((x / (z - y)) - -1.0) * t;
	} else {
		tmp = ((x - y) / z) * t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x - y) / (z - y)
	tmp = 0
	if (t_1 <= -200.0) or not (t_1 <= 0.5):
		tmp = ((x / (z - y)) - -1.0) * t
	else:
		tmp = ((x - y) / z) * t
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if ((t_1 <= -200.0) || !(t_1 <= 0.5))
		tmp = Float64(Float64(Float64(x / Float64(z - y)) - -1.0) * t);
	else
		tmp = Float64(Float64(Float64(x - y) / z) * t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x - y) / (z - y);
	tmp = 0.0;
	if ((t_1 <= -200.0) || ~((t_1 <= 0.5)))
		tmp = ((x / (z - y)) - -1.0) * t;
	else
		tmp = ((x - y) / z) * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -200.0], N[Not[LessEqual[t$95$1, 0.5]], $MachinePrecision]], N[(N[(N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision] * t), $MachinePrecision], N[(N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision] * t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -200 or 0.5 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 98.7%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{x - y}}{z - y} \cdot t \]

      2. lift--.f64 N/A

         \[\leadsto \frac{x - y}{\color{blue}{z - y}} \cdot t \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x - y}{z - y}} \cdot t \]

      4. div-sub N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 98.7

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

   4. Applied rewrites 98.7%

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

   5. Taylor expanded in y around inf

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

   7. Applied rewrites 98.4%

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

   if -200 < (/.f64 (-.f64 x y) (-.f64 z y)) < 0.5

   1. Initial program 94.0%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 90.9%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq -200 \lor \neg \left(\frac{x - y}{z - y} \leq 0.5\right):\\ \;\;\;\;\left(\frac{x}{z - y} - -1\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{z} \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 71.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y}{z - y}\\ \mathbf{if}\;t\_1 \leq 0.5 \lor \neg \left(t\_1 \leq 5\right):\\ \;\;\;\;\frac{x}{z} \cdot t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{z}{y}, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x y) (- z y))))
   (if (or (<= t_1 0.5) (not (<= t_1 5.0))) (* (/ x z) t) (fma t (/ z y) t))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - y) / (z - y);
	double tmp;
	if ((t_1 <= 0.5) || !(t_1 <= 5.0)) {
		tmp = (x / z) * t;
	} else {
		tmp = fma(t, (z / y), t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if ((t_1 <= 0.5) || !(t_1 <= 5.0))
		tmp = Float64(Float64(x / z) * t);
	else
		tmp = fma(t, Float64(z / y), t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, 0.5], N[Not[LessEqual[t$95$1, 5.0]], $MachinePrecision]], N[(N[(x / z), $MachinePrecision] * t), $MachinePrecision], N[(t * N[(z / y), $MachinePrecision] + t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 0.5 or 5 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 95.7%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x}{z}} \cdot t \]
   4. Step-by-step derivation

      1. lower-/.f64 55.5

         \[\leadsto \frac{x}{\color{blue}{z}} \cdot t \]

   5. Applied rewrites 55.5%

      \[\leadsto \color{blue}{\frac{x}{z}} \cdot t \]

   if 0.5 < (/.f64 (-.f64 x y) (-.f64 z y)) < 5

   1. Initial program 99.9%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{x - y}}{z - y} \cdot t \]

      2. lift--.f64 N/A

         \[\leadsto \frac{x - y}{\color{blue}{z - y}} \cdot t \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x - y}{z - y}} \cdot t \]

      4. div-sub N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 3.6%

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

   8. Taylor expanded in y around inf

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

      1. frac-sub N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

         \[\leadsto t \cdot \frac{z + -1 \cdot x}{y} + t \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{\frac{z + -1 \cdot x}{y}}, t\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(t, \frac{z + -1 \cdot x}{\color{blue}{y}}, t\right) \]

      7. +-commutative N/A

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

      8. lower-fma.f64 99.7

         \[\leadsto \mathsf{fma}\left(t, \frac{\mathsf{fma}\left(-1, x, z\right)}{y}, t\right) \]

   10. Applied rewrites 99.7%

       \[\leadsto \color{blue}{\mathsf{fma}\left(t, \frac{\mathsf{fma}\left(-1, x, z\right)}{y}, t\right)} \]

   11. Taylor expanded in x around 0

       \[\leadsto \mathsf{fma}\left(t, \frac{z}{y}, t\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 96.6%

       \[\leadsto \mathsf{fma}\left(t, \frac{z}{y}, t\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq 0.5 \lor \neg \left(\frac{x - y}{z - y} \leq 5\right):\\ \;\;\;\;\frac{x}{z} \cdot t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{z}{y}, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 80.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y}{z - y}\\ \mathbf{if}\;t\_1 \leq 0.5:\\ \;\;\;\;\frac{\left(x - y\right) \cdot t}{z}\\ \mathbf{elif}\;t\_1 \leq 5:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{z}{y}, t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x y) (- z y))))
   (if (<= t_1 0.5)
     (/ (* (- x y) t) z)
     (if (<= t_1 5.0) (fma t (/ z y) t) (* (/ x z) t)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - y) / (z - y);
	double tmp;
	if (t_1 <= 0.5) {
		tmp = ((x - y) * t) / z;
	} else if (t_1 <= 5.0) {
		tmp = fma(t, (z / y), t);
	} else {
		tmp = (x / z) * t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_1 <= 0.5)
		tmp = Float64(Float64(Float64(x - y) * t) / z);
	elseif (t_1 <= 5.0)
		tmp = fma(t, Float64(z / y), t);
	else
		tmp = Float64(Float64(x / z) * t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.5], N[(N[(N[(x - y), $MachinePrecision] * t), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$1, 5.0], N[(t * N[(z / y), $MachinePrecision] + t), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * t), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 0.5

   1. Initial program 95.2%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift--.f64 73.2

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

   5. Applied rewrites 73.2%

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

   if 0.5 < (/.f64 (-.f64 x y) (-.f64 z y)) < 5

   1. Initial program 99.9%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{x - y}}{z - y} \cdot t \]

      2. lift--.f64 N/A

         \[\leadsto \frac{x - y}{\color{blue}{z - y}} \cdot t \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x - y}{z - y}} \cdot t \]

      4. div-sub N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 3.6%

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

   8. Taylor expanded in y around inf

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

      1. frac-sub N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

         \[\leadsto t \cdot \frac{z + -1 \cdot x}{y} + t \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{\frac{z + -1 \cdot x}{y}}, t\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(t, \frac{z + -1 \cdot x}{\color{blue}{y}}, t\right) \]

      7. +-commutative N/A

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

      8. lower-fma.f64 99.7

         \[\leadsto \mathsf{fma}\left(t, \frac{\mathsf{fma}\left(-1, x, z\right)}{y}, t\right) \]

   10. Applied rewrites 99.7%

       \[\leadsto \color{blue}{\mathsf{fma}\left(t, \frac{\mathsf{fma}\left(-1, x, z\right)}{y}, t\right)} \]

   11. Taylor expanded in x around 0

       \[\leadsto \mathsf{fma}\left(t, \frac{z}{y}, t\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 96.6%

       \[\leadsto \mathsf{fma}\left(t, \frac{z}{y}, t\right) \]

   if 5 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 97.4%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x}{z}} \cdot t \]
   4. Step-by-step derivation

      1. lower-/.f64 59.4

         \[\leadsto \frac{x}{\color{blue}{z}} \cdot t \]

   5. Applied rewrites 59.4%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq 0.5:\\ \;\;\;\;\frac{\left(x - y\right) \cdot t}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 5:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{z}{y}, t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 71.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y}{z - y}\\ \mathbf{if}\;t\_1 \leq 0.4 \lor \neg \left(t\_1 \leq 5\right):\\ \;\;\;\;\frac{x}{z} \cdot t\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x y) (- z y))))
   (if (or (<= t_1 0.4) (not (<= t_1 5.0))) (* (/ x z) t) t)))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - y) / (z - y);
	double tmp;
	if ((t_1 <= 0.4) || !(t_1 <= 5.0)) {
		tmp = (x / z) * t;
	} else {
		tmp = t;
	}
	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)
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) :: t_1
    real(8) :: tmp
    t_1 = (x - y) / (z - y)
    if ((t_1 <= 0.4d0) .or. (.not. (t_1 <= 5.0d0))) then
        tmp = (x / z) * t
    else
        tmp = t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x - y) / (z - y);
	double tmp;
	if ((t_1 <= 0.4) || !(t_1 <= 5.0)) {
		tmp = (x / z) * t;
	} else {
		tmp = t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x - y) / (z - y)
	tmp = 0
	if (t_1 <= 0.4) or not (t_1 <= 5.0):
		tmp = (x / z) * t
	else:
		tmp = t
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if ((t_1 <= 0.4) || !(t_1 <= 5.0))
		tmp = Float64(Float64(x / z) * t);
	else
		tmp = t;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x - y) / (z - y);
	tmp = 0.0;
	if ((t_1 <= 0.4) || ~((t_1 <= 5.0)))
		tmp = (x / z) * t;
	else
		tmp = t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, 0.4], N[Not[LessEqual[t$95$1, 5.0]], $MachinePrecision]], N[(N[(x / z), $MachinePrecision] * t), $MachinePrecision], t]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 0.40000000000000002 or 5 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 95.7%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x}{z}} \cdot t \]
   4. Step-by-step derivation

      1. lower-/.f64 55.8

         \[\leadsto \frac{x}{\color{blue}{z}} \cdot t \]

   5. Applied rewrites 55.8%

      \[\leadsto \color{blue}{\frac{x}{z}} \cdot t \]

   if 0.40000000000000002 < (/.f64 (-.f64 x y) (-.f64 z y)) < 5

   1. Initial program 99.9%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{t} \]
   4. Step-by-step derivation

   5. Applied rewrites 95.4%

      \[\leadsto \color{blue}{t} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq 0.4 \lor \neg \left(\frac{x - y}{z - y} \leq 5\right):\\ \;\;\;\;\frac{x}{z} \cdot t\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 69.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y}{z - y}\\ \mathbf{if}\;t\_1 \leq 0.4 \lor \neg \left(t\_1 \leq 5\right):\\ \;\;\;\;\frac{t \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x y) (- z y))))
   (if (or (<= t_1 0.4) (not (<= t_1 5.0))) (/ (* t x) z) t)))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - y) / (z - y);
	double tmp;
	if ((t_1 <= 0.4) || !(t_1 <= 5.0)) {
		tmp = (t * x) / z;
	} else {
		tmp = t;
	}
	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)
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) :: t_1
    real(8) :: tmp
    t_1 = (x - y) / (z - y)
    if ((t_1 <= 0.4d0) .or. (.not. (t_1 <= 5.0d0))) then
        tmp = (t * x) / z
    else
        tmp = t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x - y) / (z - y);
	double tmp;
	if ((t_1 <= 0.4) || !(t_1 <= 5.0)) {
		tmp = (t * x) / z;
	} else {
		tmp = t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x - y) / (z - y)
	tmp = 0
	if (t_1 <= 0.4) or not (t_1 <= 5.0):
		tmp = (t * x) / z
	else:
		tmp = t
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if ((t_1 <= 0.4) || !(t_1 <= 5.0))
		tmp = Float64(Float64(t * x) / z);
	else
		tmp = t;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x - y) / (z - y);
	tmp = 0.0;
	if ((t_1 <= 0.4) || ~((t_1 <= 5.0)))
		tmp = (t * x) / z;
	else
		tmp = t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, 0.4], N[Not[LessEqual[t$95$1, 5.0]], $MachinePrecision]], N[(N[(t * x), $MachinePrecision] / z), $MachinePrecision], t]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 0.40000000000000002 or 5 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 95.7%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{t \cdot x}{z}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{t \cdot x}{\color{blue}{z}} \]

      2. lower-*.f64 53.0

         \[\leadsto \frac{t \cdot x}{z} \]

   5. Applied rewrites 53.0%

      \[\leadsto \color{blue}{\frac{t \cdot x}{z}} \]

   if 0.40000000000000002 < (/.f64 (-.f64 x y) (-.f64 z y)) < 5

   1. Initial program 99.9%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{t} \]
   4. Step-by-step derivation

   5. Applied rewrites 95.4%

      \[\leadsto \color{blue}{t} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq 0.4 \lor \neg \left(\frac{x - y}{z - y} \leq 5\right):\\ \;\;\;\;\frac{t \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 94.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.52 \cdot 10^{+174}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot t}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{z - y} - \frac{y}{z - y}\right) \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -1.52e+174)
   (/ (* (- x y) t) z)
   (* (- (/ x (- z y)) (/ y (- z y))) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.52e+174) {
		tmp = ((x - y) * t) / z;
	} else {
		tmp = ((x / (z - y)) - (y / (z - y))) * t;
	}
	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)
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) :: tmp
    if (z <= (-1.52d+174)) then
        tmp = ((x - y) * t) / z
    else
        tmp = ((x / (z - y)) - (y / (z - y))) * t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.52e+174) {
		tmp = ((x - y) * t) / z;
	} else {
		tmp = ((x / (z - y)) - (y / (z - y))) * t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -1.52e+174:
		tmp = ((x - y) * t) / z
	else:
		tmp = ((x / (z - y)) - (y / (z - y))) * t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -1.52e+174)
		tmp = Float64(Float64(Float64(x - y) * t) / z);
	else
		tmp = Float64(Float64(Float64(x / Float64(z - y)) - Float64(y / Float64(z - y))) * t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -1.52e+174)
		tmp = ((x - y) * t) / z;
	else
		tmp = ((x / (z - y)) - (y / (z - y))) * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -1.52e+174], N[(N[(N[(x - y), $MachinePrecision] * t), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision] - N[(y / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.52000000000000004e174

   1. Initial program 75.2%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift--.f64 95.2

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

   5. Applied rewrites 95.2%

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

   if -1.52000000000000004e174 < z

   1. Initial program 99.0%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{x - y}}{z - y} \cdot t \]

      2. lift--.f64 N/A

         \[\leadsto \frac{x - y}{\color{blue}{z - y}} \cdot t \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x - y}{z - y}} \cdot t \]

      4. div-sub N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 99.0

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

   4. Applied rewrites 99.0%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.52 \cdot 10^{+174}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot t}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{z - y} - \frac{y}{z - y}\right) \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 94.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.52 \cdot 10^{+174}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot t}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{z - y} \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -1.52e+174) (/ (* (- x y) t) z) (* (/ (- x y) (- z y)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.52e+174) {
		tmp = ((x - y) * t) / z;
	} else {
		tmp = ((x - y) / (z - y)) * t;
	}
	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)
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) :: tmp
    if (z <= (-1.52d+174)) then
        tmp = ((x - y) * t) / z
    else
        tmp = ((x - y) / (z - y)) * t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.52e+174) {
		tmp = ((x - y) * t) / z;
	} else {
		tmp = ((x - y) / (z - y)) * t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -1.52e+174:
		tmp = ((x - y) * t) / z
	else:
		tmp = ((x - y) / (z - y)) * t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -1.52e+174)
		tmp = Float64(Float64(Float64(x - y) * t) / z);
	else
		tmp = Float64(Float64(Float64(x - y) / Float64(z - y)) * t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -1.52e+174)
		tmp = ((x - y) * t) / z;
	else
		tmp = ((x - y) / (z - y)) * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -1.52e+174], N[(N[(N[(x - y), $MachinePrecision] * t), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.52000000000000004e174

   1. Initial program 75.2%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift--.f64 95.2

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

   5. Applied rewrites 95.2%

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

   if -1.52000000000000004e174 < z

   1. Initial program 99.0%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.52 \cdot 10^{+174}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot t}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{z - y} \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 36.6% accurate, 23.0× speedup

Math

\[\begin{array}{l} \\ t \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 t)

C

double code(double x, double y, double z, double t) {
	return t;
}

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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = t
end function

Java

public static double code(double x, double y, double z, double t) {
	return t;
}

Python

def code(x, y, z, t):
	return t

Julia

function code(x, y, z, t)
	return t
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = t;
end

Wolfram

code[x_, y_, z_, t_] := t

Derivation

1. Initial program 97.1%

   \[\frac{x - y}{z - y} \cdot t \]
2. Add Preprocessing

3. Taylor expanded in y around inf

   \[\leadsto \color{blue}{t} \]
4. Step-by-step derivation

5. Applied rewrites 33.5%

   \[\leadsto \color{blue}{t} \]

6. Final simplification 33.5%

   \[\leadsto t \]
7. Add Preprocessing

Developer Target 1: 96.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{t}{\frac{z - y}{x - y}} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (/ t (/ (- z y) (- x y))))

C

double code(double x, double y, double z, double t) {
	return t / ((z - y) / (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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = t / ((z - y) / (x - y))
end function

Java

public static double code(double x, double y, double z, double t) {
	return t / ((z - y) / (x - y));
}

Python

def code(x, y, z, t):
	return t / ((z - y) / (x - y))

Julia

function code(x, y, z, t)
	return Float64(t / Float64(Float64(z - y) / Float64(x - y)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = t / ((z - y) / (x - y));
end

Wolfram

code[x_, y_, z_, t_] := N[(t / N[(N[(z - y), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2025064 
(FPCore (x y z t)
  :name "Numeric.Signal.Multichannel:$cput from hsignal-0.2.7.1"
  :precision binary64

  :alt
  (! :herbie-platform default (/ t (/ (- z y) (- x y))))

  (* (/ (- x y) (- z y)) t))