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

Percentage Accurate: 96.0% → 99.3%

Time: 4.5s

Alternatives: 12

Speedup: 0.8×

• 20.4% of points produce a very large (infinite) output. You may want to add a precondition.

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: 99.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.55 \cdot 10^{-12} \lor \neg \left(z \leq 2.3 \cdot 10^{-142}\right):\\ \;\;\;\;\mathsf{fma}\left(\left(y - 1\right) \cdot x, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot z, x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -2.55e-12) (not (<= z 2.3e-142)))
   (fma (* (- y 1.0) x) z x)
   (fma (* y z) x x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.55e-12) || !(z <= 2.3e-142)) {
		tmp = fma(((y - 1.0) * x), z, x);
	} else {
		tmp = fma((y * z), x, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -2.55e-12) || !(z <= 2.3e-142))
		tmp = fma(Float64(Float64(y - 1.0) * x), z, x);
	else
		tmp = fma(Float64(y * z), x, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -2.55e-12], N[Not[LessEqual[z, 2.3e-142]], $MachinePrecision]], N[(N[(N[(y - 1.0), $MachinePrecision] * x), $MachinePrecision] * z + x), $MachinePrecision], N[(N[(y * z), $MachinePrecision] * x + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.54999999999999984e-12 or 2.30000000000000002e-142 < z

   1. Initial program 92.4%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. flip-- N/A

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

   4. Applied rewrites 99.9%

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

   if -2.54999999999999984e-12 < z < 2.30000000000000002e-142

   1. Initial program 99.9%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. flip-- N/A

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

   4. Applied rewrites 91.7%

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

   5. Taylor expanded in y around inf

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

   7. Applied rewrites 91.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 99.9

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

   9. Applied rewrites 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.55 \cdot 10^{-12} \lor \neg \left(z \leq 2.3 \cdot 10^{-142}\right):\\ \;\;\;\;\mathsf{fma}\left(\left(y - 1\right) \cdot x, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot z, x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - y \leq -100 \lor \neg \left(1 - y \leq 1\right):\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-x, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= (- 1.0 y) -100.0) (not (<= (- 1.0 y) 1.0)))
   (fma (* y x) z x)
   (fma (- x) z x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (((1.0 - y) <= -100.0) || !((1.0 - y) <= 1.0)) {
		tmp = fma((y * x), z, x);
	} else {
		tmp = fma(-x, z, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((Float64(1.0 - y) <= -100.0) || !(Float64(1.0 - y) <= 1.0))
		tmp = fma(Float64(y * x), z, x);
	else
		tmp = fma(Float64(-x), z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[N[(1.0 - y), $MachinePrecision], -100.0], N[Not[LessEqual[N[(1.0 - y), $MachinePrecision], 1.0]], $MachinePrecision]], N[(N[(y * x), $MachinePrecision] * z + x), $MachinePrecision], N[((-x) * z + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) y) < -100 or 1 < (-.f64 #s(literal 1 binary64) y)

   1. Initial program 90.8%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. flip-- N/A

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

   4. Applied rewrites 92.8%

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

   5. Taylor expanded in y around inf

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

   7. Applied rewrites 92.7%

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

   if -100 < (-.f64 #s(literal 1 binary64) y) < 1

   1. Initial program 100.0%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. flip-- N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 98.9

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

   7. Applied rewrites 98.9%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - y \leq -100 \lor \neg \left(1 - y \leq 1\right):\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-x, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - y \leq -100 \lor \neg \left(1 - y \leq 1\right):\\ \;\;\;\;\mathsf{fma}\left(x \cdot z, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-x, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= (- 1.0 y) -100.0) (not (<= (- 1.0 y) 1.0)))
   (fma (* x z) y x)
   (fma (- x) z x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (((1.0 - y) <= -100.0) || !((1.0 - y) <= 1.0)) {
		tmp = fma((x * z), y, x);
	} else {
		tmp = fma(-x, z, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((Float64(1.0 - y) <= -100.0) || !(Float64(1.0 - y) <= 1.0))
		tmp = fma(Float64(x * z), y, x);
	else
		tmp = fma(Float64(-x), z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[N[(1.0 - y), $MachinePrecision], -100.0], N[Not[LessEqual[N[(1.0 - y), $MachinePrecision], 1.0]], $MachinePrecision]], N[(N[(x * z), $MachinePrecision] * y + x), $MachinePrecision], N[((-x) * z + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) y) < -100 or 1 < (-.f64 #s(literal 1 binary64) y)

   1. Initial program 90.8%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. flip-- N/A

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

   4. Applied rewrites 92.8%

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

   5. Taylor expanded in y around inf

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

   7. Applied rewrites 92.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 91.1

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

   9. Applied rewrites 91.1%

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

   if -100 < (-.f64 #s(literal 1 binary64) y) < 1

   1. Initial program 100.0%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. flip-- N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 98.9

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

   7. Applied rewrites 98.9%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - y \leq -100 \lor \neg \left(1 - y \leq 1\right):\\ \;\;\;\;\mathsf{fma}\left(x \cdot z, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-x, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - y \leq -100:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, z, x\right)\\ \mathbf{elif}\;1 - y \leq 1:\\ \;\;\;\;\mathsf{fma}\left(-x, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot z, x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (- 1.0 y) -100.0)
   (fma (* y x) z x)
   (if (<= (- 1.0 y) 1.0) (fma (- x) z x) (fma (* y z) x x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((1.0 - y) <= -100.0) {
		tmp = fma((y * x), z, x);
	} else if ((1.0 - y) <= 1.0) {
		tmp = fma(-x, z, x);
	} else {
		tmp = fma((y * z), x, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(1.0 - y) <= -100.0)
		tmp = fma(Float64(y * x), z, x);
	elseif (Float64(1.0 - y) <= 1.0)
		tmp = fma(Float64(-x), z, x);
	else
		tmp = fma(Float64(y * z), x, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(1.0 - y), $MachinePrecision], -100.0], N[(N[(y * x), $MachinePrecision] * z + x), $MachinePrecision], If[LessEqual[N[(1.0 - y), $MachinePrecision], 1.0], N[((-x) * z + x), $MachinePrecision], N[(N[(y * z), $MachinePrecision] * x + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 #s(literal 1 binary64) y) < -100

   1. Initial program 89.6%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. flip-- N/A

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

   4. Applied rewrites 94.4%

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

   5. Taylor expanded in y around inf

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

   7. Applied rewrites 94.1%

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

   if -100 < (-.f64 #s(literal 1 binary64) y) < 1

   1. Initial program 100.0%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. flip-- N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 98.9

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

   7. Applied rewrites 98.9%

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

   if 1 < (-.f64 #s(literal 1 binary64) y)

   1. Initial program 92.4%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. flip-- N/A

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

   4. Applied rewrites 90.7%

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

   5. Taylor expanded in y around inf

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

   7. Applied rewrites 90.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 92.4

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

   9. Applied rewrites 92.4%

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

Alternative 5: 83.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - y \leq -100 \lor \neg \left(1 - y \leq 2 \cdot 10^{+134}\right):\\ \;\;\;\;\left(x \cdot y\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-x, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= (- 1.0 y) -100.0) (not (<= (- 1.0 y) 2e+134)))
   (* (* x y) z)
   (fma (- x) z x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (((1.0 - y) <= -100.0) || !((1.0 - y) <= 2e+134)) {
		tmp = (x * y) * z;
	} else {
		tmp = fma(-x, z, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((Float64(1.0 - y) <= -100.0) || !(Float64(1.0 - y) <= 2e+134))
		tmp = Float64(Float64(x * y) * z);
	else
		tmp = fma(Float64(-x), z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[N[(1.0 - y), $MachinePrecision], -100.0], N[Not[LessEqual[N[(1.0 - y), $MachinePrecision], 2e+134]], $MachinePrecision]], N[(N[(x * y), $MachinePrecision] * z), $MachinePrecision], N[((-x) * z + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) y) < -100 or 1.99999999999999984e134 < (-.f64 #s(literal 1 binary64) y)

   1. Initial program 88.9%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. flip-- N/A

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

   4. Applied rewrites 92.4%

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

   5. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift--.f64 68.3

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

   7. Applied rewrites 68.3%

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

      1. associate-*l* 68.3

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

      2. *-commutative 68.3

         \[\leadsto \left(z \cdot x\right) \cdot \left(y - 1\right) \]

      3. associate-*l* 68.3

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

          \[\leadsto \left(\left(y - 1\right) \cdot x\right) \cdot z \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\left(y - 1\right) \cdot x\right) \cdot z \]

      12. lift--.f64 75.5

          \[\leadsto \left(\left(y - 1\right) \cdot x\right) \cdot z \]

   9. Applied rewrites 75.5%

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

   10. Taylor expanded in y around inf

       \[\leadsto \left(x \cdot y\right) \cdot z \]
   11. Step-by-step derivation

       1. lower-*.f64 75.3

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

   12. Applied rewrites 75.3%

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

   if -100 < (-.f64 #s(literal 1 binary64) y) < 1.99999999999999984e134

   1. Initial program 100.0%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. flip-- N/A

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 94.9

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

   7. Applied rewrites 94.9%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - y \leq -100 \lor \neg \left(1 - y \leq 2 \cdot 10^{+134}\right):\\ \;\;\;\;\left(x \cdot y\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-x, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -540000 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;\left(\left(y - 1\right) \cdot x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot z, x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -540000.0) (not (<= z 1.0)))
   (* (* (- y 1.0) x) z)
   (fma (* y z) x x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -540000.0) || !(z <= 1.0)) {
		tmp = ((y - 1.0) * x) * z;
	} else {
		tmp = fma((y * z), x, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -540000.0) || !(z <= 1.0))
		tmp = Float64(Float64(Float64(y - 1.0) * x) * z);
	else
		tmp = fma(Float64(y * z), x, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -540000.0], N[Not[LessEqual[z, 1.0]], $MachinePrecision]], N[(N[(N[(y - 1.0), $MachinePrecision] * x), $MachinePrecision] * z), $MachinePrecision], N[(N[(y * z), $MachinePrecision] * x + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.4e5 or 1 < z

   1. Initial program 90.0%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. flip-- N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift--.f64 99.7

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

   7. Applied rewrites 99.7%

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

      1. associate-*l* 99.7

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

      2. *-commutative 99.7

         \[\leadsto \left(z \cdot x\right) \cdot \left(y - 1\right) \]

      3. associate-*l* 99.7

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

          \[\leadsto \left(\left(y - 1\right) \cdot x\right) \cdot z \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\left(y - 1\right) \cdot x\right) \cdot z \]

      12. lift--.f64 99.8

          \[\leadsto \left(\left(y - 1\right) \cdot x\right) \cdot z \]

   9. Applied rewrites 99.8%

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

   if -5.4e5 < z < 1

   1. Initial program 99.9%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. flip-- N/A

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

   4. Applied rewrites 93.8%

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

   5. Taylor expanded in y around inf

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

   7. Applied rewrites 92.3%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 98.4

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

   9. Applied rewrites 98.4%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -540000 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;\left(\left(y - 1\right) \cdot x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot z, x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -540000 \lor \neg \left(z \leq 1.72\right):\\ \;\;\;\;\left(z \cdot x\right) \cdot \left(y - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot z, x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -540000.0) (not (<= z 1.72)))
   (* (* z x) (- y 1.0))
   (fma (* y z) x x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -540000.0) || !(z <= 1.72)) {
		tmp = (z * x) * (y - 1.0);
	} else {
		tmp = fma((y * z), x, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -540000.0) || !(z <= 1.72))
		tmp = Float64(Float64(z * x) * Float64(y - 1.0));
	else
		tmp = fma(Float64(y * z), x, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -540000.0], N[Not[LessEqual[z, 1.72]], $MachinePrecision]], N[(N[(z * x), $MachinePrecision] * N[(y - 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(y * z), $MachinePrecision] * x + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.4e5 or 1.71999999999999997 < z

   1. Initial program 89.9%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. flip-- N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift--.f64 99.7

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

   7. Applied rewrites 99.7%

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

   if -5.4e5 < z < 1.71999999999999997

   1. Initial program 99.9%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. flip-- N/A

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

   4. Applied rewrites 93.9%

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

   5. Taylor expanded in y around inf

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

   7. Applied rewrites 92.4%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 98.4

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

   9. Applied rewrites 98.4%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -540000 \lor \neg \left(z \leq 1.72\right):\\ \;\;\;\;\left(z \cdot x\right) \cdot \left(y - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot z, x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.8 \cdot 10^{-80}:\\ \;\;\;\;\mathsf{fma}\left(\left(y - 1\right) \cdot x, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(y - 1\right) \cdot z, x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 4.8e-80) (fma (* (- y 1.0) x) z x) (fma (* (- y 1.0) z) x x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 4.8e-80) {
		tmp = fma(((y - 1.0) * x), z, x);
	} else {
		tmp = fma(((y - 1.0) * z), x, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 4.8e-80)
		tmp = fma(Float64(Float64(y - 1.0) * x), z, x);
	else
		tmp = fma(Float64(Float64(y - 1.0) * z), x, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 4.8e-80], N[(N[(N[(y - 1.0), $MachinePrecision] * x), $MachinePrecision] * z + x), $MachinePrecision], N[(N[(N[(y - 1.0), $MachinePrecision] * z), $MachinePrecision] * x + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.7999999999999998e-80

   1. Initial program 93.8%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. flip-- N/A

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

   4. Applied rewrites 96.7%

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

   if 4.7999999999999998e-80 < x

   1. Initial program 99.9%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. flip-- N/A

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

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(y - 1\right) \cdot z, x, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 65.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -540000 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;x \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -540000.0) (not (<= z 1.0))) (* x (- z)) x))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -540000.0) || !(z <= 1.0)) {
		tmp = x * -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-540000.0d0)) .or. (.not. (z <= 1.0d0))) then
        tmp = x * -z
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -540000.0) || !(z <= 1.0)) {
		tmp = x * -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -540000.0) or not (z <= 1.0):
		tmp = x * -z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -540000.0) || !(z <= 1.0))
		tmp = Float64(x * Float64(-z));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -540000.0) || ~((z <= 1.0)))
		tmp = x * -z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -540000.0], N[Not[LessEqual[z, 1.0]], $MachinePrecision]], N[(x * (-z)), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -5.4e5 or 1 < z

   1. Initial program 90.0%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 89.9

         \[\leadsto x \cdot \left(\left(y - 1\right) \cdot z\right) \]

   5. Applied rewrites 89.9%

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

   6. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

         \[\leadsto x \cdot \left(\mathsf{neg}\left(z\right)\right) \]

      2. lower-neg.f64 57.4

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

   8. Applied rewrites 57.4%

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

   if -5.4e5 < z < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 70.5%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -540000 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;x \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 66.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 95.5%

   \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. flip-- N/A

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

4. Applied rewrites 96.5%

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

5. Taylor expanded in y around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 65.6

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

7. Applied rewrites 65.6%

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

Alternative 11: 66.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.5%

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

3. Taylor expanded in y around 0

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

5. Applied rewrites 65.6%

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

Alternative 12: 38.8% accurate, 17.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 95.5%

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

3. Taylor expanded in z around 0

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

5. Applied rewrites 40.9%

   \[\leadsto \color{blue}{x} \]
6. 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 2025051 
(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))))