Data.Colour.RGBSpace.HSV:hsv from colour-2.3.3, J

Percentage Accurate: 96.0% → 99.9%

Time: 5.8s

Alternatives: 8

Speedup: 0.7×

Specification

Math

\[\begin{array}{l} \\ x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* x (- 1.0 (* (- 1.0 y) z))))

C

double code(double x, double y, double z) {
	return x * (1.0 - ((1.0 - 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 = x * (1.0d0 - ((1.0d0 - y) * z))
end function

Java

public static double code(double x, double y, double z) {
	return x * (1.0 - ((1.0 - y) * z));
}

Python

def code(x, y, z):
	return x * (1.0 - ((1.0 - y) * z))

Julia

function code(x, y, z)
	return Float64(x * Float64(1.0 - Float64(Float64(1.0 - y) * z)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x * (1.0 - ((1.0 - y) * z));
end

Wolfram

code[x_, y_, z_] := N[(x * N[(1.0 - N[(N[(1.0 - y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 96.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* x (- 1.0 (* (- 1.0 y) z))))

C

double code(double x, double y, double z) {
	return x * (1.0 - ((1.0 - 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 = x * (1.0d0 - ((1.0d0 - y) * z))
end function

Java

public static double code(double x, double y, double z) {
	return x * (1.0 - ((1.0 - y) * z));
}

Python

def code(x, y, z):
	return x * (1.0 - ((1.0 - y) * z))

Julia

function code(x, y, z)
	return Float64(x * Float64(1.0 - Float64(Float64(1.0 - y) * z)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x * (1.0 - ((1.0 - y) * z));
end

Wolfram

code[x_, y_, z_] := N[(x * N[(1.0 - N[(N[(1.0 - y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \left(1 - y\right) \cdot z\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(y \cdot x\right) \cdot z\\ \mathbf{elif}\;t\_0 \leq 10^{+201}:\\ \;\;\;\;\mathsf{fma}\left(\left(-1 + y\right) \cdot z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y - 1\right) \cdot \left(z \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (* (- 1.0 y) z))))
   (if (<= t_0 (- INFINITY))
     (* (* y x) z)
     (if (<= t_0 1e+201) (fma (* (+ -1.0 y) z) x x) (* (- y 1.0) (* z x))))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 - ((1.0 - y) * z);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (y * x) * z;
	} else if (t_0 <= 1e+201) {
		tmp = fma(((-1.0 + y) * z), x, x);
	} else {
		tmp = (y - 1.0) * (z * x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(1.0 - Float64(Float64(1.0 - y) * z))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(y * x) * z);
	elseif (t_0 <= 1e+201)
		tmp = fma(Float64(Float64(-1.0 + y) * z), x, x);
	else
		tmp = Float64(Float64(y - 1.0) * Float64(z * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(1.0 - N[(N[(1.0 - y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(y * x), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[t$95$0, 1e+201], N[(N[(N[(-1.0 + y), $MachinePrecision] * z), $MachinePrecision] * x + x), $MachinePrecision], N[(N[(y - 1.0), $MachinePrecision] * N[(z * x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 #s(literal 1 binary64) (*.f64 (-.f64 #s(literal 1 binary64) y) z)) < -inf.0

   1. Initial program 57.8%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 99.8

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

   5. Applied rewrites 99.8%

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

   7. Applied rewrites 99.9%

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

   if -inf.0 < (-.f64 #s(literal 1 binary64) (*.f64 (-.f64 #s(literal 1 binary64) y) z)) < 1.00000000000000004e201

   1. Initial program 99.9%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 99.9%

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

   if 1.00000000000000004e201 < (-.f64 #s(literal 1 binary64) (*.f64 (-.f64 #s(literal 1 binary64) y) z))

   1. Initial program 84.9%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 84.9%

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

   6. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. remove-double-neg N/A

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

      4. mul-1-neg N/A

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

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

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

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

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

      7. mul-1-neg N/A

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

      8. mul-1-neg N/A

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

      9. *-commutative N/A

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

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

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

      11. associate-*l* N/A

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

      12. mul-1-neg N/A

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

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

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

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

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

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

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

      16. lower-*.f64 N/A

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

      17. remove-double-neg N/A

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

      18. lower--.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 100.0

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

   8. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\left(y - 1\right) \cdot \left(z \cdot x\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 88.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{+36}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y\\ \mathbf{elif}\;y \leq 5.1 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(-z, x, x\right)\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{+145}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot y, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot x\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.4e+36)
   (* (* z x) y)
   (if (<= y 5.1e-9)
     (fma (- z) x x)
     (if (<= y 3.8e+145) (fma (* z y) x x) (* (* y x) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.4e+36) {
		tmp = (z * x) * y;
	} else if (y <= 5.1e-9) {
		tmp = fma(-z, x, x);
	} else if (y <= 3.8e+145) {
		tmp = fma((z * y), x, x);
	} else {
		tmp = (y * x) * z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.4e+36)
		tmp = Float64(Float64(z * x) * y);
	elseif (y <= 5.1e-9)
		tmp = fma(Float64(-z), x, x);
	elseif (y <= 3.8e+145)
		tmp = fma(Float64(z * y), x, x);
	else
		tmp = Float64(Float64(y * x) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.4e+36], N[(N[(z * x), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[y, 5.1e-9], N[((-z) * x + x), $MachinePrecision], If[LessEqual[y, 3.8e+145], N[(N[(z * y), $MachinePrecision] * x + x), $MachinePrecision], N[(N[(y * x), $MachinePrecision] * z), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -2.39999999999999992e36

   1. Initial program 86.9%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 90.6

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

   5. Applied rewrites 90.6%

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

   if -2.39999999999999992e36 < y < 5.10000000000000017e-9

   1. Initial program 100.0%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(-1 \cdot z, x, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 99.5%

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

   if 5.10000000000000017e-9 < y < 3.80000000000000012e145

   1. Initial program 99.9%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(-1 \cdot z, x, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 51.5%

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

   9. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(y \cdot z, x, x\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 99.4%

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

   if 3.80000000000000012e145 < y

   1. Initial program 79.0%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 81.6

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

   5. Applied rewrites 81.6%

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

   7. Applied rewrites 90.3%

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

Alternative 3: 98.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.05 \lor \neg \left(z \leq 0.075\right):\\ \;\;\;\;\left(y - 1\right) \cdot \left(z \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot y, x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.05) (not (<= z 0.075)))
   (* (- y 1.0) (* z x))
   (fma (* z y) x x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.05) || !(z <= 0.075)) {
		tmp = (y - 1.0) * (z * x);
	} else {
		tmp = fma((z * y), x, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.05) || !(z <= 0.075))
		tmp = Float64(Float64(y - 1.0) * Float64(z * x));
	else
		tmp = fma(Float64(z * y), x, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.05], N[Not[LessEqual[z, 0.075]], $MachinePrecision]], N[(N[(y - 1.0), $MachinePrecision] * N[(z * x), $MachinePrecision]), $MachinePrecision], N[(N[(z * y), $MachinePrecision] * x + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.05000000000000004 or 0.0749999999999999972 < z

   1. Initial program 87.4%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 87.4%

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

   6. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. remove-double-neg N/A

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

      4. mul-1-neg N/A

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

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

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

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

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

      7. mul-1-neg N/A

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

      8. mul-1-neg N/A

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

      9. *-commutative N/A

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

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

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

      11. associate-*l* N/A

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

      12. mul-1-neg N/A

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

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

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

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

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

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

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

      16. lower-*.f64 N/A

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

      17. remove-double-neg N/A

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

      18. lower--.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 99.0

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

   8. Applied rewrites 99.0%

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

   if -1.05000000000000004 < z < 0.0749999999999999972

   1. Initial program 99.9%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(-1 \cdot z, x, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 72.0%

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

   9. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(y \cdot z, x, x\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 99.5%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.05 \lor \neg \left(z \leq 0.075\right):\\ \;\;\;\;\left(y - 1\right) \cdot \left(z \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot y, x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{+36} \lor \neg \left(y \leq 1000000000\right):\\ \;\;\;\;\left(y \cdot x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -2.4e+36) (not (<= y 1000000000.0)))
   (* (* y x) z)
   (fma (- z) x x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.4e+36) || !(y <= 1000000000.0)) {
		tmp = (y * x) * z;
	} else {
		tmp = fma(-z, x, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -2.4e+36) || !(y <= 1000000000.0))
		tmp = Float64(Float64(y * x) * z);
	else
		tmp = fma(Float64(-z), x, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -2.4e+36], N[Not[LessEqual[y, 1000000000.0]], $MachinePrecision]], N[(N[(y * x), $MachinePrecision] * z), $MachinePrecision], N[((-z) * x + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.39999999999999992e36 or 1e9 < y

   1. Initial program 86.6%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 80.2

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

   5. Applied rewrites 80.2%

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

   7. Applied rewrites 82.8%

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

   if -2.39999999999999992e36 < y < 1e9

   1. Initial program 100.0%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(-1 \cdot z, x, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 99.5%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{+36} \lor \neg \left(y \leq 1000000000\right):\\ \;\;\;\;\left(y \cdot x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{+36}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y\\ \mathbf{elif}\;y \leq 1000000000:\\ \;\;\;\;\mathsf{fma}\left(-z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot x\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.4e+36)
   (* (* z x) y)
   (if (<= y 1000000000.0) (fma (- z) x x) (* (* y x) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.4e+36) {
		tmp = (z * x) * y;
	} else if (y <= 1000000000.0) {
		tmp = fma(-z, x, x);
	} else {
		tmp = (y * x) * z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.4e+36)
		tmp = Float64(Float64(z * x) * y);
	elseif (y <= 1000000000.0)
		tmp = fma(Float64(-z), x, x);
	else
		tmp = Float64(Float64(y * x) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.4e+36], N[(N[(z * x), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[y, 1000000000.0], N[((-z) * x + x), $MachinePrecision], N[(N[(y * x), $MachinePrecision] * z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.39999999999999992e36

   1. Initial program 86.9%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 90.6

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

   5. Applied rewrites 90.6%

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

   if -2.39999999999999992e36 < y < 1e9

   1. Initial program 100.0%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(-1 \cdot z, x, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 99.5%

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

   if 1e9 < y

   1. Initial program 86.4%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 70.7

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

   5. Applied rewrites 70.7%

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

   7. Applied rewrites 76.3%

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

Alternative 6: 65.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.000215 \lor \neg \left(z \leq 0.075\right):\\ \;\;\;\;x \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -0.000215) (not (<= z 0.075))) (* x (- z)) (* x 1.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -0.000215) || !(z <= 0.075)) {
		tmp = x * -z;
	} else {
		tmp = x * 1.0;
	}
	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) :: tmp
    if ((z <= (-0.000215d0)) .or. (.not. (z <= 0.075d0))) then
        tmp = x * -z
    else
        tmp = x * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -0.000215) || !(z <= 0.075)) {
		tmp = x * -z;
	} else {
		tmp = x * 1.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -0.000215) or not (z <= 0.075):
		tmp = x * -z
	else:
		tmp = x * 1.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -0.000215) || !(z <= 0.075))
		tmp = Float64(x * Float64(-z));
	else
		tmp = Float64(x * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -0.000215) || ~((z <= 0.075)))
		tmp = x * -z;
	else
		tmp = x * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -0.000215], N[Not[LessEqual[z, 0.075]], $MachinePrecision]], N[(x * (-z)), $MachinePrecision], N[(x * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.14999999999999995e-4 or 0.0749999999999999972 < z

   1. Initial program 87.6%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

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

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

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

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

      3. remove-double-neg N/A

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

      4. mul-1-neg N/A

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

      5. metadata-eval N/A

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

      6. mul-1-neg N/A

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

      7. distribute-neg-in N/A

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

      8. associate-*r* N/A

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

      9. distribute-lft1-in N/A

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

      10. +-commutative N/A

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

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

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

      12. metadata-eval N/A

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

      13. *-lft-identity N/A

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

      14. *-commutative N/A

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

      15. mul-1-neg N/A

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

      16. *-commutative N/A

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

      17. associate-*r* N/A

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

      18. lower-*.f64 N/A

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

   5. Applied rewrites 86.7%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 52.1%

      \[\leadsto x \cdot \left(-z\right) \]

   if -2.14999999999999995e-4 < z < 0.0749999999999999972

   1. Initial program 99.9%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. lower--.f64 73.1

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

   5. Applied rewrites 73.1%

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

   6. Taylor expanded in z around 0

      \[\leadsto x \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 72.7%

      \[\leadsto x \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.000215 \lor \neg \left(z \leq 0.075\right):\\ \;\;\;\;x \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 66.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 93.8%

   \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

5. Applied rewrites 93.8%

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

6. Taylor expanded in y around 0

   \[\leadsto \mathsf{fma}\left(-1 \cdot z, x, x\right) \]
7. Step-by-step derivation

8. Applied rewrites 63.1%

   \[\leadsto \mathsf{fma}\left(-z, x, x\right) \]
9. Add Preprocessing

Alternative 8: 38.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ x \cdot 1 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* x 1.0))

C

double code(double x, double y, double z) {
	return x * 1.0;
}

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 = x * 1.0d0
end function

Java

public static double code(double x, double y, double z) {
	return x * 1.0;
}

Python

def code(x, y, z):
	return x * 1.0

Julia

function code(x, y, z)
	return Float64(x * 1.0)
end

MATLAB

function tmp = code(x, y, z)
	tmp = x * 1.0;
end

Wolfram

code[x_, y_, z_] := N[(x * 1.0), $MachinePrecision]

Derivation

1. Initial program 93.8%

   \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
2. Add Preprocessing

3. Taylor expanded in y around 0

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

   1. lower--.f64 63.1

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

5. Applied rewrites 63.1%

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

6. Taylor expanded in z around 0

   \[\leadsto x \cdot \color{blue}{1} \]
7. Step-by-step derivation

8. Applied rewrites 38.2%

   \[\leadsto x \cdot \color{blue}{1} \]
9. Add Preprocessing

Developer Target 1: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(1 - \left(1 - y\right) \cdot z\right)\\ t_1 := x + \left(1 - y\right) \cdot \left(\left(-z\right) \cdot x\right)\\ \mathbf{if}\;t\_0 < -1.618195973607049 \cdot 10^{+50}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 < 3.892237649663903 \cdot 10^{+134}:\\ \;\;\;\;\left(x \cdot y\right) \cdot z - \left(x \cdot z - x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (- 1.0 (* (- 1.0 y) z))))
        (t_1 (+ x (* (- 1.0 y) (* (- z) x)))))
   (if (< t_0 -1.618195973607049e+50)
     t_1
     (if (< t_0 3.892237649663903e+134) (- (* (* x y) z) (- (* x z) x)) t_1))))

C

double code(double x, double y, double z) {
	double t_0 = x * (1.0 - ((1.0 - y) * z));
	double t_1 = x + ((1.0 - y) * (-z * x));
	double tmp;
	if (t_0 < -1.618195973607049e+50) {
		tmp = t_1;
	} else if (t_0 < 3.892237649663903e+134) {
		tmp = ((x * y) * z) - ((x * z) - x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
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) :: t_1
    real(8) :: tmp
    t_0 = x * (1.0d0 - ((1.0d0 - y) * z))
    t_1 = x + ((1.0d0 - y) * (-z * x))
    if (t_0 < (-1.618195973607049d+50)) then
        tmp = t_1
    else if (t_0 < 3.892237649663903d+134) then
        tmp = ((x * y) * z) - ((x * z) - x)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x * (1.0 - ((1.0 - y) * z));
	double t_1 = x + ((1.0 - y) * (-z * x));
	double tmp;
	if (t_0 < -1.618195973607049e+50) {
		tmp = t_1;
	} else if (t_0 < 3.892237649663903e+134) {
		tmp = ((x * y) * z) - ((x * z) - x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * (1.0 - ((1.0 - y) * z))
	t_1 = x + ((1.0 - y) * (-z * x))
	tmp = 0
	if t_0 < -1.618195973607049e+50:
		tmp = t_1
	elif t_0 < 3.892237649663903e+134:
		tmp = ((x * y) * z) - ((x * z) - x)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(1.0 - Float64(Float64(1.0 - y) * z)))
	t_1 = Float64(x + Float64(Float64(1.0 - y) * Float64(Float64(-z) * x)))
	tmp = 0.0
	if (t_0 < -1.618195973607049e+50)
		tmp = t_1;
	elseif (t_0 < 3.892237649663903e+134)
		tmp = Float64(Float64(Float64(x * y) * z) - Float64(Float64(x * z) - x));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * (1.0 - ((1.0 - y) * z));
	t_1 = x + ((1.0 - y) * (-z * x));
	tmp = 0.0;
	if (t_0 < -1.618195973607049e+50)
		tmp = t_1;
	elseif (t_0 < 3.892237649663903e+134)
		tmp = ((x * y) * z) - ((x * z) - x);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(1.0 - N[(N[(1.0 - y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x + N[(N[(1.0 - y), $MachinePrecision] * N[((-z) * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[t$95$0, -1.618195973607049e+50], t$95$1, If[Less[t$95$0, 3.892237649663903e+134], N[(N[(N[(x * y), $MachinePrecision] * z), $MachinePrecision] - N[(N[(x * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Reproduce

herbie shell --seed 2024359 
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSV:hsv from colour-2.3.3, J"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< (* x (- 1 (* (- 1 y) z))) -161819597360704900000000000000000000000000000000000) (+ x (* (- 1 y) (* (- z) x))) (if (< (* x (- 1 (* (- 1 y) z))) 389223764966390300000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- (* (* x y) z) (- (* x z) x)) (+ x (* (- 1 y) (* (- z) x))))))

  (* x (- 1.0 (* (- 1.0 y) z))))