Graphics.Rendering.Chart.Plot.AreaSpots:renderAreaSpots4D from Chart-1.5.3

Percentage Accurate: 84.3% → 96.9%

Time: 8.9s

Alternatives: 15

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

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)) / (t - z)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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)) / (t - z)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
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 = ((y - z) / (t - z)) * x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.7%

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

4. Applied rewrites 97.2%

   \[\leadsto \color{blue}{\frac{y - z}{t - z} \cdot x} \]
5. Add Preprocessing

Alternative 2: 59.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot \left(y - z\right)}{t - z}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+92}:\\ \;\;\;\;\frac{x}{t - z} \cdot y\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-308}:\\ \;\;\;\;\frac{\left(y - z\right) \cdot x}{t}\\ \mathbf{elif}\;t\_1 \leq 1:\\ \;\;\;\;\frac{z}{t - z} \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{t - y}{z}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* x (- y z)) (- t z))))
   (if (<= t_1 -5e+92)
     (* (/ x (- t z)) y)
     (if (<= t_1 -2e-308)
       (/ (* (- y z) x) t)
       (if (<= t_1 1.0) (* (/ z (- t z)) (- x)) (fma x (/ (- t y) z) x))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * (y - z)) / (t - z);
	double tmp;
	if (t_1 <= -5e+92) {
		tmp = (x / (t - z)) * y;
	} else if (t_1 <= -2e-308) {
		tmp = ((y - z) * x) / t;
	} else if (t_1 <= 1.0) {
		tmp = (z / (t - z)) * -x;
	} else {
		tmp = fma(x, ((t - y) / z), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * Float64(y - z)) / Float64(t - z))
	tmp = 0.0
	if (t_1 <= -5e+92)
		tmp = Float64(Float64(x / Float64(t - z)) * y);
	elseif (t_1 <= -2e-308)
		tmp = Float64(Float64(Float64(y - z) * x) / t);
	elseif (t_1 <= 1.0)
		tmp = Float64(Float64(z / Float64(t - z)) * Float64(-x));
	else
		tmp = fma(x, Float64(Float64(t - y) / z), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * N[(y - z), $MachinePrecision]), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+92], N[(N[(x / N[(t - z), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$1, -2e-308], N[(N[(N[(y - z), $MachinePrecision] * x), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[t$95$1, 1.0], N[(N[(z / N[(t - z), $MachinePrecision]), $MachinePrecision] * (-x)), $MachinePrecision], N[(x * N[(N[(t - y), $MachinePrecision] / z), $MachinePrecision] + x), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z)) < -5.00000000000000022e92

   1. Initial program 65.4%

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

   3. Taylor expanded in y around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 56.9

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

   5. Applied rewrites 56.9%

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

   if -5.00000000000000022e92 < (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z)) < -1.9999999999999998e-308

   1. Initial program 99.6%

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

   4. Applied rewrites 97.1%

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

   5. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 56.2

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

   7. Applied rewrites 56.2%

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

   if -1.9999999999999998e-308 < (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z)) < 1

   1. Initial program 92.4%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 74.8

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

   5. Applied rewrites 74.8%

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

   if 1 < (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z))

   1. Initial program 69.7%

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

   3. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. distribute-neg-frac N/A

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

      7. lower-/.f64 N/A

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

      8. *-lft-identity N/A

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

      9. metadata-eval N/A

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

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

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. mul-1-neg N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(z\right)\right)}\right)\right) + \left(\mathsf{neg}\left(y\right)\right)}{z} \cdot x \]

      14. remove-double-neg N/A

          \[\leadsto \frac{\color{blue}{z} + \left(\mathsf{neg}\left(y\right)\right)}{z} \cdot x \]

      15. mul-1-neg N/A

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

      16. metadata-eval N/A

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

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

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

      18. *-lft-identity N/A

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

      19. lower--.f64 75.0

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

   5. Applied rewrites 75.0%

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

   6. Taylor expanded in y around 0

      \[\leadsto 1 \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 33.5%

      \[\leadsto 1 \cdot x \]

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 75.7%

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

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y - z\right)}{t - z} \leq -5 \cdot 10^{+92}:\\ \;\;\;\;\frac{x}{t - z} \cdot y\\ \mathbf{elif}\;\frac{x \cdot \left(y - z\right)}{t - z} \leq -2 \cdot 10^{-308}:\\ \;\;\;\;\frac{\left(y - z\right) \cdot x}{t}\\ \mathbf{elif}\;\frac{x \cdot \left(y - z\right)}{t - z} \leq 1:\\ \;\;\;\;\frac{z}{t - z} \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{t - y}{z}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 56.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot \left(y - z\right)}{t - z}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+77}:\\ \;\;\;\;\frac{x}{t - z} \cdot y\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{y - z}{t} \cdot x\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-48}:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(z - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* x (- y z)) (- t z))))
   (if (<= t_1 -2e+77)
     (* (/ x (- t z)) y)
     (if (<= t_1 0.0)
       (* (/ (- y z) t) x)
       (if (<= t_1 5e-48) (* 1.0 x) (* (/ x z) (- z y)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * (y - z)) / (t - z);
	double tmp;
	if (t_1 <= -2e+77) {
		tmp = (x / (t - z)) * y;
	} else if (t_1 <= 0.0) {
		tmp = ((y - z) / t) * x;
	} else if (t_1 <= 5e-48) {
		tmp = 1.0 * x;
	} else {
		tmp = (x / z) * (z - y);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
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)) / (t - z)
    if (t_1 <= (-2d+77)) then
        tmp = (x / (t - z)) * y
    else if (t_1 <= 0.0d0) then
        tmp = ((y - z) / t) * x
    else if (t_1 <= 5d-48) then
        tmp = 1.0d0 * x
    else
        tmp = (x / z) * (z - y)
    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)) / (t - z);
	double tmp;
	if (t_1 <= -2e+77) {
		tmp = (x / (t - z)) * y;
	} else if (t_1 <= 0.0) {
		tmp = ((y - z) / t) * x;
	} else if (t_1 <= 5e-48) {
		tmp = 1.0 * x;
	} else {
		tmp = (x / z) * (z - y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * (y - z)) / (t - z)
	tmp = 0
	if t_1 <= -2e+77:
		tmp = (x / (t - z)) * y
	elif t_1 <= 0.0:
		tmp = ((y - z) / t) * x
	elif t_1 <= 5e-48:
		tmp = 1.0 * x
	else:
		tmp = (x / z) * (z - y)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * Float64(y - z)) / Float64(t - z))
	tmp = 0.0
	if (t_1 <= -2e+77)
		tmp = Float64(Float64(x / Float64(t - z)) * y);
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(Float64(y - z) / t) * x);
	elseif (t_1 <= 5e-48)
		tmp = Float64(1.0 * x);
	else
		tmp = Float64(Float64(x / z) * Float64(z - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * (y - z)) / (t - z);
	tmp = 0.0;
	if (t_1 <= -2e+77)
		tmp = (x / (t - z)) * y;
	elseif (t_1 <= 0.0)
		tmp = ((y - z) / t) * x;
	elseif (t_1 <= 5e-48)
		tmp = 1.0 * x;
	else
		tmp = (x / z) * (z - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * N[(y - z), $MachinePrecision]), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+77], N[(N[(x / N[(t - z), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(N[(y - z), $MachinePrecision] / t), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$1, 5e-48], N[(1.0 * x), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * N[(z - y), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z)) < -1.99999999999999997e77

   1. Initial program 67.3%

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

   3. Taylor expanded in y around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 59.4

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

   5. Applied rewrites 59.4%

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

   if -1.99999999999999997e77 < (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z)) < 0.0

   1. Initial program 94.2%

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

   3. Taylor expanded in t around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 61.5

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

   5. Applied rewrites 61.5%

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

   if 0.0 < (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z)) < 4.9999999999999999e-48

   1. Initial program 99.4%

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

   3. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. distribute-neg-frac N/A

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

      7. lower-/.f64 N/A

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

      8. *-lft-identity N/A

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

      9. metadata-eval N/A

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

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

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. mul-1-neg N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(z\right)\right)}\right)\right) + \left(\mathsf{neg}\left(y\right)\right)}{z} \cdot x \]

      14. remove-double-neg N/A

          \[\leadsto \frac{\color{blue}{z} + \left(\mathsf{neg}\left(y\right)\right)}{z} \cdot x \]

      15. mul-1-neg N/A

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

      16. metadata-eval N/A

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

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

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

      18. *-lft-identity N/A

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

      19. lower--.f64 50.8

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

   5. Applied rewrites 50.8%

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

   6. Taylor expanded in y around 0

      \[\leadsto 1 \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 45.0%

      \[\leadsto 1 \cdot x \]

   if 4.9999999999999999e-48 < (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z))

   1. Initial program 73.2%

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

   3. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. distribute-neg-frac N/A

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

      7. lower-/.f64 N/A

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

      8. *-lft-identity N/A

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

      9. metadata-eval N/A

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

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

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. mul-1-neg N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(z\right)\right)}\right)\right) + \left(\mathsf{neg}\left(y\right)\right)}{z} \cdot x \]

      14. remove-double-neg N/A

          \[\leadsto \frac{\color{blue}{z} + \left(\mathsf{neg}\left(y\right)\right)}{z} \cdot x \]

      15. mul-1-neg N/A

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

      16. metadata-eval N/A

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

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

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

      18. *-lft-identity N/A

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

      19. lower--.f64 73.1

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

   5. Applied rewrites 73.1%

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

   7. Applied rewrites 74.5%

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

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y - z\right)}{t - z} \leq -2 \cdot 10^{+77}:\\ \;\;\;\;\frac{x}{t - z} \cdot y\\ \mathbf{elif}\;\frac{x \cdot \left(y - z\right)}{t - z} \leq 0:\\ \;\;\;\;\frac{y - z}{t} \cdot x\\ \mathbf{elif}\;\frac{x \cdot \left(y - z\right)}{t - z} \leq 5 \cdot 10^{-48}:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(z - y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 55.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot \left(y - z\right)}{t - z}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+92}:\\ \;\;\;\;\frac{x}{t - z} \cdot y\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{\left(y - z\right) \cdot x}{t}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-48}:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(z - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* x (- y z)) (- t z))))
   (if (<= t_1 -5e+92)
     (* (/ x (- t z)) y)
     (if (<= t_1 0.0)
       (/ (* (- y z) x) t)
       (if (<= t_1 5e-48) (* 1.0 x) (* (/ x z) (- z y)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * (y - z)) / (t - z);
	double tmp;
	if (t_1 <= -5e+92) {
		tmp = (x / (t - z)) * y;
	} else if (t_1 <= 0.0) {
		tmp = ((y - z) * x) / t;
	} else if (t_1 <= 5e-48) {
		tmp = 1.0 * x;
	} else {
		tmp = (x / z) * (z - y);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
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)) / (t - z)
    if (t_1 <= (-5d+92)) then
        tmp = (x / (t - z)) * y
    else if (t_1 <= 0.0d0) then
        tmp = ((y - z) * x) / t
    else if (t_1 <= 5d-48) then
        tmp = 1.0d0 * x
    else
        tmp = (x / z) * (z - y)
    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)) / (t - z);
	double tmp;
	if (t_1 <= -5e+92) {
		tmp = (x / (t - z)) * y;
	} else if (t_1 <= 0.0) {
		tmp = ((y - z) * x) / t;
	} else if (t_1 <= 5e-48) {
		tmp = 1.0 * x;
	} else {
		tmp = (x / z) * (z - y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * (y - z)) / (t - z)
	tmp = 0
	if t_1 <= -5e+92:
		tmp = (x / (t - z)) * y
	elif t_1 <= 0.0:
		tmp = ((y - z) * x) / t
	elif t_1 <= 5e-48:
		tmp = 1.0 * x
	else:
		tmp = (x / z) * (z - y)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * Float64(y - z)) / Float64(t - z))
	tmp = 0.0
	if (t_1 <= -5e+92)
		tmp = Float64(Float64(x / Float64(t - z)) * y);
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(Float64(y - z) * x) / t);
	elseif (t_1 <= 5e-48)
		tmp = Float64(1.0 * x);
	else
		tmp = Float64(Float64(x / z) * Float64(z - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * (y - z)) / (t - z);
	tmp = 0.0;
	if (t_1 <= -5e+92)
		tmp = (x / (t - z)) * y;
	elseif (t_1 <= 0.0)
		tmp = ((y - z) * x) / t;
	elseif (t_1 <= 5e-48)
		tmp = 1.0 * x;
	else
		tmp = (x / z) * (z - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * N[(y - z), $MachinePrecision]), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+92], N[(N[(x / N[(t - z), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(N[(y - z), $MachinePrecision] * x), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[t$95$1, 5e-48], N[(1.0 * x), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * N[(z - y), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z)) < -5.00000000000000022e92

   1. Initial program 65.4%

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

   3. Taylor expanded in y around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 56.9

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

   5. Applied rewrites 56.9%

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

   if -5.00000000000000022e92 < (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z)) < 0.0

   1. Initial program 94.3%

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

   4. Applied rewrites 97.8%

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

   5. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 61.1

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

   7. Applied rewrites 61.1%

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

   if 0.0 < (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z)) < 4.9999999999999999e-48

   1. Initial program 99.4%

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

   3. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. distribute-neg-frac N/A

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

      7. lower-/.f64 N/A

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

      8. *-lft-identity N/A

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

      9. metadata-eval N/A

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

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

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. mul-1-neg N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(z\right)\right)}\right)\right) + \left(\mathsf{neg}\left(y\right)\right)}{z} \cdot x \]

      14. remove-double-neg N/A

          \[\leadsto \frac{\color{blue}{z} + \left(\mathsf{neg}\left(y\right)\right)}{z} \cdot x \]

      15. mul-1-neg N/A

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

      16. metadata-eval N/A

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

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

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

      18. *-lft-identity N/A

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

      19. lower--.f64 50.8

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

   5. Applied rewrites 50.8%

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

   6. Taylor expanded in y around 0

      \[\leadsto 1 \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 45.0%

      \[\leadsto 1 \cdot x \]

   if 4.9999999999999999e-48 < (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z))

   1. Initial program 73.2%

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

   3. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. distribute-neg-frac N/A

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

      7. lower-/.f64 N/A

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

      8. *-lft-identity N/A

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

      9. metadata-eval N/A

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

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

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. mul-1-neg N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(z\right)\right)}\right)\right) + \left(\mathsf{neg}\left(y\right)\right)}{z} \cdot x \]

      14. remove-double-neg N/A

          \[\leadsto \frac{\color{blue}{z} + \left(\mathsf{neg}\left(y\right)\right)}{z} \cdot x \]

      15. mul-1-neg N/A

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

      16. metadata-eval N/A

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

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

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

      18. *-lft-identity N/A

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

      19. lower--.f64 73.1

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

   5. Applied rewrites 73.1%

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

   7. Applied rewrites 74.5%

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

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y - z\right)}{t - z} \leq -5 \cdot 10^{+92}:\\ \;\;\;\;\frac{x}{t - z} \cdot y\\ \mathbf{elif}\;\frac{x \cdot \left(y - z\right)}{t - z} \leq 0:\\ \;\;\;\;\frac{\left(y - z\right) \cdot x}{t}\\ \mathbf{elif}\;\frac{x \cdot \left(y - z\right)}{t - z} \leq 5 \cdot 10^{-48}:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(z - y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 56.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot \left(y - z\right)}{t - z}\\ t_2 := \frac{x}{z} \cdot \left(z - y\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+114}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{\left(y - z\right) \cdot x}{t}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-48}:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* x (- y z)) (- t z))) (t_2 (* (/ x z) (- z y))))
   (if (<= t_1 -1e+114)
     t_2
     (if (<= t_1 0.0) (/ (* (- y z) x) t) (if (<= t_1 5e-48) (* 1.0 x) t_2)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * (y - z)) / (t - z);
	double t_2 = (x / z) * (z - y);
	double tmp;
	if (t_1 <= -1e+114) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = ((y - z) * x) / t;
	} else if (t_1 <= 5e-48) {
		tmp = 1.0 * x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
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)) / (t - z)
    t_2 = (x / z) * (z - y)
    if (t_1 <= (-1d+114)) then
        tmp = t_2
    else if (t_1 <= 0.0d0) then
        tmp = ((y - z) * x) / t
    else if (t_1 <= 5d-48) then
        tmp = 1.0d0 * x
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x * (y - z)) / (t - z);
	double t_2 = (x / z) * (z - y);
	double tmp;
	if (t_1 <= -1e+114) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = ((y - z) * x) / t;
	} else if (t_1 <= 5e-48) {
		tmp = 1.0 * x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * (y - z)) / (t - z)
	t_2 = (x / z) * (z - y)
	tmp = 0
	if t_1 <= -1e+114:
		tmp = t_2
	elif t_1 <= 0.0:
		tmp = ((y - z) * x) / t
	elif t_1 <= 5e-48:
		tmp = 1.0 * x
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * Float64(y - z)) / Float64(t - z))
	t_2 = Float64(Float64(x / z) * Float64(z - y))
	tmp = 0.0
	if (t_1 <= -1e+114)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(Float64(y - z) * x) / t);
	elseif (t_1 <= 5e-48)
		tmp = Float64(1.0 * x);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * (y - z)) / (t - z);
	t_2 = (x / z) * (z - y);
	tmp = 0.0;
	if (t_1 <= -1e+114)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = ((y - z) * x) / t;
	elseif (t_1 <= 5e-48)
		tmp = 1.0 * x;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * N[(y - z), $MachinePrecision]), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x / z), $MachinePrecision] * N[(z - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+114], t$95$2, If[LessEqual[t$95$1, 0.0], N[(N[(N[(y - z), $MachinePrecision] * x), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[t$95$1, 5e-48], N[(1.0 * x), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z)) < -1e114 or 4.9999999999999999e-48 < (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z))

   1. Initial program 67.9%

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

   3. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. distribute-neg-frac N/A

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

      7. lower-/.f64 N/A

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

      8. *-lft-identity N/A

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

      9. metadata-eval N/A

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

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

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. mul-1-neg N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(z\right)\right)}\right)\right) + \left(\mathsf{neg}\left(y\right)\right)}{z} \cdot x \]

      14. remove-double-neg N/A

          \[\leadsto \frac{\color{blue}{z} + \left(\mathsf{neg}\left(y\right)\right)}{z} \cdot x \]

      15. mul-1-neg N/A

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

      16. metadata-eval N/A

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

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

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

      18. *-lft-identity N/A

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

      19. lower--.f64 75.6

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

   5. Applied rewrites 75.6%

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

   7. Applied rewrites 77.3%

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

   if -1e114 < (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z)) < 0.0

   1. Initial program 94.6%

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

   4. Applied rewrites 97.9%

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

   5. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 61.5

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

   7. Applied rewrites 61.5%

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

   if 0.0 < (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z)) < 4.9999999999999999e-48

   1. Initial program 99.4%

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

   3. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. distribute-neg-frac N/A

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

      7. lower-/.f64 N/A

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

      8. *-lft-identity N/A

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

      9. metadata-eval N/A

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

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

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. mul-1-neg N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(z\right)\right)}\right)\right) + \left(\mathsf{neg}\left(y\right)\right)}{z} \cdot x \]

      14. remove-double-neg N/A

          \[\leadsto \frac{\color{blue}{z} + \left(\mathsf{neg}\left(y\right)\right)}{z} \cdot x \]

      15. mul-1-neg N/A

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

      16. metadata-eval N/A

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

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

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

      18. *-lft-identity N/A

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

      19. lower--.f64 50.8

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

   5. Applied rewrites 50.8%

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

   6. Taylor expanded in y around 0

      \[\leadsto 1 \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 45.0%

      \[\leadsto 1 \cdot x \]
3. Recombined 3 regimes into one program.

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y - z\right)}{t - z} \leq -1 \cdot 10^{+114}:\\ \;\;\;\;\frac{x}{z} \cdot \left(z - y\right)\\ \mathbf{elif}\;\frac{x \cdot \left(y - z\right)}{t - z} \leq 0:\\ \;\;\;\;\frac{\left(y - z\right) \cdot x}{t}\\ \mathbf{elif}\;\frac{x \cdot \left(y - z\right)}{t - z} \leq 5 \cdot 10^{-48}:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(z - y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 57.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot \left(y - z\right)}{t - z}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+77}:\\ \;\;\;\;\frac{x}{t - z} \cdot y\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{y - z}{t} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{z - y}{z} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* x (- y z)) (- t z))))
   (if (<= t_1 -2e+77)
     (* (/ x (- t z)) y)
     (if (<= t_1 0.0) (* (/ (- y z) t) x) (* (/ (- z y) z) x)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * (y - z)) / (t - z);
	double tmp;
	if (t_1 <= -2e+77) {
		tmp = (x / (t - z)) * y;
	} else if (t_1 <= 0.0) {
		tmp = ((y - z) / t) * x;
	} else {
		tmp = ((z - y) / z) * x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
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)) / (t - z)
    if (t_1 <= (-2d+77)) then
        tmp = (x / (t - z)) * y
    else if (t_1 <= 0.0d0) then
        tmp = ((y - z) / t) * x
    else
        tmp = ((z - y) / z) * x
    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)) / (t - z);
	double tmp;
	if (t_1 <= -2e+77) {
		tmp = (x / (t - z)) * y;
	} else if (t_1 <= 0.0) {
		tmp = ((y - z) / t) * x;
	} else {
		tmp = ((z - y) / z) * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * (y - z)) / (t - z)
	tmp = 0
	if t_1 <= -2e+77:
		tmp = (x / (t - z)) * y
	elif t_1 <= 0.0:
		tmp = ((y - z) / t) * x
	else:
		tmp = ((z - y) / z) * x
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * Float64(y - z)) / Float64(t - z))
	tmp = 0.0
	if (t_1 <= -2e+77)
		tmp = Float64(Float64(x / Float64(t - z)) * y);
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(Float64(y - z) / t) * x);
	else
		tmp = Float64(Float64(Float64(z - y) / z) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * (y - z)) / (t - z);
	tmp = 0.0;
	if (t_1 <= -2e+77)
		tmp = (x / (t - z)) * y;
	elseif (t_1 <= 0.0)
		tmp = ((y - z) / t) * x;
	else
		tmp = ((z - y) / z) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * N[(y - z), $MachinePrecision]), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+77], N[(N[(x / N[(t - z), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(N[(y - z), $MachinePrecision] / t), $MachinePrecision] * x), $MachinePrecision], N[(N[(N[(z - y), $MachinePrecision] / z), $MachinePrecision] * x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z)) < -1.99999999999999997e77

   1. Initial program 67.3%

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

   3. Taylor expanded in y around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 59.4

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

   5. Applied rewrites 59.4%

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

   if -1.99999999999999997e77 < (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z)) < 0.0

   1. Initial program 94.2%

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

   3. Taylor expanded in t around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 61.5

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

   5. Applied rewrites 61.5%

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

   if 0.0 < (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z))

   1. Initial program 84.4%

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

   3. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. distribute-neg-frac N/A

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

      7. lower-/.f64 N/A

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

      8. *-lft-identity N/A

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

      9. metadata-eval N/A

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

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

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. mul-1-neg N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(z\right)\right)}\right)\right) + \left(\mathsf{neg}\left(y\right)\right)}{z} \cdot x \]

      14. remove-double-neg N/A

          \[\leadsto \frac{\color{blue}{z} + \left(\mathsf{neg}\left(y\right)\right)}{z} \cdot x \]

      15. mul-1-neg N/A

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

      16. metadata-eval N/A

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

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

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

      18. *-lft-identity N/A

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

      19. lower--.f64 63.5

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

   5. Applied rewrites 63.5%

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

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y - z\right)}{t - z} \leq -2 \cdot 10^{+77}:\\ \;\;\;\;\frac{x}{t - z} \cdot y\\ \mathbf{elif}\;\frac{x \cdot \left(y - z\right)}{t - z} \leq 0:\\ \;\;\;\;\frac{y - z}{t} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{z - y}{z} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 59.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.4 \cdot 10^{+104}:\\ \;\;\;\;1 \cdot x\\ \mathbf{elif}\;z \leq -1.6 \cdot 10^{+68}:\\ \;\;\;\;\frac{-y}{z} \cdot x\\ \mathbf{elif}\;z \leq -1.12 \cdot 10^{-16}:\\ \;\;\;\;\frac{\left(-z\right) \cdot x}{t}\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{-15}:\\ \;\;\;\;\frac{y \cdot x}{t}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -4.4e+104)
   (* 1.0 x)
   (if (<= z -1.6e+68)
     (* (/ (- y) z) x)
     (if (<= z -1.12e-16)
       (/ (* (- z) x) t)
       (if (<= z 3.8e-15) (/ (* y x) t) (* 1.0 x))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -4.4e+104) {
		tmp = 1.0 * x;
	} else if (z <= -1.6e+68) {
		tmp = (-y / z) * x;
	} else if (z <= -1.12e-16) {
		tmp = (-z * x) / t;
	} else if (z <= 3.8e-15) {
		tmp = (y * x) / t;
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
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 <= (-4.4d+104)) then
        tmp = 1.0d0 * x
    else if (z <= (-1.6d+68)) then
        tmp = (-y / z) * x
    else if (z <= (-1.12d-16)) then
        tmp = (-z * x) / t
    else if (z <= 3.8d-15) then
        tmp = (y * x) / t
    else
        tmp = 1.0d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -4.4e+104) {
		tmp = 1.0 * x;
	} else if (z <= -1.6e+68) {
		tmp = (-y / z) * x;
	} else if (z <= -1.12e-16) {
		tmp = (-z * x) / t;
	} else if (z <= 3.8e-15) {
		tmp = (y * x) / t;
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -4.4e+104:
		tmp = 1.0 * x
	elif z <= -1.6e+68:
		tmp = (-y / z) * x
	elif z <= -1.12e-16:
		tmp = (-z * x) / t
	elif z <= 3.8e-15:
		tmp = (y * x) / t
	else:
		tmp = 1.0 * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -4.4e+104)
		tmp = Float64(1.0 * x);
	elseif (z <= -1.6e+68)
		tmp = Float64(Float64(Float64(-y) / z) * x);
	elseif (z <= -1.12e-16)
		tmp = Float64(Float64(Float64(-z) * x) / t);
	elseif (z <= 3.8e-15)
		tmp = Float64(Float64(y * x) / t);
	else
		tmp = Float64(1.0 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -4.4e+104)
		tmp = 1.0 * x;
	elseif (z <= -1.6e+68)
		tmp = (-y / z) * x;
	elseif (z <= -1.12e-16)
		tmp = (-z * x) / t;
	elseif (z <= 3.8e-15)
		tmp = (y * x) / t;
	else
		tmp = 1.0 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -4.4e+104], N[(1.0 * x), $MachinePrecision], If[LessEqual[z, -1.6e+68], N[(N[((-y) / z), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[z, -1.12e-16], N[(N[((-z) * x), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[z, 3.8e-15], N[(N[(y * x), $MachinePrecision] / t), $MachinePrecision], N[(1.0 * x), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -4.40000000000000001e104 or 3.8000000000000002e-15 < z

   1. Initial program 70.0%

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

   3. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. distribute-neg-frac N/A

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

      7. lower-/.f64 N/A

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

      8. *-lft-identity N/A

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

      9. metadata-eval N/A

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

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

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. mul-1-neg N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(z\right)\right)}\right)\right) + \left(\mathsf{neg}\left(y\right)\right)}{z} \cdot x \]

      14. remove-double-neg N/A

          \[\leadsto \frac{\color{blue}{z} + \left(\mathsf{neg}\left(y\right)\right)}{z} \cdot x \]

      15. mul-1-neg N/A

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

      16. metadata-eval N/A

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

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

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

      18. *-lft-identity N/A

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

      19. lower--.f64 84.4

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

   5. Applied rewrites 84.4%

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

   6. Taylor expanded in y around 0

      \[\leadsto 1 \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 67.2%

      \[\leadsto 1 \cdot x \]

   if -4.40000000000000001e104 < z < -1.59999999999999997e68

   1. Initial program 91.2%

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

   3. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. distribute-neg-frac N/A

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

      7. lower-/.f64 N/A

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

      8. *-lft-identity N/A

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

      9. metadata-eval N/A

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

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

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. mul-1-neg N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(z\right)\right)}\right)\right) + \left(\mathsf{neg}\left(y\right)\right)}{z} \cdot x \]

      14. remove-double-neg N/A

          \[\leadsto \frac{\color{blue}{z} + \left(\mathsf{neg}\left(y\right)\right)}{z} \cdot x \]

      15. mul-1-neg N/A

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

      16. metadata-eval N/A

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

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

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

      18. *-lft-identity N/A

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

      19. lower--.f64 82.2

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

   5. Applied rewrites 82.2%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 65.1%

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

   if -1.59999999999999997e68 < z < -1.12e-16

   1. Initial program 95.0%

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 71.0

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

   7. Applied rewrites 71.0%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 53.2%

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

   if -1.12e-16 < z < 3.8000000000000002e-15

   1. Initial program 95.1%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 60.4

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

   5. Applied rewrites 60.4%

      \[\leadsto \color{blue}{\frac{y \cdot x}{t}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.4 \cdot 10^{+104}:\\ \;\;\;\;1 \cdot x\\ \mathbf{elif}\;z \leq -1.6 \cdot 10^{+68}:\\ \;\;\;\;\frac{-y}{z} \cdot x\\ \mathbf{elif}\;z \leq -1.12 \cdot 10^{-16}:\\ \;\;\;\;\frac{\left(-z\right) \cdot x}{t}\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{-15}:\\ \;\;\;\;\frac{y \cdot x}{t}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x, \frac{t - y}{z}, x\right)\\ \mathbf{if}\;z \leq -7 \cdot 10^{+60}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -2.1 \cdot 10^{-30}:\\ \;\;\;\;\frac{\left(y - z\right) \cdot x}{t}\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{-28}:\\ \;\;\;\;\frac{y \cdot x}{t - z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma x (/ (- t y) z) x)))
   (if (<= z -7e+60)
     t_1
     (if (<= z -2.1e-30)
       (/ (* (- y z) x) t)
       (if (<= z 1.4e-28) (/ (* y x) (- t z)) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(x, ((t - y) / z), x);
	double tmp;
	if (z <= -7e+60) {
		tmp = t_1;
	} else if (z <= -2.1e-30) {
		tmp = ((y - z) * x) / t;
	} else if (z <= 1.4e-28) {
		tmp = (y * x) / (t - z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(x, Float64(Float64(t - y) / z), x)
	tmp = 0.0
	if (z <= -7e+60)
		tmp = t_1;
	elseif (z <= -2.1e-30)
		tmp = Float64(Float64(Float64(y - z) * x) / t);
	elseif (z <= 1.4e-28)
		tmp = Float64(Float64(y * x) / Float64(t - z));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[(N[(t - y), $MachinePrecision] / z), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[z, -7e+60], t$95$1, If[LessEqual[z, -2.1e-30], N[(N[(N[(y - z), $MachinePrecision] * x), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[z, 1.4e-28], N[(N[(y * x), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -7.0000000000000004e60 or 1.3999999999999999e-28 < z

   1. Initial program 72.2%

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

   3. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. distribute-neg-frac N/A

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

      7. lower-/.f64 N/A

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

      8. *-lft-identity N/A

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

      9. metadata-eval N/A

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

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

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. mul-1-neg N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(z\right)\right)}\right)\right) + \left(\mathsf{neg}\left(y\right)\right)}{z} \cdot x \]

      14. remove-double-neg N/A

          \[\leadsto \frac{\color{blue}{z} + \left(\mathsf{neg}\left(y\right)\right)}{z} \cdot x \]

      15. mul-1-neg N/A

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

      16. metadata-eval N/A

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

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

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

      18. *-lft-identity N/A

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

      19. lower--.f64 84.7

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

   5. Applied rewrites 84.7%

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

   6. Taylor expanded in y around 0

      \[\leadsto 1 \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 62.3%

      \[\leadsto 1 \cdot x \]

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 85.2%

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

   if -7.0000000000000004e60 < z < -2.1000000000000002e-30

   1. Initial program 99.5%

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 71.7

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

   7. Applied rewrites 71.7%

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

   if -2.1000000000000002e-30 < z < 1.3999999999999999e-28

   1. Initial program 94.8%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 78.2

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

   5. Applied rewrites 78.2%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{+60}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{t - y}{z}, x\right)\\ \mathbf{elif}\;z \leq -2.1 \cdot 10^{-30}:\\ \;\;\;\;\frac{\left(y - z\right) \cdot x}{t}\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{-28}:\\ \;\;\;\;\frac{y \cdot x}{t - z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{t - y}{z}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - y}{z} \cdot x\\ \mathbf{if}\;z \leq -7 \cdot 10^{+60}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -2.1 \cdot 10^{-30}:\\ \;\;\;\;\frac{\left(y - z\right) \cdot x}{t}\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{-28}:\\ \;\;\;\;\frac{y \cdot x}{t - z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ (- z y) z) x)))
   (if (<= z -7e+60)
     t_1
     (if (<= z -2.1e-30)
       (/ (* (- y z) x) t)
       (if (<= z 1.4e-28) (/ (* y x) (- t z)) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((z - y) / z) * x;
	double tmp;
	if (z <= -7e+60) {
		tmp = t_1;
	} else if (z <= -2.1e-30) {
		tmp = ((y - z) * x) / t;
	} else if (z <= 1.4e-28) {
		tmp = (y * x) / (t - z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
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 = ((z - y) / z) * x
    if (z <= (-7d+60)) then
        tmp = t_1
    else if (z <= (-2.1d-30)) then
        tmp = ((y - z) * x) / t
    else if (z <= 1.4d-28) then
        tmp = (y * x) / (t - z)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = ((z - y) / z) * x;
	double tmp;
	if (z <= -7e+60) {
		tmp = t_1;
	} else if (z <= -2.1e-30) {
		tmp = ((y - z) * x) / t;
	} else if (z <= 1.4e-28) {
		tmp = (y * x) / (t - z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = ((z - y) / z) * x
	tmp = 0
	if z <= -7e+60:
		tmp = t_1
	elif z <= -2.1e-30:
		tmp = ((y - z) * x) / t
	elif z <= 1.4e-28:
		tmp = (y * x) / (t - z)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(z - y) / z) * x)
	tmp = 0.0
	if (z <= -7e+60)
		tmp = t_1;
	elseif (z <= -2.1e-30)
		tmp = Float64(Float64(Float64(y - z) * x) / t);
	elseif (z <= 1.4e-28)
		tmp = Float64(Float64(y * x) / Float64(t - z));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = ((z - y) / z) * x;
	tmp = 0.0;
	if (z <= -7e+60)
		tmp = t_1;
	elseif (z <= -2.1e-30)
		tmp = ((y - z) * x) / t;
	elseif (z <= 1.4e-28)
		tmp = (y * x) / (t - z);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(z - y), $MachinePrecision] / z), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[z, -7e+60], t$95$1, If[LessEqual[z, -2.1e-30], N[(N[(N[(y - z), $MachinePrecision] * x), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[z, 1.4e-28], N[(N[(y * x), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -7.0000000000000004e60 or 1.3999999999999999e-28 < z

   1. Initial program 72.2%

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

   3. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. distribute-neg-frac N/A

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

      7. lower-/.f64 N/A

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

      8. *-lft-identity N/A

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

      9. metadata-eval N/A

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

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

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. mul-1-neg N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(z\right)\right)}\right)\right) + \left(\mathsf{neg}\left(y\right)\right)}{z} \cdot x \]

      14. remove-double-neg N/A

          \[\leadsto \frac{\color{blue}{z} + \left(\mathsf{neg}\left(y\right)\right)}{z} \cdot x \]

      15. mul-1-neg N/A

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

      16. metadata-eval N/A

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

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

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

      18. *-lft-identity N/A

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

      19. lower--.f64 84.7

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

   5. Applied rewrites 84.7%

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

   if -7.0000000000000004e60 < z < -2.1000000000000002e-30

   1. Initial program 99.5%

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 71.7

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

   7. Applied rewrites 71.7%

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

   if -2.1000000000000002e-30 < z < 1.3999999999999999e-28

   1. Initial program 94.8%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 78.2

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

   5. Applied rewrites 78.2%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{+60}:\\ \;\;\;\;\frac{z - y}{z} \cdot x\\ \mathbf{elif}\;z \leq -2.1 \cdot 10^{-30}:\\ \;\;\;\;\frac{\left(y - z\right) \cdot x}{t}\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{-28}:\\ \;\;\;\;\frac{y \cdot x}{t - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{z - y}{z} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 59.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.5 \cdot 10^{+87}:\\ \;\;\;\;1 \cdot x\\ \mathbf{elif}\;z \leq -1.12 \cdot 10^{-16}:\\ \;\;\;\;\frac{\left(-z\right) \cdot x}{t}\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{-15}:\\ \;\;\;\;\frac{y \cdot x}{t}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -3.5e+87)
   (* 1.0 x)
   (if (<= z -1.12e-16)
     (/ (* (- z) x) t)
     (if (<= z 3.8e-15) (/ (* y x) t) (* 1.0 x)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -3.5e+87) {
		tmp = 1.0 * x;
	} else if (z <= -1.12e-16) {
		tmp = (-z * x) / t;
	} else if (z <= 3.8e-15) {
		tmp = (y * x) / t;
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
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 <= (-3.5d+87)) then
        tmp = 1.0d0 * x
    else if (z <= (-1.12d-16)) then
        tmp = (-z * x) / t
    else if (z <= 3.8d-15) then
        tmp = (y * x) / t
    else
        tmp = 1.0d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -3.5e+87) {
		tmp = 1.0 * x;
	} else if (z <= -1.12e-16) {
		tmp = (-z * x) / t;
	} else if (z <= 3.8e-15) {
		tmp = (y * x) / t;
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -3.5e+87:
		tmp = 1.0 * x
	elif z <= -1.12e-16:
		tmp = (-z * x) / t
	elif z <= 3.8e-15:
		tmp = (y * x) / t
	else:
		tmp = 1.0 * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -3.5e+87)
		tmp = Float64(1.0 * x);
	elseif (z <= -1.12e-16)
		tmp = Float64(Float64(Float64(-z) * x) / t);
	elseif (z <= 3.8e-15)
		tmp = Float64(Float64(y * x) / t);
	else
		tmp = Float64(1.0 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -3.5e+87)
		tmp = 1.0 * x;
	elseif (z <= -1.12e-16)
		tmp = (-z * x) / t;
	elseif (z <= 3.8e-15)
		tmp = (y * x) / t;
	else
		tmp = 1.0 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -3.5e+87], N[(1.0 * x), $MachinePrecision], If[LessEqual[z, -1.12e-16], N[(N[((-z) * x), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[z, 3.8e-15], N[(N[(y * x), $MachinePrecision] / t), $MachinePrecision], N[(1.0 * x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.49999999999999986e87 or 3.8000000000000002e-15 < z

   1. Initial program 71.1%

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

   3. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. distribute-neg-frac N/A

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

      7. lower-/.f64 N/A

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

      8. *-lft-identity N/A

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

      9. metadata-eval N/A

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

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

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. mul-1-neg N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(z\right)\right)}\right)\right) + \left(\mathsf{neg}\left(y\right)\right)}{z} \cdot x \]

      14. remove-double-neg N/A

          \[\leadsto \frac{\color{blue}{z} + \left(\mathsf{neg}\left(y\right)\right)}{z} \cdot x \]

      15. mul-1-neg N/A

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

      16. metadata-eval N/A

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

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

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

      18. *-lft-identity N/A

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

      19. lower--.f64 84.9

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

   5. Applied rewrites 84.9%

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

   6. Taylor expanded in y around 0

      \[\leadsto 1 \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 66.6%

      \[\leadsto 1 \cdot x \]

   if -3.49999999999999986e87 < z < -1.12e-16

   1. Initial program 92.8%

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 62.1

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

   7. Applied rewrites 62.1%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 43.6%

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

   if -1.12e-16 < z < 3.8000000000000002e-15

   1. Initial program 95.1%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 60.4

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

   5. Applied rewrites 60.4%

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

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.5 \cdot 10^{+87}:\\ \;\;\;\;1 \cdot x\\ \mathbf{elif}\;z \leq -1.12 \cdot 10^{-16}:\\ \;\;\;\;\frac{\left(-z\right) \cdot x}{t}\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{-15}:\\ \;\;\;\;\frac{y \cdot x}{t}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 89.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.9 \cdot 10^{+98}:\\ \;\;\;\;x - \frac{x}{z} \cdot \left(y - t\right)\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{+151}:\\ \;\;\;\;\frac{x}{t - z} \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{t - y}{z}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -2.9e+98)
   (- x (* (/ x z) (- y t)))
   (if (<= z 3.3e+151) (* (/ x (- t z)) (- y z)) (fma x (/ (- t y) z) x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -2.9e+98) {
		tmp = x - ((x / z) * (y - t));
	} else if (z <= 3.3e+151) {
		tmp = (x / (t - z)) * (y - z);
	} else {
		tmp = fma(x, ((t - y) / z), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -2.9e+98)
		tmp = Float64(x - Float64(Float64(x / z) * Float64(y - t)));
	elseif (z <= 3.3e+151)
		tmp = Float64(Float64(x / Float64(t - z)) * Float64(y - z));
	else
		tmp = fma(x, Float64(Float64(t - y) / z), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -2.9e+98], N[(x - N[(N[(x / z), $MachinePrecision] * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.3e+151], N[(N[(x / N[(t - z), $MachinePrecision]), $MachinePrecision] * N[(y - z), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(t - y), $MachinePrecision] / z), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.9000000000000001e98

   1. Initial program 63.8%

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

   3. Taylor expanded in z around inf

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

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

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. associate-+l- N/A

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

      8. div-sub N/A

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

      9. lower--.f64 N/A

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

      10. div-sub N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. associate-/l* N/A

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

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

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower--.f64 94.1

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

   5. Applied rewrites 94.1%

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

   if -2.9000000000000001e98 < z < 3.30000000000000025e151

   1. Initial program 94.5%

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

   4. Applied rewrites 93.7%

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

   if 3.30000000000000025e151 < z

   1. Initial program 60.2%

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

   3. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. distribute-neg-frac N/A

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

      7. lower-/.f64 N/A

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

      8. *-lft-identity N/A

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

      9. metadata-eval N/A

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

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

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. mul-1-neg N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(z\right)\right)}\right)\right) + \left(\mathsf{neg}\left(y\right)\right)}{z} \cdot x \]

      14. remove-double-neg N/A

          \[\leadsto \frac{\color{blue}{z} + \left(\mathsf{neg}\left(y\right)\right)}{z} \cdot x \]

      15. mul-1-neg N/A

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

      16. metadata-eval N/A

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

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

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

      18. *-lft-identity N/A

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

      19. lower--.f64 93.4

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

   5. Applied rewrites 93.4%

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

   6. Taylor expanded in y around 0

      \[\leadsto 1 \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 69.3%

      \[\leadsto 1 \cdot x \]

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 93.8%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.9 \cdot 10^{+98}:\\ \;\;\;\;x - \frac{x}{z} \cdot \left(y - t\right)\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{+151}:\\ \;\;\;\;\frac{x}{t - z} \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{t - y}{z}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 66.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.5 \cdot 10^{+87} \lor \neg \left(z \leq 6.8 \cdot 10^{+143}\right):\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(y - z\right) \cdot x}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -3.5e+87) (not (<= z 6.8e+143))) (* 1.0 x) (/ (* (- y z) x) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -3.5e+87) || !(z <= 6.8e+143)) {
		tmp = 1.0 * x;
	} else {
		tmp = ((y - z) * x) / 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 <= (-3.5d+87)) .or. (.not. (z <= 6.8d+143))) then
        tmp = 1.0d0 * x
    else
        tmp = ((y - z) * x) / t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -3.5e+87) || !(z <= 6.8e+143)) {
		tmp = 1.0 * x;
	} else {
		tmp = ((y - z) * x) / t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -3.5e+87) or not (z <= 6.8e+143):
		tmp = 1.0 * x
	else:
		tmp = ((y - z) * x) / t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -3.5e+87) || !(z <= 6.8e+143))
		tmp = Float64(1.0 * x);
	else
		tmp = Float64(Float64(Float64(y - z) * x) / t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -3.5e+87) || ~((z <= 6.8e+143)))
		tmp = 1.0 * x;
	else
		tmp = ((y - z) * x) / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -3.5e+87], N[Not[LessEqual[z, 6.8e+143]], $MachinePrecision]], N[(1.0 * x), $MachinePrecision], N[(N[(N[(y - z), $MachinePrecision] * x), $MachinePrecision] / t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.49999999999999986e87 or 6.79999999999999964e143 < z

   1. Initial program 63.1%

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

   3. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. distribute-neg-frac N/A

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

      7. lower-/.f64 N/A

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

      8. *-lft-identity N/A

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

      9. metadata-eval N/A

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

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

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. mul-1-neg N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(z\right)\right)}\right)\right) + \left(\mathsf{neg}\left(y\right)\right)}{z} \cdot x \]

      14. remove-double-neg N/A

          \[\leadsto \frac{\color{blue}{z} + \left(\mathsf{neg}\left(y\right)\right)}{z} \cdot x \]

      15. mul-1-neg N/A

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

      16. metadata-eval N/A

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

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

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

      18. *-lft-identity N/A

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

      19. lower--.f64 92.5

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

   5. Applied rewrites 92.5%

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

   6. Taylor expanded in y around 0

      \[\leadsto 1 \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 76.4%

      \[\leadsto 1 \cdot x \]

   if -3.49999999999999986e87 < z < 6.79999999999999964e143

   1. Initial program 94.9%

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

   4. Applied rewrites 95.9%

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

   5. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 65.5

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

   7. Applied rewrites 65.5%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.5 \cdot 10^{+87} \lor \neg \left(z \leq 6.8 \cdot 10^{+143}\right):\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(y - z\right) \cdot x}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 60.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.4 \cdot 10^{-17} \lor \neg \left(z \leq 3.8 \cdot 10^{-15}\right):\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -6.4e-17) (not (<= z 3.8e-15))) (* 1.0 x) (/ (* y x) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -6.4e-17) || !(z <= 3.8e-15)) {
		tmp = 1.0 * x;
	} else {
		tmp = (y * x) / 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 <= (-6.4d-17)) .or. (.not. (z <= 3.8d-15))) then
        tmp = 1.0d0 * x
    else
        tmp = (y * x) / t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -6.4e-17) || !(z <= 3.8e-15)) {
		tmp = 1.0 * x;
	} else {
		tmp = (y * x) / t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -6.4e-17) or not (z <= 3.8e-15):
		tmp = 1.0 * x
	else:
		tmp = (y * x) / t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -6.4e-17) || !(z <= 3.8e-15))
		tmp = Float64(1.0 * x);
	else
		tmp = Float64(Float64(y * x) / t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -6.4e-17) || ~((z <= 3.8e-15)))
		tmp = 1.0 * x;
	else
		tmp = (y * x) / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -6.4e-17], N[Not[LessEqual[z, 3.8e-15]], $MachinePrecision]], N[(1.0 * x), $MachinePrecision], N[(N[(y * x), $MachinePrecision] / t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -6.4000000000000005e-17 or 3.8000000000000002e-15 < z

   1. Initial program 75.6%

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

   3. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. distribute-neg-frac N/A

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

      7. lower-/.f64 N/A

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

      8. *-lft-identity N/A

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

      9. metadata-eval N/A

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

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

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. mul-1-neg N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(z\right)\right)}\right)\right) + \left(\mathsf{neg}\left(y\right)\right)}{z} \cdot x \]

      14. remove-double-neg N/A

          \[\leadsto \frac{\color{blue}{z} + \left(\mathsf{neg}\left(y\right)\right)}{z} \cdot x \]

      15. mul-1-neg N/A

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

      16. metadata-eval N/A

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

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

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

      18. *-lft-identity N/A

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

      19. lower--.f64 76.4

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

   5. Applied rewrites 76.4%

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

   6. Taylor expanded in y around 0

      \[\leadsto 1 \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 57.5%

      \[\leadsto 1 \cdot x \]

   if -6.4000000000000005e-17 < z < 3.8000000000000002e-15

   1. Initial program 95.1%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 60.9

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

   5. Applied rewrites 60.9%

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

4. Final simplification 59.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.4 \cdot 10^{-17} \lor \neg \left(z \leq 3.8 \cdot 10^{-15}\right):\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 60.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.2 \cdot 10^{-52} \lor \neg \left(z \leq 3.8 \cdot 10^{-15}\right):\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -4.2e-52) (not (<= z 3.8e-15))) (* 1.0 x) (* (/ x t) y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -4.2e-52) || !(z <= 3.8e-15)) {
		tmp = 1.0 * x;
	} else {
		tmp = (x / t) * y;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
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 <= (-4.2d-52)) .or. (.not. (z <= 3.8d-15))) then
        tmp = 1.0d0 * x
    else
        tmp = (x / t) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -4.2e-52) || !(z <= 3.8e-15)) {
		tmp = 1.0 * x;
	} else {
		tmp = (x / t) * y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -4.2e-52) or not (z <= 3.8e-15):
		tmp = 1.0 * x
	else:
		tmp = (x / t) * y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -4.2e-52) || !(z <= 3.8e-15))
		tmp = Float64(1.0 * x);
	else
		tmp = Float64(Float64(x / t) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -4.2e-52) || ~((z <= 3.8e-15)))
		tmp = 1.0 * x;
	else
		tmp = (x / t) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -4.2e-52], N[Not[LessEqual[z, 3.8e-15]], $MachinePrecision]], N[(1.0 * x), $MachinePrecision], N[(N[(x / t), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.1999999999999997e-52 or 3.8000000000000002e-15 < z

   1. Initial program 77.5%

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

   3. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. distribute-neg-frac N/A

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

      7. lower-/.f64 N/A

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

      8. *-lft-identity N/A

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

      9. metadata-eval N/A

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

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

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. mul-1-neg N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(z\right)\right)}\right)\right) + \left(\mathsf{neg}\left(y\right)\right)}{z} \cdot x \]

      14. remove-double-neg N/A

          \[\leadsto \frac{\color{blue}{z} + \left(\mathsf{neg}\left(y\right)\right)}{z} \cdot x \]

      15. mul-1-neg N/A

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

      16. metadata-eval N/A

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

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

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

      18. *-lft-identity N/A

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

      19. lower--.f64 74.3

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

   5. Applied rewrites 74.3%

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

   6. Taylor expanded in y around 0

      \[\leadsto 1 \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 54.5%

      \[\leadsto 1 \cdot x \]

   if -4.1999999999999997e-52 < z < 3.8000000000000002e-15

   1. Initial program 94.6%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 63.7

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

   5. Applied rewrites 63.7%

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

   7. Applied rewrites 60.0%

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

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.2 \cdot 10^{-52} \lor \neg \left(z \leq 3.8 \cdot 10^{-15}\right):\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 35.3% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ 1 \cdot x \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (* 1.0 x))

C

double code(double x, double y, double z, double t) {
	return 1.0 * x;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
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 = 1.0d0 * x
end function

Java

public static double code(double x, double y, double z, double t) {
	return 1.0 * x;
}

Python

def code(x, y, z, t):
	return 1.0 * x

Julia

function code(x, y, z, t)
	return Float64(1.0 * x)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = 1.0 * x;
end

Wolfram

code[x_, y_, z_, t_] := N[(1.0 * x), $MachinePrecision]

Derivation

1. Initial program 84.7%

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

3. Taylor expanded in t around 0

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

   1. mul-1-neg N/A

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

   2. associate-/l* N/A

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

   3. *-commutative N/A

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

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

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

   5. lower-*.f64 N/A

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

   6. distribute-neg-frac N/A

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

   7. lower-/.f64 N/A

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

   8. *-lft-identity N/A

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

   9. metadata-eval N/A

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

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

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

   11. +-commutative N/A

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

   12. distribute-neg-in N/A

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

   13. mul-1-neg N/A

       \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(z\right)\right)}\right)\right) + \left(\mathsf{neg}\left(y\right)\right)}{z} \cdot x \]

   14. remove-double-neg N/A

       \[\leadsto \frac{\color{blue}{z} + \left(\mathsf{neg}\left(y\right)\right)}{z} \cdot x \]

   15. mul-1-neg N/A

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

   16. metadata-eval N/A

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

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

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

   18. *-lft-identity N/A

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

   19. lower--.f64 55.4

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

5. Applied rewrites 55.4%

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

6. Taylor expanded in y around 0

   \[\leadsto 1 \cdot x \]
7. Step-by-step derivation

8. Applied rewrites 35.2%

   \[\leadsto 1 \cdot x \]

9. Final simplification 35.2%

   \[\leadsto 1 \cdot x \]
10. Add Preprocessing

Developer Target 1: 96.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

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 / ((t - z) / (y - z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2025017 
(FPCore (x y z t)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderAreaSpots4D from Chart-1.5.3"
  :precision binary64

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

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