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

Percentage Accurate: 95.7% → 98.7%

Time: 7.4s

Alternatives: 7

Speedup: 0.7×

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x * (1.0d0 - (y * z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 95.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x * (1.0d0 - (y * z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \cdot z \leq -5 \cdot 10^{+61}:\\ \;\;\;\;\left(\left(-x\right) \cdot z\right) \cdot y\\ \mathbf{elif}\;y \cdot z \leq 5 \cdot 10^{+154}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(-z, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(-x\right) \cdot y, z, x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* y z) -5e+61)
   (* (* (- x) z) y)
   (if (<= (* y z) 5e+154) (* x (fma (- z) y 1.0)) (fma (* (- x) y) z x))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if ((y * z) <= -5e+61) {
		tmp = (-x * z) * y;
	} else if ((y * z) <= 5e+154) {
		tmp = x * fma(-z, y, 1.0);
	} else {
		tmp = fma((-x * y), z, x);
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (Float64(y * z) <= -5e+61)
		tmp = Float64(Float64(Float64(-x) * z) * y);
	elseif (Float64(y * z) <= 5e+154)
		tmp = Float64(x * fma(Float64(-z), y, 1.0));
	else
		tmp = fma(Float64(Float64(-x) * y), z, x);
	end
	return tmp
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[N[(y * z), $MachinePrecision], -5e+61], N[(N[((-x) * z), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[N[(y * z), $MachinePrecision], 5e+154], N[(x * N[((-z) * y + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[((-x) * y), $MachinePrecision] * z + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 y z) < -5.00000000000000018e61

   1. Initial program 89.5%

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

   4. Applied rewrites 28.2%

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

   5. Taylor expanded in y around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z\right)} \]
   6. 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 0.7

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

   7. Applied rewrites 0.7%

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

   8. Taylor expanded in z around -inf

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

      1. mul-1-neg N/A

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

      2. associate-*r* N/A

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

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

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. rem-square-sqrt N/A

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

      7. mul-1-neg N/A

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

      8. remove-double-neg N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 0.6

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

   10. Applied rewrites 0.6%

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

   11. Taylor expanded in y around -inf

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

       1. mul-1-neg N/A

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

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

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

       3. *-commutative N/A

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

       4. associate-*r* N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-neg.f64 96.3

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

   13. Applied rewrites 96.3%

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

   if -5.00000000000000018e61 < (*.f64 y z) < 5.00000000000000004e154

   1. Initial program 99.9%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

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

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

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

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

      8. lower-fma.f64 N/A

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

      9. lower-neg.f64 99.9

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

   4. Applied rewrites 99.9%

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

   if 5.00000000000000004e154 < (*.f64 y z)

   1. Initial program 76.3%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. associate-*r* N/A

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

      8. *-rgt-identity N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-neg.f64 99.9

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

   4. Applied rewrites 99.9%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot z \leq -5 \cdot 10^{+61}:\\ \;\;\;\;\left(\left(-x\right) \cdot z\right) \cdot y\\ \mathbf{elif}\;y \cdot z \leq 5 \cdot 10^{+154}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(-z, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(-x\right) \cdot y, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} t_0 := \left(\left(-x\right) \cdot z\right) \cdot y\\ \mathbf{if}\;y \cdot z \leq -2:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \cdot z \leq 1:\\ \;\;\;\;x \cdot 1\\ \mathbf{elif}\;y \cdot z \leq 2 \cdot 10^{+200}:\\ \;\;\;\;x \cdot \left(\left(-y\right) \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (* (- x) z) y)))
   (if (<= (* y z) -2.0)
     t_0
     (if (<= (* y z) 1.0)
       (* x 1.0)
       (if (<= (* y z) 2e+200) (* x (* (- y) z)) t_0)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double t_0 = (-x * z) * y;
	double tmp;
	if ((y * z) <= -2.0) {
		tmp = t_0;
	} else if ((y * z) <= 1.0) {
		tmp = x * 1.0;
	} else if ((y * z) <= 2e+200) {
		tmp = x * (-y * z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (-x * z) * y
    if ((y * z) <= (-2.0d0)) then
        tmp = t_0
    else if ((y * z) <= 1.0d0) then
        tmp = x * 1.0d0
    else if ((y * z) <= 2d+200) then
        tmp = x * (-y * z)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double t_0 = (-x * z) * y;
	double tmp;
	if ((y * z) <= -2.0) {
		tmp = t_0;
	} else if ((y * z) <= 1.0) {
		tmp = x * 1.0;
	} else if ((y * z) <= 2e+200) {
		tmp = x * (-y * z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	t_0 = (-x * z) * y
	tmp = 0
	if (y * z) <= -2.0:
		tmp = t_0
	elif (y * z) <= 1.0:
		tmp = x * 1.0
	elif (y * z) <= 2e+200:
		tmp = x * (-y * z)
	else:
		tmp = t_0
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	t_0 = Float64(Float64(Float64(-x) * z) * y)
	tmp = 0.0
	if (Float64(y * z) <= -2.0)
		tmp = t_0;
	elseif (Float64(y * z) <= 1.0)
		tmp = Float64(x * 1.0);
	elseif (Float64(y * z) <= 2e+200)
		tmp = Float64(x * Float64(Float64(-y) * z));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	t_0 = (-x * z) * y;
	tmp = 0.0;
	if ((y * z) <= -2.0)
		tmp = t_0;
	elseif ((y * z) <= 1.0)
		tmp = x * 1.0;
	elseif ((y * z) <= 2e+200)
		tmp = x * (-y * z);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := Block[{t$95$0 = N[(N[((-x) * z), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[N[(y * z), $MachinePrecision], -2.0], t$95$0, If[LessEqual[N[(y * z), $MachinePrecision], 1.0], N[(x * 1.0), $MachinePrecision], If[LessEqual[N[(y * z), $MachinePrecision], 2e+200], N[(x * N[((-y) * z), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 y z) < -2 or 1.9999999999999999e200 < (*.f64 y z)

   1. Initial program 87.3%

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

   4. Applied rewrites 24.0%

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

   5. Taylor expanded in y around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z\right)} \]
   6. 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 0.8

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

   7. Applied rewrites 0.8%

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

   8. Taylor expanded in z around -inf

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

      1. mul-1-neg N/A

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

      2. associate-*r* N/A

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

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

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. rem-square-sqrt N/A

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

      7. mul-1-neg N/A

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

      8. remove-double-neg N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 0.8

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

   10. Applied rewrites 0.8%

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

   11. Taylor expanded in y around -inf

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

       1. mul-1-neg N/A

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

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

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

       3. *-commutative N/A

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

       4. associate-*r* N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-neg.f64 94.0

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

   13. Applied rewrites 94.0%

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

   if -2 < (*.f64 y z) < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 97.4%

      \[\leadsto x \cdot \color{blue}{1} \]

   if 1 < (*.f64 y z) < 1.9999999999999999e200

   1. Initial program 99.8%

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

   3. Taylor expanded in y around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 98.9

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

   5. Applied rewrites 98.9%

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

Alternative 3: 98.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \cdot z \leq -5 \cdot 10^{+61} \lor \neg \left(y \cdot z \leq 2 \cdot 10^{+200}\right):\\ \;\;\;\;\left(\left(-x\right) \cdot z\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(-z, y, 1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (or (<= (* y z) -5e+61) (not (<= (* y z) 2e+200)))
   (* (* (- x) z) y)
   (* x (fma (- z) y 1.0))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (((y * z) <= -5e+61) || !((y * z) <= 2e+200)) {
		tmp = (-x * z) * y;
	} else {
		tmp = x * fma(-z, y, 1.0);
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if ((Float64(y * z) <= -5e+61) || !(Float64(y * z) <= 2e+200))
		tmp = Float64(Float64(Float64(-x) * z) * y);
	else
		tmp = Float64(x * fma(Float64(-z), y, 1.0));
	end
	return tmp
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[Or[LessEqual[N[(y * z), $MachinePrecision], -5e+61], N[Not[LessEqual[N[(y * z), $MachinePrecision], 2e+200]], $MachinePrecision]], N[(N[((-x) * z), $MachinePrecision] * y), $MachinePrecision], N[(x * N[((-z) * y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y z) < -5.00000000000000018e61 or 1.9999999999999999e200 < (*.f64 y z)

   1. Initial program 84.3%

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

   4. Applied rewrites 19.6%

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

   5. Taylor expanded in y around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z\right)} \]
   6. 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 0.5

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

   7. Applied rewrites 0.5%

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

   8. Taylor expanded in z around -inf

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

      1. mul-1-neg N/A

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

      2. associate-*r* N/A

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

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

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. rem-square-sqrt N/A

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

      7. mul-1-neg N/A

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

      8. remove-double-neg N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 0.5

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

   10. Applied rewrites 0.5%

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

   11. Taylor expanded in y around -inf

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

       1. mul-1-neg N/A

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

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

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

       3. *-commutative N/A

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

       4. associate-*r* N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-neg.f64 97.4

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

   13. Applied rewrites 97.4%

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

   if -5.00000000000000018e61 < (*.f64 y z) < 1.9999999999999999e200

   1. Initial program 99.9%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

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

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

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

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

      8. lower-fma.f64 N/A

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

      9. lower-neg.f64 99.9

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

   4. Applied rewrites 99.9%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot z \leq -5 \cdot 10^{+61} \lor \neg \left(y \cdot z \leq 2 \cdot 10^{+200}\right):\\ \;\;\;\;\left(\left(-x\right) \cdot z\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(-z, y, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \cdot z \leq -5 \cdot 10^{+61} \lor \neg \left(y \cdot z \leq 2 \cdot 10^{+200}\right):\\ \;\;\;\;\left(\left(-x\right) \cdot z\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - y \cdot z\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (or (<= (* y z) -5e+61) (not (<= (* y z) 2e+200)))
   (* (* (- x) z) y)
   (* x (- 1.0 (* y z)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (((y * z) <= -5e+61) || !((y * z) <= 2e+200)) {
		tmp = (-x * z) * y;
	} else {
		tmp = x * (1.0 - (y * z));
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (((y * z) <= (-5d+61)) .or. (.not. ((y * z) <= 2d+200))) then
        tmp = (-x * z) * y
    else
        tmp = x * (1.0d0 - (y * z))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (((y * z) <= -5e+61) || !((y * z) <= 2e+200)) {
		tmp = (-x * z) * y;
	} else {
		tmp = x * (1.0 - (y * z));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if ((y * z) <= -5e+61) or not ((y * z) <= 2e+200):
		tmp = (-x * z) * y
	else:
		tmp = x * (1.0 - (y * z))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if ((Float64(y * z) <= -5e+61) || !(Float64(y * z) <= 2e+200))
		tmp = Float64(Float64(Float64(-x) * z) * y);
	else
		tmp = Float64(x * Float64(1.0 - Float64(y * z)));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (((y * z) <= -5e+61) || ~(((y * z) <= 2e+200)))
		tmp = (-x * z) * y;
	else
		tmp = x * (1.0 - (y * z));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[Or[LessEqual[N[(y * z), $MachinePrecision], -5e+61], N[Not[LessEqual[N[(y * z), $MachinePrecision], 2e+200]], $MachinePrecision]], N[(N[((-x) * z), $MachinePrecision] * y), $MachinePrecision], N[(x * N[(1.0 - N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y z) < -5.00000000000000018e61 or 1.9999999999999999e200 < (*.f64 y z)

   1. Initial program 84.3%

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

   4. Applied rewrites 19.6%

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

   5. Taylor expanded in y around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z\right)} \]
   6. 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 0.5

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

   7. Applied rewrites 0.5%

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

   8. Taylor expanded in z around -inf

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

      1. mul-1-neg N/A

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

      2. associate-*r* N/A

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

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

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. rem-square-sqrt N/A

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

      7. mul-1-neg N/A

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

      8. remove-double-neg N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 0.5

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

   10. Applied rewrites 0.5%

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

   11. Taylor expanded in y around -inf

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

       1. mul-1-neg N/A

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

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

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

       3. *-commutative N/A

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

       4. associate-*r* N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-neg.f64 97.4

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

   13. Applied rewrites 97.4%

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

   if -5.00000000000000018e61 < (*.f64 y z) < 1.9999999999999999e200

   1. Initial program 99.9%

      \[x \cdot \left(1 - y \cdot z\right) \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot z \leq -5 \cdot 10^{+61} \lor \neg \left(y \cdot z \leq 2 \cdot 10^{+200}\right):\\ \;\;\;\;\left(\left(-x\right) \cdot z\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - y \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \cdot z \leq -2 \lor \neg \left(y \cdot z \leq 1\right):\\ \;\;\;\;\left(\left(-x\right) \cdot z\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (or (<= (* y z) -2.0) (not (<= (* y z) 1.0)))
   (* (* (- x) z) y)
   (* x 1.0)))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (((y * z) <= -2.0) || !((y * z) <= 1.0)) {
		tmp = (-x * z) * y;
	} else {
		tmp = x * 1.0;
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (((y * z) <= (-2.0d0)) .or. (.not. ((y * z) <= 1.0d0))) then
        tmp = (-x * z) * y
    else
        tmp = x * 1.0d0
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (((y * z) <= -2.0) || !((y * z) <= 1.0)) {
		tmp = (-x * z) * y;
	} else {
		tmp = x * 1.0;
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if ((y * z) <= -2.0) or not ((y * z) <= 1.0):
		tmp = (-x * z) * y
	else:
		tmp = x * 1.0
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if ((Float64(y * z) <= -2.0) || !(Float64(y * z) <= 1.0))
		tmp = Float64(Float64(Float64(-x) * z) * y);
	else
		tmp = Float64(x * 1.0);
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (((y * z) <= -2.0) || ~(((y * z) <= 1.0)))
		tmp = (-x * z) * y;
	else
		tmp = x * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[Or[LessEqual[N[(y * z), $MachinePrecision], -2.0], N[Not[LessEqual[N[(y * z), $MachinePrecision], 1.0]], $MachinePrecision]], N[(N[((-x) * z), $MachinePrecision] * y), $MachinePrecision], N[(x * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y z) < -2 or 1 < (*.f64 y z)

   1. Initial program 90.1%

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

   4. Applied rewrites 18.8%

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

   5. Taylor expanded in y around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z\right)} \]
   6. 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 1.0

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

   7. Applied rewrites 1.0%

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

   8. Taylor expanded in z around -inf

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

      1. mul-1-neg N/A

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

      2. associate-*r* N/A

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

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

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. rem-square-sqrt N/A

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

      7. mul-1-neg N/A

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

      8. remove-double-neg N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 1.1

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

   10. Applied rewrites 1.1%

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

   11. Taylor expanded in y around -inf

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

       1. mul-1-neg N/A

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

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

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

       3. *-commutative N/A

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

       4. associate-*r* N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-neg.f64 92.7

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

   13. Applied rewrites 92.7%

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

   if -2 < (*.f64 y z) < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 97.4%

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

4. Final simplification 95.1%

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

Alternative 6: 95.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 3 \cdot 10^{-26}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot z, -y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(-z, y, 1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= x 3e-26) (fma (* x z) (- y) x) (* x (fma (- z) y 1.0))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (x <= 3e-26) {
		tmp = fma((x * z), -y, x);
	} else {
		tmp = x * fma(-z, y, 1.0);
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (x <= 3e-26)
		tmp = fma(Float64(x * z), Float64(-y), x);
	else
		tmp = Float64(x * fma(Float64(-z), y, 1.0));
	end
	return tmp
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[x, 3e-26], N[(N[(x * z), $MachinePrecision] * (-y) + x), $MachinePrecision], N[(x * N[((-z) * y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.00000000000000012e-26

   1. Initial program 93.3%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-rgt-identity N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 94.5

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

   4. Applied rewrites 94.5%

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

   if 3.00000000000000012e-26 < x

   1. Initial program 99.9%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

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

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

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

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

      8. lower-fma.f64 N/A

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

      9. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

Alternative 7: 50.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ x \cdot 1 \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z) :precision binary64 (* x 1.0))

C

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

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x * 1.0d0
end function

Java

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

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	return x * 1.0

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	return Float64(x * 1.0)
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp = code(x, y, z)
	tmp = x * 1.0;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := N[(x * 1.0), $MachinePrecision]

Derivation

1. Initial program 95.1%

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

3. Taylor expanded in y around 0

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

5. Applied rewrites 51.5%

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

Reproduce

herbie shell --seed 2024329 
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSV:hsv from colour-2.3.3, I"
  :precision binary64
  (* x (- 1.0 (* y z))))