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

Percentage Accurate: 96.1% → 97.2%

Time: 10.7s

Alternatives: 15

Speedup: 0.5×

• 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

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 - ((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: 96.1% 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

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 - ((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: 97.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - y \leq -15000000000:\\ \;\;\;\;\left(x \cdot z\right) \cdot \left(y + -2\right) + x \cdot \left(1 + z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \left(y + -1\right), x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (- 1.0 y) -15000000000.0)
   (+ (* (* x z) (+ y -2.0)) (* x (+ 1.0 z)))
   (fma (* z (+ y -1.0)) x x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((1.0 - y) <= -15000000000.0) {
		tmp = ((x * z) * (y + -2.0)) + (x * (1.0 + z));
	} else {
		tmp = fma((z * (y + -1.0)), x, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(1.0 - y) <= -15000000000.0)
		tmp = Float64(Float64(Float64(x * z) * Float64(y + -2.0)) + Float64(x * Float64(1.0 + z)));
	else
		tmp = fma(Float64(z * Float64(y + -1.0)), x, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(1.0 - y), $MachinePrecision], -15000000000.0], N[(N[(N[(x * z), $MachinePrecision] * N[(y + -2.0), $MachinePrecision]), $MachinePrecision] + N[(x * N[(1.0 + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z * N[(y + -1.0), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 1 y) < -1.5e10

   1. Initial program 87.9%

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

      1. *-commutative 87.9%

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

      2. flip-- 60.8%

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

      3. associate-*l/ 60.7%

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

      4. metadata-eval 60.7%

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

      5. pow2 60.7%

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

      6. +-commutative 60.7%

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

      7. fma-define 60.7%

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

   4. Applied egg-rr 60.7%

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

   5. Taylor expanded in y around inf 72.0%

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

      1. sub-neg 72.0%

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

      2. mul-1-neg 72.0%

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

      3. remove-double-neg 72.0%

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

      4. +-commutative 72.0%

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

      5. *-commutative 72.0%

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

      6. associate-*l* 82.5%

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

      7. *-commutative 82.5%

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

      8. distribute-lft-out 98.3%

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

   7. Simplified 98.3%

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

   if -1.5e10 < (-.f64 1 y)

   1. Initial program 98.9%

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

   3. Taylor expanded in z around 0 98.9%

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

      1. +-commutative 98.9%

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

      2. *-commutative 98.9%

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

      3. fma-define 99.0%

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

      4. sub-neg 99.0%

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

      5. metadata-eval 99.0%

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

   5. Applied egg-rr 99.0%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - y \leq -15000000000:\\ \;\;\;\;\left(x \cdot z\right) \cdot \left(y + -2\right) + x \cdot \left(1 + z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \left(y + -1\right), x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 64.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(-z\right)\\ \mathbf{if}\;z \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{-59}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{+103}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (- z))))
   (if (<= z -1.0)
     t_0
     (if (<= z 1.25e-59) x (if (<= z 3.2e+103) (* x (* y z)) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = x * -z;
	double tmp;
	if (z <= -1.0) {
		tmp = t_0;
	} else if (z <= 1.25e-59) {
		tmp = x;
	} else if (z <= 3.2e+103) {
		tmp = x * (y * z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

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
    if (z <= (-1.0d0)) then
        tmp = t_0
    else if (z <= 1.25d-59) then
        tmp = x
    else if (z <= 3.2d+103) then
        tmp = x * (y * z)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x * -z;
	double tmp;
	if (z <= -1.0) {
		tmp = t_0;
	} else if (z <= 1.25e-59) {
		tmp = x;
	} else if (z <= 3.2e+103) {
		tmp = x * (y * z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * -z
	tmp = 0
	if z <= -1.0:
		tmp = t_0
	elif z <= 1.25e-59:
		tmp = x
	elif z <= 3.2e+103:
		tmp = x * (y * z)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(-z))
	tmp = 0.0
	if (z <= -1.0)
		tmp = t_0;
	elseif (z <= 1.25e-59)
		tmp = x;
	elseif (z <= 3.2e+103)
		tmp = Float64(x * Float64(y * z));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * -z;
	tmp = 0.0;
	if (z <= -1.0)
		tmp = t_0;
	elseif (z <= 1.25e-59)
		tmp = x;
	elseif (z <= 3.2e+103)
		tmp = x * (y * z);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * (-z)), $MachinePrecision]}, If[LessEqual[z, -1.0], t$95$0, If[LessEqual[z, 1.25e-59], x, If[LessEqual[z, 3.2e+103], N[(x * N[(y * z), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1 or 3.19999999999999993e103 < z

   1. Initial program 92.3%

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

   3. Taylor expanded in z around inf 90.2%

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

   4. Taylor expanded in y around 0 57.6%

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

      1. mul-1-neg 57.6%

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

      2. distribute-rgt-neg-in 57.6%

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

   6. Simplified 57.6%

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

   if -1 < z < 1.25e-59

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0 82.8%

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

   if 1.25e-59 < z < 3.19999999999999993e103

   1. Initial program 99.8%

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

   3. Taylor expanded in y around inf 59.3%

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

      1. *-commutative 59.3%

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

   5. Simplified 59.3%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;x \cdot \left(-z\right)\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{-59}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{+103}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - y \leq -15000000000:\\ \;\;\;\;\left(x \cdot z\right) \cdot \left(y + -2\right) + x \cdot \left(1 + z\right)\\ \mathbf{else}:\\ \;\;\;\;x + x \cdot \left(z \cdot \left(y + -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (- 1.0 y) -15000000000.0)
   (+ (* (* x z) (+ y -2.0)) (* x (+ 1.0 z)))
   (+ x (* x (* z (+ y -1.0))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((1.0 - y) <= -15000000000.0) {
		tmp = ((x * z) * (y + -2.0)) + (x * (1.0 + z));
	} else {
		tmp = x + (x * (z * (y + -1.0)));
	}
	return tmp;
}

Fortran

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 ((1.0d0 - y) <= (-15000000000.0d0)) then
        tmp = ((x * z) * (y + (-2.0d0))) + (x * (1.0d0 + z))
    else
        tmp = x + (x * (z * (y + (-1.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((1.0 - y) <= -15000000000.0) {
		tmp = ((x * z) * (y + -2.0)) + (x * (1.0 + z));
	} else {
		tmp = x + (x * (z * (y + -1.0)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (1.0 - y) <= -15000000000.0:
		tmp = ((x * z) * (y + -2.0)) + (x * (1.0 + z))
	else:
		tmp = x + (x * (z * (y + -1.0)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(1.0 - y) <= -15000000000.0)
		tmp = Float64(Float64(Float64(x * z) * Float64(y + -2.0)) + Float64(x * Float64(1.0 + z)));
	else
		tmp = Float64(x + Float64(x * Float64(z * Float64(y + -1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((1.0 - y) <= -15000000000.0)
		tmp = ((x * z) * (y + -2.0)) + (x * (1.0 + z));
	else
		tmp = x + (x * (z * (y + -1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(1.0 - y), $MachinePrecision], -15000000000.0], N[(N[(N[(x * z), $MachinePrecision] * N[(y + -2.0), $MachinePrecision]), $MachinePrecision] + N[(x * N[(1.0 + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(x * N[(z * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 1 y) < -1.5e10

   1. Initial program 87.9%

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

      1. *-commutative 87.9%

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

      2. flip-- 60.8%

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

      3. associate-*l/ 60.7%

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

      4. metadata-eval 60.7%

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

      5. pow2 60.7%

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

      6. +-commutative 60.7%

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

      7. fma-define 60.7%

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

   4. Applied egg-rr 60.7%

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

   5. Taylor expanded in y around inf 72.0%

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

      1. sub-neg 72.0%

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

      2. mul-1-neg 72.0%

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

      3. remove-double-neg 72.0%

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

      4. +-commutative 72.0%

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

      5. *-commutative 72.0%

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

      6. associate-*l* 82.5%

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

      7. *-commutative 82.5%

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

      8. distribute-lft-out 98.3%

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

   7. Simplified 98.3%

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

   if -1.5e10 < (-.f64 1 y)

   1. Initial program 98.9%

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

   3. Taylor expanded in z around 0 98.9%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - y \leq -15000000000:\\ \;\;\;\;\left(x \cdot z\right) \cdot \left(y + -2\right) + x \cdot \left(1 + z\right)\\ \mathbf{else}:\\ \;\;\;\;x + x \cdot \left(z \cdot \left(y + -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(1 - y\right) \cdot z \leq -5 \cdot 10^{+73}:\\ \;\;\;\;\left(x \cdot z\right) \cdot \left(y + -1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + z \cdot \left(y + -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* (- 1.0 y) z) -5e+73)
   (* (* x z) (+ y -1.0))
   (* x (+ 1.0 (* z (+ y -1.0))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (((1.0 - y) * z) <= -5e+73) {
		tmp = (x * z) * (y + -1.0);
	} else {
		tmp = x * (1.0 + (z * (y + -1.0)));
	}
	return tmp;
}

Fortran

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 (((1.0d0 - y) * z) <= (-5d+73)) then
        tmp = (x * z) * (y + (-1.0d0))
    else
        tmp = x * (1.0d0 + (z * (y + (-1.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (((1.0 - y) * z) <= -5e+73) {
		tmp = (x * z) * (y + -1.0);
	} else {
		tmp = x * (1.0 + (z * (y + -1.0)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if ((1.0 - y) * z) <= -5e+73:
		tmp = (x * z) * (y + -1.0)
	else:
		tmp = x * (1.0 + (z * (y + -1.0)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(1.0 - y) * z) <= -5e+73)
		tmp = Float64(Float64(x * z) * Float64(y + -1.0));
	else
		tmp = Float64(x * Float64(1.0 + Float64(z * Float64(y + -1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (((1.0 - y) * z) <= -5e+73)
		tmp = (x * z) * (y + -1.0);
	else
		tmp = x * (1.0 + (z * (y + -1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(1.0 - y), $MachinePrecision] * z), $MachinePrecision], -5e+73], N[(N[(x * z), $MachinePrecision] * N[(y + -1.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(1.0 + N[(z * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 1 y) z) < -4.99999999999999976e73

   1. Initial program 91.2%

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

   3. Taylor expanded in z around 0 91.2%

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

      1. +-commutative 91.2%

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

      2. *-commutative 91.2%

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

      3. fma-define 91.2%

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

      4. sub-neg 91.2%

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

      5. metadata-eval 91.2%

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

   5. Applied egg-rr 91.2%

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

   6. Taylor expanded in z around inf 91.2%

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

      1. *-commutative 91.2%

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

      2. sub-neg 91.2%

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

      3. metadata-eval 91.2%

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

      4. associate-*r* 98.4%

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

      5. *-commutative 98.4%

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

      6. associate-*l* 99.9%

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

   8. Simplified 99.9%

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

   if -4.99999999999999976e73 < (*.f64 (-.f64 1 y) z)

   1. Initial program 97.9%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - y\right) \cdot z \leq -5 \cdot 10^{+73}:\\ \;\;\;\;\left(x \cdot z\right) \cdot \left(y + -1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + z \cdot \left(y + -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -170000 \lor \neg \left(z \leq 3.8 \cdot 10^{-60}\right):\\ \;\;\;\;x \cdot \left(z \cdot \left(y + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x - x \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -170000.0) (not (<= z 3.8e-60)))
   (* x (* z (+ y -1.0)))
   (- x (* x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -170000.0) || !(z <= 3.8e-60)) {
		tmp = x * (z * (y + -1.0));
	} else {
		tmp = x - (x * z);
	}
	return tmp;
}

Fortran

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 ((z <= (-170000.0d0)) .or. (.not. (z <= 3.8d-60))) then
        tmp = x * (z * (y + (-1.0d0)))
    else
        tmp = x - (x * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -170000.0) || !(z <= 3.8e-60)) {
		tmp = x * (z * (y + -1.0));
	} else {
		tmp = x - (x * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -170000.0) or not (z <= 3.8e-60):
		tmp = x * (z * (y + -1.0))
	else:
		tmp = x - (x * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -170000.0) || !(z <= 3.8e-60))
		tmp = Float64(x * Float64(z * Float64(y + -1.0)));
	else
		tmp = Float64(x - Float64(x * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -170000.0) || ~((z <= 3.8e-60)))
		tmp = x * (z * (y + -1.0));
	else
		tmp = x - (x * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -170000.0], N[Not[LessEqual[z, 3.8e-60]], $MachinePrecision]], N[(x * N[(z * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(x * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.7e5 or 3.79999999999999994e-60 < z

   1. Initial program 93.6%

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

   3. Taylor expanded in z around inf 89.6%

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

   if -1.7e5 < z < 3.79999999999999994e-60

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0 100.0%

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

   4. Taylor expanded in y around 0 82.6%

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

      1. mul-1-neg 5.0%

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

      2. distribute-rgt-neg-in 5.0%

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

   6. Simplified 82.6%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -170000 \lor \neg \left(z \leq 3.8 \cdot 10^{-60}\right):\\ \;\;\;\;x \cdot \left(z \cdot \left(y + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x - x \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 87.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -170000 \lor \neg \left(z \leq 0.00045\right):\\ \;\;\;\;z \cdot \left(x \cdot \left(y + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x - x \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -170000.0) (not (<= z 0.00045)))
   (* z (* x (+ y -1.0)))
   (- x (* x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -170000.0) || !(z <= 0.00045)) {
		tmp = z * (x * (y + -1.0));
	} else {
		tmp = x - (x * z);
	}
	return tmp;
}

Fortran

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 ((z <= (-170000.0d0)) .or. (.not. (z <= 0.00045d0))) then
        tmp = z * (x * (y + (-1.0d0)))
    else
        tmp = x - (x * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -170000.0) || !(z <= 0.00045)) {
		tmp = z * (x * (y + -1.0));
	} else {
		tmp = x - (x * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -170000.0) or not (z <= 0.00045):
		tmp = z * (x * (y + -1.0))
	else:
		tmp = x - (x * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -170000.0) || !(z <= 0.00045))
		tmp = Float64(z * Float64(x * Float64(y + -1.0)));
	else
		tmp = Float64(x - Float64(x * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -170000.0) || ~((z <= 0.00045)))
		tmp = z * (x * (y + -1.0));
	else
		tmp = x - (x * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -170000.0], N[Not[LessEqual[z, 0.00045]], $MachinePrecision]], N[(z * N[(x * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(x * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.7e5 or 4.4999999999999999e-4 < z

   1. Initial program 93.1%

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

   3. Taylor expanded in z around inf 92.6%

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

      1. *-commutative 92.6%

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

      2. associate-*l* 99.4%

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

      3. sub-neg 99.4%

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

      4. remove-double-neg 99.4%

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

      5. distribute-neg-in 99.4%

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

      6. +-commutative 99.4%

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

      7. sub-neg 99.4%

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

      8. distribute-lft-neg-out 99.4%

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

      9. *-commutative 99.4%

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

      10. distribute-rgt-neg-out 99.4%

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

      11. neg-sub0 99.4%

          \[\leadsto z \cdot \left(x \cdot \color{blue}{\left(0 - \left(1 - y\right)\right)}\right) \]

      12. associate--r- 99.4%

          \[\leadsto z \cdot \left(x \cdot \color{blue}{\left(\left(0 - 1\right) + y\right)}\right) \]

      13. metadata-eval 99.4%

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

      14. +-commutative 99.4%

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

   5. Simplified 99.4%

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

   if -1.7e5 < z < 4.4999999999999999e-4

   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 99.9%

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

   4. Taylor expanded in y around 0 79.2%

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

      1. mul-1-neg 4.9%

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

      2. distribute-rgt-neg-in 4.9%

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

   6. Simplified 79.2%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -170000 \lor \neg \left(z \leq 0.00045\right):\\ \;\;\;\;z \cdot \left(x \cdot \left(y + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x - x \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 87.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -210000:\\ \;\;\;\;z \cdot \left(x \cdot \left(y + -1\right)\right)\\ \mathbf{elif}\;z \leq 1.08 \cdot 10^{-5}:\\ \;\;\;\;x - x \cdot z\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot x - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -210000.0)
   (* z (* x (+ y -1.0)))
   (if (<= z 1.08e-5) (- x (* x z)) (* z (- (* y x) x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -210000.0) {
		tmp = z * (x * (y + -1.0));
	} else if (z <= 1.08e-5) {
		tmp = x - (x * z);
	} else {
		tmp = z * ((y * x) - x);
	}
	return tmp;
}

Fortran

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 (z <= (-210000.0d0)) then
        tmp = z * (x * (y + (-1.0d0)))
    else if (z <= 1.08d-5) then
        tmp = x - (x * z)
    else
        tmp = z * ((y * x) - x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -210000.0) {
		tmp = z * (x * (y + -1.0));
	} else if (z <= 1.08e-5) {
		tmp = x - (x * z);
	} else {
		tmp = z * ((y * x) - x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -210000.0:
		tmp = z * (x * (y + -1.0))
	elif z <= 1.08e-5:
		tmp = x - (x * z)
	else:
		tmp = z * ((y * x) - x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -210000.0)
		tmp = Float64(z * Float64(x * Float64(y + -1.0)));
	elseif (z <= 1.08e-5)
		tmp = Float64(x - Float64(x * z));
	else
		tmp = Float64(z * Float64(Float64(y * x) - x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -210000.0)
		tmp = z * (x * (y + -1.0));
	elseif (z <= 1.08e-5)
		tmp = x - (x * z);
	else
		tmp = z * ((y * x) - x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -210000.0], N[(z * N[(x * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.08e-5], N[(x - N[(x * z), $MachinePrecision]), $MachinePrecision], N[(z * N[(N[(y * x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.1e5

   1. Initial program 94.6%

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

   3. Taylor expanded in z around inf 93.7%

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

      1. *-commutative 93.7%

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

      2. associate-*l* 99.0%

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

      3. sub-neg 99.0%

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

      4. remove-double-neg 99.0%

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

      5. distribute-neg-in 99.0%

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

      6. +-commutative 99.0%

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

      7. sub-neg 99.0%

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

      8. distribute-lft-neg-out 99.0%

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

      9. *-commutative 99.0%

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

      10. distribute-rgt-neg-out 99.0%

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

      11. neg-sub0 99.0%

          \[\leadsto z \cdot \left(x \cdot \color{blue}{\left(0 - \left(1 - y\right)\right)}\right) \]

      12. associate--r- 99.0%

          \[\leadsto z \cdot \left(x \cdot \color{blue}{\left(\left(0 - 1\right) + y\right)}\right) \]

      13. metadata-eval 99.0%

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

      14. +-commutative 99.0%

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

   5. Simplified 99.0%

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

   if -2.1e5 < z < 1.07999999999999999e-5

   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 99.9%

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

   4. Taylor expanded in y around 0 79.2%

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

      1. mul-1-neg 4.9%

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

      2. distribute-rgt-neg-in 4.9%

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

   6. Simplified 79.2%

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

   if 1.07999999999999999e-5 < z

   1. Initial program 91.5%

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

   3. Taylor expanded in z around inf 91.5%

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

      1. *-commutative 91.5%

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

      2. associate-*l* 99.8%

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

      3. sub-neg 99.8%

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

      4. remove-double-neg 99.8%

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

      5. distribute-neg-in 99.8%

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

      6. +-commutative 99.8%

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

      7. sub-neg 99.8%

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

      8. distribute-lft-neg-out 99.8%

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

      9. *-commutative 99.8%

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

      10. distribute-rgt-neg-out 99.8%

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

      11. neg-sub0 99.8%

          \[\leadsto z \cdot \left(x \cdot \color{blue}{\left(0 - \left(1 - y\right)\right)}\right) \]

      12. associate--r- 99.8%

          \[\leadsto z \cdot \left(x \cdot \color{blue}{\left(\left(0 - 1\right) + y\right)}\right) \]

      13. metadata-eval 99.8%

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

      14. +-commutative 99.8%

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

   5. Simplified 99.8%

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

      1. distribute-rgt-in 99.9%

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

      2. mul-1-neg 99.9%

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

   7. Applied egg-rr 99.9%

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

   8. Taylor expanded in z around 0 99.9%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -210000:\\ \;\;\;\;z \cdot \left(x \cdot \left(y + -1\right)\right)\\ \mathbf{elif}\;z \leq 1.08 \cdot 10^{-5}:\\ \;\;\;\;x - x \cdot z\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot x - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 87.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -130000:\\ \;\;\;\;\left(x \cdot z\right) \cdot \left(y + -1\right)\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{-6}:\\ \;\;\;\;x - x \cdot z\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot x - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -130000.0)
   (* (* x z) (+ y -1.0))
   (if (<= z 1.35e-6) (- x (* x z)) (* z (- (* y x) x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -130000.0) {
		tmp = (x * z) * (y + -1.0);
	} else if (z <= 1.35e-6) {
		tmp = x - (x * z);
	} else {
		tmp = z * ((y * x) - x);
	}
	return tmp;
}

Fortran

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 (z <= (-130000.0d0)) then
        tmp = (x * z) * (y + (-1.0d0))
    else if (z <= 1.35d-6) then
        tmp = x - (x * z)
    else
        tmp = z * ((y * x) - x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -130000.0) {
		tmp = (x * z) * (y + -1.0);
	} else if (z <= 1.35e-6) {
		tmp = x - (x * z);
	} else {
		tmp = z * ((y * x) - x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -130000.0:
		tmp = (x * z) * (y + -1.0)
	elif z <= 1.35e-6:
		tmp = x - (x * z)
	else:
		tmp = z * ((y * x) - x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -130000.0)
		tmp = Float64(Float64(x * z) * Float64(y + -1.0));
	elseif (z <= 1.35e-6)
		tmp = Float64(x - Float64(x * z));
	else
		tmp = Float64(z * Float64(Float64(y * x) - x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -130000.0)
		tmp = (x * z) * (y + -1.0);
	elseif (z <= 1.35e-6)
		tmp = x - (x * z);
	else
		tmp = z * ((y * x) - x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -130000.0], N[(N[(x * z), $MachinePrecision] * N[(y + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.35e-6], N[(x - N[(x * z), $MachinePrecision]), $MachinePrecision], N[(z * N[(N[(y * x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.3e5

   1. Initial program 94.6%

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

   3. Taylor expanded in z around 0 94.6%

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

      1. +-commutative 94.6%

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

      2. *-commutative 94.6%

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

      3. fma-define 94.6%

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

      4. sub-neg 94.6%

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

      5. metadata-eval 94.6%

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

   5. Applied egg-rr 94.6%

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

   6. Taylor expanded in z around inf 93.7%

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

      1. *-commutative 93.7%

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

      2. sub-neg 93.7%

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

      3. metadata-eval 93.7%

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

      4. associate-*r* 99.0%

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

      5. *-commutative 99.0%

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

      6. associate-*l* 99.0%

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

   8. Simplified 99.0%

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

   if -1.3e5 < z < 1.34999999999999999e-6

   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 99.9%

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

   4. Taylor expanded in y around 0 79.2%

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

      1. mul-1-neg 4.9%

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

      2. distribute-rgt-neg-in 4.9%

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

   6. Simplified 79.2%

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

   if 1.34999999999999999e-6 < z

   1. Initial program 91.5%

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

   3. Taylor expanded in z around inf 91.5%

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

      1. *-commutative 91.5%

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

      2. associate-*l* 99.8%

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

      3. sub-neg 99.8%

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

      4. remove-double-neg 99.8%

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

      5. distribute-neg-in 99.8%

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

      6. +-commutative 99.8%

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

      7. sub-neg 99.8%

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

      8. distribute-lft-neg-out 99.8%

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

      9. *-commutative 99.8%

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

      10. distribute-rgt-neg-out 99.8%

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

      11. neg-sub0 99.8%

          \[\leadsto z \cdot \left(x \cdot \color{blue}{\left(0 - \left(1 - y\right)\right)}\right) \]

      12. associate--r- 99.8%

          \[\leadsto z \cdot \left(x \cdot \color{blue}{\left(\left(0 - 1\right) + y\right)}\right) \]

      13. metadata-eval 99.8%

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

      14. +-commutative 99.8%

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

   5. Simplified 99.8%

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

      1. distribute-rgt-in 99.9%

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

      2. mul-1-neg 99.9%

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

   7. Applied egg-rr 99.9%

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

   8. Taylor expanded in z around 0 99.9%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -130000:\\ \;\;\;\;\left(x \cdot z\right) \cdot \left(y + -1\right)\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{-6}:\\ \;\;\;\;x - x \cdot z\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot x - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.15:\\ \;\;\;\;\left(x \cdot z\right) \cdot \left(y + -1\right)\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;x + x \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot x - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.15)
   (* (* x z) (+ y -1.0))
   (if (<= z 1.0) (+ x (* x (* y z))) (* z (- (* y x) x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.15) {
		tmp = (x * z) * (y + -1.0);
	} else if (z <= 1.0) {
		tmp = x + (x * (y * z));
	} else {
		tmp = z * ((y * x) - x);
	}
	return tmp;
}

Fortran

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 (z <= (-1.15d0)) then
        tmp = (x * z) * (y + (-1.0d0))
    else if (z <= 1.0d0) then
        tmp = x + (x * (y * z))
    else
        tmp = z * ((y * x) - x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.15) {
		tmp = (x * z) * (y + -1.0);
	} else if (z <= 1.0) {
		tmp = x + (x * (y * z));
	} else {
		tmp = z * ((y * x) - x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.15:
		tmp = (x * z) * (y + -1.0)
	elif z <= 1.0:
		tmp = x + (x * (y * z))
	else:
		tmp = z * ((y * x) - x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.15)
		tmp = Float64(Float64(x * z) * Float64(y + -1.0));
	elseif (z <= 1.0)
		tmp = Float64(x + Float64(x * Float64(y * z)));
	else
		tmp = Float64(z * Float64(Float64(y * x) - x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.15)
		tmp = (x * z) * (y + -1.0);
	elseif (z <= 1.0)
		tmp = x + (x * (y * z));
	else
		tmp = z * ((y * x) - x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.15], N[(N[(x * z), $MachinePrecision] * N[(y + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.0], N[(x + N[(x * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[(N[(y * x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.1499999999999999

   1. Initial program 94.9%

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

   3. Taylor expanded in z around 0 94.9%

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

      1. +-commutative 94.9%

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

      2. *-commutative 94.9%

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

      3. fma-define 94.9%

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

      4. sub-neg 94.9%

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

      5. metadata-eval 94.9%

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

   5. Applied egg-rr 94.9%

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

   6. Taylor expanded in z around inf 91.5%

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

      1. *-commutative 91.5%

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

      2. sub-neg 91.5%

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

      3. metadata-eval 91.5%

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

      4. associate-*r* 96.5%

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

      5. *-commutative 96.5%

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

      6. associate-*l* 96.6%

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

   8. Simplified 96.6%

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

   if -1.1499999999999999 < 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 99.9%

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

   4. Taylor expanded in y around inf 99.2%

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

      1. *-commutative 99.2%

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

   6. Simplified 99.2%

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

   if 1 < z

   1. Initial program 91.4%

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

   3. Taylor expanded in z around inf 91.4%

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

      1. *-commutative 91.4%

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

      2. associate-*l* 99.8%

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

      3. sub-neg 99.8%

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

      4. remove-double-neg 99.8%

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

      5. distribute-neg-in 99.8%

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

      6. +-commutative 99.8%

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

      7. sub-neg 99.8%

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

      8. distribute-lft-neg-out 99.8%

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

      9. *-commutative 99.8%

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

      10. distribute-rgt-neg-out 99.8%

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

      11. neg-sub0 99.8%

          \[\leadsto z \cdot \left(x \cdot \color{blue}{\left(0 - \left(1 - y\right)\right)}\right) \]

      12. associate--r- 99.8%

          \[\leadsto z \cdot \left(x \cdot \color{blue}{\left(\left(0 - 1\right) + y\right)}\right) \]

      13. metadata-eval 99.8%

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

      14. +-commutative 99.8%

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

   5. Simplified 99.8%

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

      1. distribute-rgt-in 99.9%

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

      2. mul-1-neg 99.9%

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

   7. Applied egg-rr 99.9%

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

   8. Taylor expanded in z around 0 99.9%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.15:\\ \;\;\;\;\left(x \cdot z\right) \cdot \left(y + -1\right)\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;x + x \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot x - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 98.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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 (z <= (-1.0d0)) then
        tmp = (x * z) * (y + (-1.0d0))
    else if (z <= 1.0d0) then
        tmp = x * (1.0d0 + (y * z))
    else
        tmp = z * ((y * x) - x)
    end if
    code = tmp
end function

Java

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

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.0:
		tmp = (x * z) * (y + -1.0)
	elif z <= 1.0:
		tmp = x * (1.0 + (y * z))
	else:
		tmp = z * ((y * x) - x)
	return tmp

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if z < -1

   1. Initial program 94.9%

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

   3. Taylor expanded in z around 0 94.9%

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

      1. +-commutative 94.9%

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

      2. *-commutative 94.9%

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

      3. fma-define 94.9%

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

      4. sub-neg 94.9%

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

      5. metadata-eval 94.9%

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

   5. Applied egg-rr 94.9%

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

   6. Taylor expanded in z around inf 91.5%

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

      1. *-commutative 91.5%

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

      2. sub-neg 91.5%

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

      3. metadata-eval 91.5%

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

      4. associate-*r* 96.5%

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

      5. *-commutative 96.5%

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

      6. associate-*l* 96.6%

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

   8. Simplified 96.6%

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

   if -1 < 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 y around inf 99.3%

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

      1. mul-1-neg 99.3%

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

      2. distribute-lft-neg-out 99.3%

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

      3. *-commutative 99.3%

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

   5. Simplified 99.3%

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

   if 1 < z

   1. Initial program 91.4%

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

   3. Taylor expanded in z around inf 91.4%

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

      1. *-commutative 91.4%

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

      2. associate-*l* 99.8%

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

      3. sub-neg 99.8%

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

      4. remove-double-neg 99.8%

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

      5. distribute-neg-in 99.8%

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

      6. +-commutative 99.8%

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

      7. sub-neg 99.8%

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

      8. distribute-lft-neg-out 99.8%

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

      9. *-commutative 99.8%

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

      10. distribute-rgt-neg-out 99.8%

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

      11. neg-sub0 99.8%

          \[\leadsto z \cdot \left(x \cdot \color{blue}{\left(0 - \left(1 - y\right)\right)}\right) \]

      12. associate--r- 99.8%

          \[\leadsto z \cdot \left(x \cdot \color{blue}{\left(\left(0 - 1\right) + y\right)}\right) \]

      13. metadata-eval 99.8%

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

      14. +-commutative 99.8%

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

   5. Simplified 99.8%

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

      1. distribute-rgt-in 99.9%

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

      2. mul-1-neg 99.9%

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

   7. Applied egg-rr 99.9%

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

   8. Taylor expanded in z around 0 99.9%

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

4. Final simplification 98.6%

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

Alternative 11: 83.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{+83} \lor \neg \left(y \leq 4.8 \cdot 10^{+14}\right):\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -8e+83) (not (<= y 4.8e+14))) (* x (* y z)) (* x (- 1.0 z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -8e+83) || !(y <= 4.8e+14)) {
		tmp = x * (y * z);
	} else {
		tmp = x * (1.0 - z);
	}
	return tmp;
}

Fortran

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 <= (-8d+83)) .or. (.not. (y <= 4.8d+14))) then
        tmp = x * (y * z)
    else
        tmp = x * (1.0d0 - z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -8e+83) || !(y <= 4.8e+14)) {
		tmp = x * (y * z);
	} else {
		tmp = x * (1.0 - z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -8e+83) or not (y <= 4.8e+14):
		tmp = x * (y * z)
	else:
		tmp = x * (1.0 - z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -8e+83) || !(y <= 4.8e+14))
		tmp = Float64(x * Float64(y * z));
	else
		tmp = Float64(x * Float64(1.0 - z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -8e+83) || ~((y <= 4.8e+14)))
		tmp = x * (y * z);
	else
		tmp = x * (1.0 - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -8e+83], N[Not[LessEqual[y, 4.8e+14]], $MachinePrecision]], N[(x * N[(y * z), $MachinePrecision]), $MachinePrecision], N[(x * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -8.00000000000000025e83 or 4.8e14 < y

   1. Initial program 92.4%

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

   3. Taylor expanded in y around inf 72.5%

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

      1. *-commutative 72.5%

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

   5. Simplified 72.5%

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

   if -8.00000000000000025e83 < y < 4.8e14

   1. Initial program 98.7%

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

   3. Taylor expanded in y around 0 92.1%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{+83} \lor \neg \left(y \leq 4.8 \cdot 10^{+14}\right):\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 84.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{+83}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;y \leq 15000000000:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.3e+83)
   (* x (* y z))
   (if (<= y 15000000000.0) (* x (- 1.0 z)) (* y (* x z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.3e+83) {
		tmp = x * (y * z);
	} else if (y <= 15000000000.0) {
		tmp = x * (1.0 - z);
	} else {
		tmp = y * (x * z);
	}
	return tmp;
}

Fortran

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 <= (-2.3d+83)) then
        tmp = x * (y * z)
    else if (y <= 15000000000.0d0) then
        tmp = x * (1.0d0 - z)
    else
        tmp = y * (x * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.3e+83) {
		tmp = x * (y * z);
	} else if (y <= 15000000000.0) {
		tmp = x * (1.0 - z);
	} else {
		tmp = y * (x * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -2.3e+83:
		tmp = x * (y * z)
	elif y <= 15000000000.0:
		tmp = x * (1.0 - z)
	else:
		tmp = y * (x * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.3e+83)
		tmp = Float64(x * Float64(y * z));
	elseif (y <= 15000000000.0)
		tmp = Float64(x * Float64(1.0 - z));
	else
		tmp = Float64(y * Float64(x * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -2.3e+83)
		tmp = x * (y * z);
	elseif (y <= 15000000000.0)
		tmp = x * (1.0 - z);
	else
		tmp = y * (x * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.3e+83], N[(x * N[(y * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 15000000000.0], N[(x * N[(1.0 - z), $MachinePrecision]), $MachinePrecision], N[(y * N[(x * z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.29999999999999995e83

   1. Initial program 95.2%

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

   3. Taylor expanded in y around inf 82.2%

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

      1. *-commutative 82.2%

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

   5. Simplified 82.2%

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

   if -2.29999999999999995e83 < y < 1.5e10

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 93.3%

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

   if 1.5e10 < y

   1. Initial program 87.9%

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

   3. Taylor expanded in z around 0 87.9%

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

      1. +-commutative 87.9%

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

      2. *-commutative 87.9%

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

      3. fma-define 87.9%

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

      4. sub-neg 87.9%

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

      5. metadata-eval 87.9%

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

   5. Applied egg-rr 87.9%

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

   6. Taylor expanded in y around inf 63.1%

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

      1. *-commutative 63.1%

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

      2. associate-*r* 73.0%

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

      3. *-commutative 73.0%

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

   8. Simplified 73.0%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{+83}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;y \leq 15000000000:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 84.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{+88}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;y \leq 15500000000:\\ \;\;\;\;x - x \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.7e+88)
   (* x (* y z))
   (if (<= y 15500000000.0) (- x (* x z)) (* y (* x z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.7e+88) {
		tmp = x * (y * z);
	} else if (y <= 15500000000.0) {
		tmp = x - (x * z);
	} else {
		tmp = y * (x * z);
	}
	return tmp;
}

Fortran

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 <= (-1.7d+88)) then
        tmp = x * (y * z)
    else if (y <= 15500000000.0d0) then
        tmp = x - (x * z)
    else
        tmp = y * (x * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.7e+88) {
		tmp = x * (y * z);
	} else if (y <= 15500000000.0) {
		tmp = x - (x * z);
	} else {
		tmp = y * (x * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.7e+88:
		tmp = x * (y * z)
	elif y <= 15500000000.0:
		tmp = x - (x * z)
	else:
		tmp = y * (x * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.7e+88)
		tmp = Float64(x * Float64(y * z));
	elseif (y <= 15500000000.0)
		tmp = Float64(x - Float64(x * z));
	else
		tmp = Float64(y * Float64(x * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.7e+88)
		tmp = x * (y * z);
	elseif (y <= 15500000000.0)
		tmp = x - (x * z);
	else
		tmp = y * (x * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.7e+88], N[(x * N[(y * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 15500000000.0], N[(x - N[(x * z), $MachinePrecision]), $MachinePrecision], N[(y * N[(x * z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.70000000000000002e88

   1. Initial program 95.2%

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

   3. Taylor expanded in y around inf 82.2%

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

      1. *-commutative 82.2%

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

   5. Simplified 82.2%

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

   if -1.70000000000000002e88 < y < 1.55e10

   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 99.9%

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

   4. Taylor expanded in y around 0 93.3%

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

      1. mul-1-neg 49.1%

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

      2. distribute-rgt-neg-in 49.1%

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

   6. Simplified 93.3%

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

   if 1.55e10 < y

   1. Initial program 87.9%

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

   3. Taylor expanded in z around 0 87.9%

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

      1. +-commutative 87.9%

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

      2. *-commutative 87.9%

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

      3. fma-define 87.9%

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

      4. sub-neg 87.9%

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

      5. metadata-eval 87.9%

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

   5. Applied egg-rr 87.9%

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

   6. Taylor expanded in y around inf 63.1%

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

      1. *-commutative 63.1%

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

      2. associate-*r* 73.0%

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

      3. *-commutative 73.0%

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

   8. Simplified 73.0%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{+88}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;y \leq 15500000000:\\ \;\;\;\;x - x \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 65.1% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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 ((z <= (-1.0d0)) .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 <= -1.0) || !(z <= 1.0)) {
		tmp = x * -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if z < -1 or 1 < z

   1. Initial program 93.2%

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

   3. Taylor expanded in z around inf 91.5%

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

   4. Taylor expanded in y around 0 54.5%

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

      1. mul-1-neg 54.5%

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

      2. distribute-rgt-neg-in 54.5%

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

   6. Simplified 54.5%

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

   if -1 < 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 78.0%

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

4. Final simplification 65.0%

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

Alternative 15: 38.8% accurate, 9.0× 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

real(8) function code(x, y, z)
    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 96.2%

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

3. Taylor expanded in z around 0 36.6%

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

4. Final simplification 36.6%

   \[\leadsto x \]
5. Add Preprocessing

Developer target: 99.7% accurate, 0.2× 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

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) :: 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 2024040 
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSV:hsv from colour-2.3.3, J"
  :precision binary64

  :herbie-target
  (if (< (* x (- 1.0 (* (- 1.0 y) z))) -1.618195973607049e+50) (+ x (* (- 1.0 y) (* (- z) x))) (if (< (* x (- 1.0 (* (- 1.0 y) z))) 3.892237649663903e+134) (- (* (* x y) z) (- (* x z) x)) (+ x (* (- 1.0 y) (* (- z) x)))))

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