Data.Array.Repa.Algorithms.ColorRamp:rampColorHotToCold from repa-algorithms-3.4.0.1, B

Percentage Accurate: 99.9% → 99.8%

Time: 6.3s

Alternatives: 6

Speedup: N/A×

• 20.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (* 4.0 (- (- x y) (* z 0.5))) z))

C

double code(double x, double y, double z) {
	return (4.0 * ((x - y) - (z * 0.5))) / 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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (4.0d0 * ((x - y) - (z * 0.5d0))) / z
end function

Java

public static double code(double x, double y, double z) {
	return (4.0 * ((x - y) - (z * 0.5))) / z;
}

Python

def code(x, y, z):
	return (4.0 * ((x - y) - (z * 0.5))) / z

Julia

function code(x, y, z)
	return Float64(Float64(4.0 * Float64(Float64(x - y) - Float64(z * 0.5))) / z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (4.0 * ((x - y) - (z * 0.5))) / z;
end

Wolfram

code[x_, y_, z_] := N[(N[(4.0 * N[(N[(x - y), $MachinePrecision] - N[(z * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (* 4.0 (- (- x y) (* z 0.5))) z))

C

double code(double x, double y, double z) {
	return (4.0 * ((x - y) - (z * 0.5))) / 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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (4.0d0 * ((x - y) - (z * 0.5d0))) / z
end function

Java

public static double code(double x, double y, double z) {
	return (4.0 * ((x - y) - (z * 0.5))) / z;
}

Python

def code(x, y, z):
	return (4.0 * ((x - y) - (z * 0.5))) / z

Julia

function code(x, y, z)
	return Float64(Float64(4.0 * Float64(Float64(x - y) - Float64(z * 0.5))) / z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (4.0 * ((x - y) - (z * 0.5))) / z;
end

Wolfram

code[x_, y_, z_] := N[(N[(4.0 * N[(N[(x - y), $MachinePrecision] - N[(z * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]

Alternative 1: 99.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(y - x, \frac{-4}{z}, -2\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma (- y x) (/ -4.0 z) -2.0))

C

double code(double x, double y, double z) {
	return fma((y - x), (-4.0 / z), -2.0);
}

Julia

function code(x, y, z)
	return fma(Float64(y - x), Float64(-4.0 / z), -2.0)
end

Wolfram

code[x_, y_, z_] := N[(N[(y - x), $MachinePrecision] * N[(-4.0 / z), $MachinePrecision] + -2.0), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{-4 \cdot \frac{y + \frac{1}{2} \cdot z}{z} + 4 \cdot \frac{x}{z}} \]

5. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{-4}{z}, -2\right)} \]
6. Add Preprocessing

Alternative 2: 98.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z}\\ \mathbf{if}\;t\_0 \leq -50000000 \lor \neg \left(t\_0 \leq 40000\right):\\ \;\;\;\;\frac{\left(x - y\right) \cdot 4}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{y}{z}, -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* 4.0 (- (- x y) (* z 0.5))) z)))
   (if (or (<= t_0 -50000000.0) (not (<= t_0 40000.0)))
     (/ (* (- x y) 4.0) z)
     (fma -4.0 (/ y z) -2.0))))

C

double code(double x, double y, double z) {
	double t_0 = (4.0 * ((x - y) - (z * 0.5))) / z;
	double tmp;
	if ((t_0 <= -50000000.0) || !(t_0 <= 40000.0)) {
		tmp = ((x - y) * 4.0) / z;
	} else {
		tmp = fma(-4.0, (y / z), -2.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(4.0 * Float64(Float64(x - y) - Float64(z * 0.5))) / z)
	tmp = 0.0
	if ((t_0 <= -50000000.0) || !(t_0 <= 40000.0))
		tmp = Float64(Float64(Float64(x - y) * 4.0) / z);
	else
		tmp = fma(-4.0, Float64(y / z), -2.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(4.0 * N[(N[(x - y), $MachinePrecision] - N[(z * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -50000000.0], N[Not[LessEqual[t$95$0, 40000.0]], $MachinePrecision]], N[(N[(N[(x - y), $MachinePrecision] * 4.0), $MachinePrecision] / z), $MachinePrecision], N[(-4.0 * N[(y / z), $MachinePrecision] + -2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < -5e7 or 4e4 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z)

   1. Initial program 99.5%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(-4\right)\right)} \cdot \left(x - y\right)}{z} \]

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

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(-4 \cdot \left(x - y\right)\right)}}{z} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(4\right)\right)} \cdot \left(x - y\right)\right)}{z} \]

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

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(4 \cdot \left(x - y\right)\right)\right)}\right)}{z} \]

      5. *-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(x - y\right) \cdot 4}\right)\right)\right)}{z} \]

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

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\left(x - y\right)\right)\right) \cdot 4}\right)}{z} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{4 \cdot \left(\mathsf{neg}\left(\left(x - y\right)\right)\right)}\right)}{z} \]

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

         \[\leadsto \frac{\color{blue}{4 \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(x - y\right)\right)\right)\right)\right)}}{z} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(-4\right)\right)} \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(x - y\right)\right)\right)\right)\right)}{z} \]

      10. mul-1-neg N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(-4\right)\right) \cdot \left(\mathsf{neg}\left(\color{blue}{-1 \cdot \left(x - y\right)}\right)\right)}{z} \]

      11. *-commutative N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(-4\right)\right) \cdot \left(\mathsf{neg}\left(\color{blue}{\left(x - y\right) \cdot -1}\right)\right)}{z} \]

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

          \[\leadsto \frac{\left(\mathsf{neg}\left(-4\right)\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(x - y\right)\right)\right) \cdot -1\right)}}{z} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(-4\right)\right) \cdot \left(\left(\mathsf{neg}\left(\left(x - y\right)\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)}{z} \]

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

          \[\leadsto \frac{\left(\mathsf{neg}\left(-4\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(x - y\right)\right)\right) \cdot 1\right)\right)}}{z} \]

      15. *-inverses N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(-4\right)\right) \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(x - y\right)\right)\right) \cdot \color{blue}{\frac{z}{z}}\right)\right)}{z} \]

      16. associate-/l* N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(-4\right)\right) \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{\left(\mathsf{neg}\left(\left(x - y\right)\right)\right) \cdot z}{z}}\right)\right)}{z} \]

      17. associate-*l/ N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(-4\right)\right) \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{\mathsf{neg}\left(\left(x - y\right)\right)}{z} \cdot z}\right)\right)}{z} \]

      18. distribute-neg-frac N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(-4\right)\right) \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\frac{x - y}{z}\right)\right)} \cdot z\right)\right)}{z} \]

      19. mul-1-neg N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(-4\right)\right) \cdot \left(\mathsf{neg}\left(\color{blue}{\left(-1 \cdot \frac{x - y}{z}\right)} \cdot z\right)\right)}{z} \]

      20. mul-1-neg N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(-4\right)\right) \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\frac{x - y}{z}\right)\right)} \cdot z\right)\right)}{z} \]

      21. distribute-lft-neg-out N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(-4\right)\right) \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\frac{x - y}{z} \cdot z\right)\right)}\right)\right)}{z} \]

      22. remove-double-neg N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(-4\right)\right) \cdot \color{blue}{\left(\frac{x - y}{z} \cdot z\right)}}{z} \]

   5. Applied rewrites 99.3%

      \[\leadsto \frac{\color{blue}{\left(x - y\right) \cdot 4}}{z} \]

   if -5e7 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < 4e4

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-4 \cdot \frac{y + \frac{1}{2} \cdot z}{z}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot \left(y + \frac{1}{2} \cdot z\right)}{z}} \]

      2. distribute-rgt-in N/A

         \[\leadsto \frac{\color{blue}{y \cdot -4 + \left(\frac{1}{2} \cdot z\right) \cdot -4}}{z} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{-4 \cdot y} + \left(\frac{1}{2} \cdot z\right) \cdot -4}{z} \]

      4. div-add N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot y}{z} + \frac{\left(\frac{1}{2} \cdot z\right) \cdot -4}{z}} \]

      5. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{-4}{z} \cdot y} + \frac{\left(\frac{1}{2} \cdot z\right) \cdot -4}{z} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{-4 \cdot 1}}{z} \cdot y + \frac{\left(\frac{1}{2} \cdot z\right) \cdot -4}{z} \]

      7. associate-*r/ N/A

         \[\leadsto \color{blue}{\left(-4 \cdot \frac{1}{z}\right)} \cdot y + \frac{\left(\frac{1}{2} \cdot z\right) \cdot -4}{z} \]

      8. metadata-eval N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(4\right)\right)} \cdot \frac{1}{z}\right) \cdot y + \frac{\left(\frac{1}{2} \cdot z\right) \cdot -4}{z} \]

      9. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(4\right)\right) \cdot \frac{1}{z}\right) \cdot y + \frac{\color{blue}{-4 \cdot \left(\frac{1}{2} \cdot z\right)}}{z} \]

      10. associate-*r* N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(4\right)\right) \cdot \frac{1}{z}\right) \cdot y + \frac{\color{blue}{\left(-4 \cdot \frac{1}{2}\right) \cdot z}}{z} \]

      11. metadata-eval N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(4\right)\right) \cdot \frac{1}{z}\right) \cdot y + \frac{\color{blue}{-2} \cdot z}{z} \]

      12. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\mathsf{neg}\left(4\right)\right) \cdot \frac{1}{z}, y, \frac{-2 \cdot z}{z}\right)} \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-4} \cdot \frac{1}{z}, y, \frac{-2 \cdot z}{z}\right) \]

      14. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-4 \cdot 1}{z}}, y, \frac{-2 \cdot z}{z}\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{-4}}{z}, y, \frac{-2 \cdot z}{z}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-4}{z}}, y, \frac{-2 \cdot z}{z}\right) \]

      17. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{-4}{z}, y, \color{blue}{-2 \cdot \frac{z}{z}}\right) \]

      18. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{-4}{z}, y, \color{blue}{\left(-4 \cdot \frac{1}{2}\right)} \cdot \frac{z}{z}\right) \]

      19. *-inverses N/A

          \[\leadsto \mathsf{fma}\left(\frac{-4}{z}, y, \left(-4 \cdot \frac{1}{2}\right) \cdot \color{blue}{1}\right) \]

      20. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{-4}{z}, y, \color{blue}{-2} \cdot 1\right) \]

      21. metadata-eval 99.4

          \[\leadsto \mathsf{fma}\left(\frac{-4}{z}, y, \color{blue}{-2}\right) \]

   5. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-4}{z}, y, -2\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.4%

      \[\leadsto \mathsf{fma}\left(-4, \color{blue}{\frac{y}{z}}, -2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \leq -50000000 \lor \neg \left(\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \leq 40000\right):\\ \;\;\;\;\frac{\left(x - y\right) \cdot 4}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{y}{z}, -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 66.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z}\\ \mathbf{if}\;t\_0 \leq -50000000 \lor \neg \left(t\_0 \leq -1\right):\\ \;\;\;\;\frac{x}{z} \cdot 4\\ \mathbf{else}:\\ \;\;\;\;-2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* 4.0 (- (- x y) (* z 0.5))) z)))
   (if (or (<= t_0 -50000000.0) (not (<= t_0 -1.0))) (* (/ x z) 4.0) -2.0)))

C

double code(double x, double y, double z) {
	double t_0 = (4.0 * ((x - y) - (z * 0.5))) / z;
	double tmp;
	if ((t_0 <= -50000000.0) || !(t_0 <= -1.0)) {
		tmp = (x / z) * 4.0;
	} else {
		tmp = -2.0;
	}
	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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (4.0d0 * ((x - y) - (z * 0.5d0))) / z
    if ((t_0 <= (-50000000.0d0)) .or. (.not. (t_0 <= (-1.0d0)))) then
        tmp = (x / z) * 4.0d0
    else
        tmp = -2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (4.0 * ((x - y) - (z * 0.5))) / z;
	double tmp;
	if ((t_0 <= -50000000.0) || !(t_0 <= -1.0)) {
		tmp = (x / z) * 4.0;
	} else {
		tmp = -2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (4.0 * ((x - y) - (z * 0.5))) / z
	tmp = 0
	if (t_0 <= -50000000.0) or not (t_0 <= -1.0):
		tmp = (x / z) * 4.0
	else:
		tmp = -2.0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(4.0 * Float64(Float64(x - y) - Float64(z * 0.5))) / z)
	tmp = 0.0
	if ((t_0 <= -50000000.0) || !(t_0 <= -1.0))
		tmp = Float64(Float64(x / z) * 4.0);
	else
		tmp = -2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (4.0 * ((x - y) - (z * 0.5))) / z;
	tmp = 0.0;
	if ((t_0 <= -50000000.0) || ~((t_0 <= -1.0)))
		tmp = (x / z) * 4.0;
	else
		tmp = -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(4.0 * N[(N[(x - y), $MachinePrecision] - N[(z * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -50000000.0], N[Not[LessEqual[t$95$0, -1.0]], $MachinePrecision]], N[(N[(x / z), $MachinePrecision] * 4.0), $MachinePrecision], -2.0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < -5e7 or -1 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z)

   1. Initial program 99.5%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{x}{z} \cdot 4} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{z} \cdot 4} \]

      3. lower-/.f64 56.7

         \[\leadsto \color{blue}{\frac{x}{z}} \cdot 4 \]

   5. Applied rewrites 56.7%

      \[\leadsto \color{blue}{\frac{x}{z} \cdot 4} \]

   if -5e7 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < -1

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-2} \]
   4. Step-by-step derivation

   5. Applied rewrites 96.2%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \leq -50000000 \lor \neg \left(\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \leq -1\right):\\ \;\;\;\;\frac{x}{z} \cdot 4\\ \mathbf{else}:\\ \;\;\;\;-2\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.3 \cdot 10^{+55} \lor \neg \left(x \leq 1.7 \cdot 10^{+82}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{4}{z}, x, -2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{y}{z}, -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -4.3e+55) (not (<= x 1.7e+82)))
   (fma (/ 4.0 z) x -2.0)
   (fma -4.0 (/ y z) -2.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -4.3e+55) || !(x <= 1.7e+82)) {
		tmp = fma((4.0 / z), x, -2.0);
	} else {
		tmp = fma(-4.0, (y / z), -2.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -4.3e+55) || !(x <= 1.7e+82))
		tmp = fma(Float64(4.0 / z), x, -2.0);
	else
		tmp = fma(-4.0, Float64(y / z), -2.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -4.3e+55], N[Not[LessEqual[x, 1.7e+82]], $MachinePrecision]], N[(N[(4.0 / z), $MachinePrecision] * x + -2.0), $MachinePrecision], N[(-4.0 * N[(y / z), $MachinePrecision] + -2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.2999999999999999e55 or 1.69999999999999997e82 < x

   1. Initial program 99.1%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{4 \cdot \frac{x - \frac{1}{2} \cdot z}{z}} \]
   4. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto 4 \cdot \frac{x - \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)} \cdot z}{z} \]

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

         \[\leadsto 4 \cdot \frac{\color{blue}{x + \frac{-1}{2} \cdot z}}{z} \]

      3. div-add N/A

         \[\leadsto 4 \cdot \color{blue}{\left(\frac{x}{z} + \frac{\frac{-1}{2} \cdot z}{z}\right)} \]

      4. distribute-lft-in N/A

         \[\leadsto \color{blue}{4 \cdot \frac{x}{z} + 4 \cdot \frac{\frac{-1}{2} \cdot z}{z}} \]

      5. *-lft-identity N/A

         \[\leadsto 4 \cdot \frac{\color{blue}{1 \cdot x}}{z} + 4 \cdot \frac{\frac{-1}{2} \cdot z}{z} \]

      6. associate-*l/ N/A

         \[\leadsto 4 \cdot \color{blue}{\left(\frac{1}{z} \cdot x\right)} + 4 \cdot \frac{\frac{-1}{2} \cdot z}{z} \]

      7. associate-*l* N/A

         \[\leadsto \color{blue}{\left(4 \cdot \frac{1}{z}\right) \cdot x} + 4 \cdot \frac{\frac{-1}{2} \cdot z}{z} \]

      8. associate-/l* N/A

         \[\leadsto \left(4 \cdot \frac{1}{z}\right) \cdot x + 4 \cdot \color{blue}{\left(\frac{-1}{2} \cdot \frac{z}{z}\right)} \]

      9. *-inverses N/A

         \[\leadsto \left(4 \cdot \frac{1}{z}\right) \cdot x + 4 \cdot \left(\frac{-1}{2} \cdot \color{blue}{1}\right) \]

      10. metadata-eval N/A

          \[\leadsto \left(4 \cdot \frac{1}{z}\right) \cdot x + 4 \cdot \color{blue}{\frac{-1}{2}} \]

      11. metadata-eval N/A

          \[\leadsto \left(4 \cdot \frac{1}{z}\right) \cdot x + \color{blue}{-2} \]

      12. metadata-eval N/A

          \[\leadsto \left(4 \cdot \frac{1}{z}\right) \cdot x + \color{blue}{-2 \cdot 1} \]

      13. metadata-eval N/A

          \[\leadsto \left(4 \cdot \frac{1}{z}\right) \cdot x + \color{blue}{\left(-4 \cdot \frac{1}{2}\right)} \cdot 1 \]

      14. metadata-eval N/A

          \[\leadsto \left(4 \cdot \frac{1}{z}\right) \cdot x + \color{blue}{-2} \cdot 1 \]

      15. *-inverses N/A

          \[\leadsto \left(4 \cdot \frac{1}{z}\right) \cdot x + -2 \cdot \color{blue}{\frac{z}{z}} \]

      16. associate-/l* N/A

          \[\leadsto \left(4 \cdot \frac{1}{z}\right) \cdot x + \color{blue}{\frac{-2 \cdot z}{z}} \]

      17. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(4 \cdot \frac{1}{z}, x, \frac{-2 \cdot z}{z}\right)} \]

      18. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{4 \cdot 1}{z}}, x, \frac{-2 \cdot z}{z}\right) \]

      19. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{4}}{z}, x, \frac{-2 \cdot z}{z}\right) \]

      20. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{4}{z}}, x, \frac{-2 \cdot z}{z}\right) \]

      21. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{4}{z}, x, \color{blue}{-2 \cdot \frac{z}{z}}\right) \]

      22. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{4}{z}, x, \color{blue}{\left(-4 \cdot \frac{1}{2}\right)} \cdot \frac{z}{z}\right) \]

      23. *-inverses N/A

          \[\leadsto \mathsf{fma}\left(\frac{4}{z}, x, \left(-4 \cdot \frac{1}{2}\right) \cdot \color{blue}{1}\right) \]

      24. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{4}{z}, x, \color{blue}{-2} \cdot 1\right) \]

      25. metadata-eval 90.8

          \[\leadsto \mathsf{fma}\left(\frac{4}{z}, x, \color{blue}{-2}\right) \]

   5. Applied rewrites 90.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{4}{z}, x, -2\right)} \]

   if -4.2999999999999999e55 < x < 1.69999999999999997e82

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-4 \cdot \frac{y + \frac{1}{2} \cdot z}{z}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot \left(y + \frac{1}{2} \cdot z\right)}{z}} \]

      2. distribute-rgt-in N/A

         \[\leadsto \frac{\color{blue}{y \cdot -4 + \left(\frac{1}{2} \cdot z\right) \cdot -4}}{z} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{-4 \cdot y} + \left(\frac{1}{2} \cdot z\right) \cdot -4}{z} \]

      4. div-add N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot y}{z} + \frac{\left(\frac{1}{2} \cdot z\right) \cdot -4}{z}} \]

      5. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{-4}{z} \cdot y} + \frac{\left(\frac{1}{2} \cdot z\right) \cdot -4}{z} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{-4 \cdot 1}}{z} \cdot y + \frac{\left(\frac{1}{2} \cdot z\right) \cdot -4}{z} \]

      7. associate-*r/ N/A

         \[\leadsto \color{blue}{\left(-4 \cdot \frac{1}{z}\right)} \cdot y + \frac{\left(\frac{1}{2} \cdot z\right) \cdot -4}{z} \]

      8. metadata-eval N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(4\right)\right)} \cdot \frac{1}{z}\right) \cdot y + \frac{\left(\frac{1}{2} \cdot z\right) \cdot -4}{z} \]

      9. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(4\right)\right) \cdot \frac{1}{z}\right) \cdot y + \frac{\color{blue}{-4 \cdot \left(\frac{1}{2} \cdot z\right)}}{z} \]

      10. associate-*r* N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(4\right)\right) \cdot \frac{1}{z}\right) \cdot y + \frac{\color{blue}{\left(-4 \cdot \frac{1}{2}\right) \cdot z}}{z} \]

      11. metadata-eval N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(4\right)\right) \cdot \frac{1}{z}\right) \cdot y + \frac{\color{blue}{-2} \cdot z}{z} \]

      12. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\mathsf{neg}\left(4\right)\right) \cdot \frac{1}{z}, y, \frac{-2 \cdot z}{z}\right)} \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-4} \cdot \frac{1}{z}, y, \frac{-2 \cdot z}{z}\right) \]

      14. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-4 \cdot 1}{z}}, y, \frac{-2 \cdot z}{z}\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{-4}}{z}, y, \frac{-2 \cdot z}{z}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-4}{z}}, y, \frac{-2 \cdot z}{z}\right) \]

      17. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{-4}{z}, y, \color{blue}{-2 \cdot \frac{z}{z}}\right) \]

      18. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{-4}{z}, y, \color{blue}{\left(-4 \cdot \frac{1}{2}\right)} \cdot \frac{z}{z}\right) \]

      19. *-inverses N/A

          \[\leadsto \mathsf{fma}\left(\frac{-4}{z}, y, \left(-4 \cdot \frac{1}{2}\right) \cdot \color{blue}{1}\right) \]

      20. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{-4}{z}, y, \color{blue}{-2} \cdot 1\right) \]

      21. metadata-eval 86.7

          \[\leadsto \mathsf{fma}\left(\frac{-4}{z}, y, \color{blue}{-2}\right) \]

   5. Applied rewrites 86.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-4}{z}, y, -2\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 86.8%

      \[\leadsto \mathsf{fma}\left(-4, \color{blue}{\frac{y}{z}}, -2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.3 \cdot 10^{+55} \lor \neg \left(x \leq 1.7 \cdot 10^{+82}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{4}{z}, x, -2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{y}{z}, -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.18 \cdot 10^{+61} \lor \neg \left(x \leq 2.7 \cdot 10^{+161}\right):\\ \;\;\;\;\frac{x}{z} \cdot 4\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{y}{z}, -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.18e+61) (not (<= x 2.7e+161)))
   (* (/ x z) 4.0)
   (fma -4.0 (/ y z) -2.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.18e+61) || !(x <= 2.7e+161)) {
		tmp = (x / z) * 4.0;
	} else {
		tmp = fma(-4.0, (y / z), -2.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.18e+61) || !(x <= 2.7e+161))
		tmp = Float64(Float64(x / z) * 4.0);
	else
		tmp = fma(-4.0, Float64(y / z), -2.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.18e+61], N[Not[LessEqual[x, 2.7e+161]], $MachinePrecision]], N[(N[(x / z), $MachinePrecision] * 4.0), $MachinePrecision], N[(-4.0 * N[(y / z), $MachinePrecision] + -2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.18000000000000004e61 or 2.6999999999999998e161 < x

   1. Initial program 99.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{x}{z} \cdot 4} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{z} \cdot 4} \]

      3. lower-/.f64 82.9

         \[\leadsto \color{blue}{\frac{x}{z}} \cdot 4 \]

   5. Applied rewrites 82.9%

      \[\leadsto \color{blue}{\frac{x}{z} \cdot 4} \]

   if -1.18000000000000004e61 < x < 2.6999999999999998e161

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-4 \cdot \frac{y + \frac{1}{2} \cdot z}{z}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot \left(y + \frac{1}{2} \cdot z\right)}{z}} \]

      2. distribute-rgt-in N/A

         \[\leadsto \frac{\color{blue}{y \cdot -4 + \left(\frac{1}{2} \cdot z\right) \cdot -4}}{z} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{-4 \cdot y} + \left(\frac{1}{2} \cdot z\right) \cdot -4}{z} \]

      4. div-add N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot y}{z} + \frac{\left(\frac{1}{2} \cdot z\right) \cdot -4}{z}} \]

      5. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{-4}{z} \cdot y} + \frac{\left(\frac{1}{2} \cdot z\right) \cdot -4}{z} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{-4 \cdot 1}}{z} \cdot y + \frac{\left(\frac{1}{2} \cdot z\right) \cdot -4}{z} \]

      7. associate-*r/ N/A

         \[\leadsto \color{blue}{\left(-4 \cdot \frac{1}{z}\right)} \cdot y + \frac{\left(\frac{1}{2} \cdot z\right) \cdot -4}{z} \]

      8. metadata-eval N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(4\right)\right)} \cdot \frac{1}{z}\right) \cdot y + \frac{\left(\frac{1}{2} \cdot z\right) \cdot -4}{z} \]

      9. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(4\right)\right) \cdot \frac{1}{z}\right) \cdot y + \frac{\color{blue}{-4 \cdot \left(\frac{1}{2} \cdot z\right)}}{z} \]

      10. associate-*r* N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(4\right)\right) \cdot \frac{1}{z}\right) \cdot y + \frac{\color{blue}{\left(-4 \cdot \frac{1}{2}\right) \cdot z}}{z} \]

      11. metadata-eval N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(4\right)\right) \cdot \frac{1}{z}\right) \cdot y + \frac{\color{blue}{-2} \cdot z}{z} \]

      12. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\mathsf{neg}\left(4\right)\right) \cdot \frac{1}{z}, y, \frac{-2 \cdot z}{z}\right)} \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-4} \cdot \frac{1}{z}, y, \frac{-2 \cdot z}{z}\right) \]

      14. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-4 \cdot 1}{z}}, y, \frac{-2 \cdot z}{z}\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{-4}}{z}, y, \frac{-2 \cdot z}{z}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-4}{z}}, y, \frac{-2 \cdot z}{z}\right) \]

      17. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{-4}{z}, y, \color{blue}{-2 \cdot \frac{z}{z}}\right) \]

      18. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{-4}{z}, y, \color{blue}{\left(-4 \cdot \frac{1}{2}\right)} \cdot \frac{z}{z}\right) \]

      19. *-inverses N/A

          \[\leadsto \mathsf{fma}\left(\frac{-4}{z}, y, \left(-4 \cdot \frac{1}{2}\right) \cdot \color{blue}{1}\right) \]

      20. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{-4}{z}, y, \color{blue}{-2} \cdot 1\right) \]

      21. metadata-eval 84.4

          \[\leadsto \mathsf{fma}\left(\frac{-4}{z}, y, \color{blue}{-2}\right) \]

   5. Applied rewrites 84.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-4}{z}, y, -2\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 84.5%

      \[\leadsto \mathsf{fma}\left(-4, \color{blue}{\frac{y}{z}}, -2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.18 \cdot 10^{+61} \lor \neg \left(x \leq 2.7 \cdot 10^{+161}\right):\\ \;\;\;\;\frac{x}{z} \cdot 4\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{y}{z}, -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 34.2% accurate, 28.0× speedup

Math

\[\begin{array}{l} \\ -2 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 -2.0)

C

double code(double x, double y, double z) {
	return -2.0;
}

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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = -2.0d0
end function

Java

public static double code(double x, double y, double z) {
	return -2.0;
}

Python

def code(x, y, z):
	return -2.0

Julia

function code(x, y, z)
	return -2.0
end

MATLAB

function tmp = code(x, y, z)
	tmp = -2.0;
end

Wolfram

code[x_, y_, z_] := -2.0

Derivation

1. Initial program 99.6%

   \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
2. Add Preprocessing

3. Taylor expanded in z around inf

   \[\leadsto \color{blue}{-2} \]
4. Step-by-step derivation

5. Applied rewrites 28.7%

   \[\leadsto \color{blue}{-2} \]
6. Add Preprocessing

Developer Target 1: 98.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ 4 \cdot \frac{x}{z} - \left(2 + 4 \cdot \frac{y}{z}\right) \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (- (* 4.0 (/ x z)) (+ 2.0 (* 4.0 (/ y z)))))

C

double code(double x, double y, double z) {
	return (4.0 * (x / z)) - (2.0 + (4.0 * (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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (4.0d0 * (x / z)) - (2.0d0 + (4.0d0 * (y / z)))
end function

Java

public static double code(double x, double y, double z) {
	return (4.0 * (x / z)) - (2.0 + (4.0 * (y / z)));
}

Python

def code(x, y, z):
	return (4.0 * (x / z)) - (2.0 + (4.0 * (y / z)))

Julia

function code(x, y, z)
	return Float64(Float64(4.0 * Float64(x / z)) - Float64(2.0 + Float64(4.0 * Float64(y / z))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (4.0 * (x / z)) - (2.0 + (4.0 * (y / z)));
end

Wolfram

code[x_, y_, z_] := N[(N[(4.0 * N[(x / z), $MachinePrecision]), $MachinePrecision] - N[(2.0 + N[(4.0 * N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024359 
(FPCore (x y z)
  :name "Data.Array.Repa.Algorithms.ColorRamp:rampColorHotToCold from repa-algorithms-3.4.0.1, B"
  :precision binary64

  :alt
  (! :herbie-platform default (- (* 4 (/ x z)) (+ 2 (* 4 (/ y z)))))

  (/ (* 4.0 (- (- x y) (* z 0.5))) z))