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

Percentage Accurate: 95.9% → 96.7%

Time: 6.8s

Alternatives: 11

Speedup: 0.8×

• 20.3% 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: 95.9% 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: 96.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - y \leq 500000000:\\ \;\;\;\;x \cdot \left(1 + z \cdot \left(y + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(x \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (- 1.0 y) 500000000.0)
   (* x (+ 1.0 (* z (+ y -1.0))))
   (+ x (* y (* x z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((1.0 - y) <= 500000000.0) {
		tmp = x * (1.0 + (z * (y + -1.0)));
	} else {
		tmp = x + (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 ((1.0d0 - y) <= 500000000.0d0) then
        tmp = x * (1.0d0 + (z * (y + (-1.0d0))))
    else
        tmp = x + (y * (x * z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((1.0 - y) <= 500000000.0) {
		tmp = x * (1.0 + (z * (y + -1.0)));
	} else {
		tmp = x + (y * (x * z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (1.0 - y) <= 500000000.0:
		tmp = x * (1.0 + (z * (y + -1.0)))
	else:
		tmp = x + (y * (x * z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(1.0 - y) <= 500000000.0)
		tmp = Float64(x * Float64(1.0 + Float64(z * Float64(y + -1.0))));
	else
		tmp = Float64(x + Float64(y * Float64(x * z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((1.0 - y) <= 500000000.0)
		tmp = x * (1.0 + (z * (y + -1.0)));
	else
		tmp = x + (y * (x * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(1.0 - y), $MachinePrecision], 500000000.0], N[(x * N[(1.0 + N[(z * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(y * N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 1 y) < 5e8

   1. Initial program 99.0%

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

   if 5e8 < (-.f64 1 y)

   1. Initial program 84.0%

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

      1. flip-- 65.1%

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

      2. associate-*l/ 63.5%

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

      3. metadata-eval 63.5%

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

   3. Applied egg-rr 63.5%

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

   4. Taylor expanded in y around inf 84.0%

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

      1. associate-*r* 84.0%

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

      2. neg-mul-1 84.0%

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

   6. Simplified 84.0%

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

   7. Taylor expanded in y around 0 98.4%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - y \leq 500000000:\\ \;\;\;\;x \cdot \left(1 + z \cdot \left(y + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(x \cdot z\right)\\ \end{array} \]

Alternative 2: 97.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 95.5%

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

   1. distribute-rgt-out-- 95.5%

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

   2. *-lft-identity 95.5%

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

   3. cancel-sign-sub-inv 95.5%

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

   4. +-commutative 95.5%

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

   5. distribute-lft-neg-in 95.5%

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

   6. associate-*l* 98.3%

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

   7. fma-def 98.3%

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

   8. neg-sub0 98.3%

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

   9. associate--r- 98.3%

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

   10. metadata-eval 98.3%

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

   11. +-commutative 98.3%

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

   12. *-commutative 98.3%

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

3. Simplified 98.3%

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

4. Final simplification 98.3%

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

Alternative 3: 84.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{+69}:\\ \;\;\;\;y \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;y \leq -9.2 \cdot 10^{+49}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -105000000:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+79}:\\ \;\;\;\;x - x \cdot z\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.7e+69)
   (* y (* x z))
   (if (<= y -9.2e+49)
     x
     (if (<= y -105000000.0)
       (* x (* y z))
       (if (<= y 1.35e+79) (- x (* x z)) (* z (* y x)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.7e+69) {
		tmp = y * (x * z);
	} else if (y <= -9.2e+49) {
		tmp = x;
	} else if (y <= -105000000.0) {
		tmp = x * (y * z);
	} else if (y <= 1.35e+79) {
		tmp = x - (x * z);
	} else {
		tmp = z * (y * 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 (y <= (-2.7d+69)) then
        tmp = y * (x * z)
    else if (y <= (-9.2d+49)) then
        tmp = x
    else if (y <= (-105000000.0d0)) then
        tmp = x * (y * z)
    else if (y <= 1.35d+79) then
        tmp = x - (x * z)
    else
        tmp = z * (y * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.7e+69) {
		tmp = y * (x * z);
	} else if (y <= -9.2e+49) {
		tmp = x;
	} else if (y <= -105000000.0) {
		tmp = x * (y * z);
	} else if (y <= 1.35e+79) {
		tmp = x - (x * z);
	} else {
		tmp = z * (y * x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -2.7e+69:
		tmp = y * (x * z)
	elif y <= -9.2e+49:
		tmp = x
	elif y <= -105000000.0:
		tmp = x * (y * z)
	elif y <= 1.35e+79:
		tmp = x - (x * z)
	else:
		tmp = z * (y * x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.7e+69)
		tmp = Float64(y * Float64(x * z));
	elseif (y <= -9.2e+49)
		tmp = x;
	elseif (y <= -105000000.0)
		tmp = Float64(x * Float64(y * z));
	elseif (y <= 1.35e+79)
		tmp = Float64(x - Float64(x * z));
	else
		tmp = Float64(z * Float64(y * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -2.7e+69)
		tmp = y * (x * z);
	elseif (y <= -9.2e+49)
		tmp = x;
	elseif (y <= -105000000.0)
		tmp = x * (y * z);
	elseif (y <= 1.35e+79)
		tmp = x - (x * z);
	else
		tmp = z * (y * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.7e+69], N[(y * N[(x * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -9.2e+49], x, If[LessEqual[y, -105000000.0], N[(x * N[(y * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.35e+79], N[(x - N[(x * z), $MachinePrecision]), $MachinePrecision], N[(z * N[(y * x), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if y < -2.6999999999999998e69

   1. Initial program 77.8%

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

   2. Taylor expanded in y around inf 85.8%

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

   if -2.6999999999999998e69 < y < -9.20000000000000008e49

   1. Initial program 100.0%

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

   2. Taylor expanded in z around 0 100.0%

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

   if -9.20000000000000008e49 < y < -1.05e8

   1. Initial program 99.6%

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

   2. Taylor expanded in y around inf 76.5%

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

      1. *-commutative 76.5%

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

   4. Simplified 76.5%

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

   if -1.05e8 < y < 1.35e79

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 94.1%

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

      1. sub-neg 94.1%

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

      2. +-commutative 94.1%

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

      3. distribute-rgt1-in 94.1%

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

      4. cancel-sign-sub-inv 94.1%

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

   4. Simplified 94.1%

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

   if 1.35e79 < y

   1. Initial program 95.4%

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

   2. Taylor expanded in y around inf 71.3%

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

      1. associate-*r* 73.5%

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

      2. *-commutative 73.5%

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

      3. associate-*l* 75.7%

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

   4. Simplified 75.7%

      \[\leadsto \color{blue}{z \cdot \left(y \cdot x\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{+69}:\\ \;\;\;\;y \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;y \leq -9.2 \cdot 10^{+49}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -105000000:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+79}:\\ \;\;\;\;x - x \cdot z\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot x\right)\\ \end{array} \]

Alternative 4: 84.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{+69}:\\ \;\;\;\;y \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;y \leq -5.8 \cdot 10^{+48}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -4000000:\\ \;\;\;\;x \cdot \left(z \cdot \left(y + -1\right)\right)\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{+79}:\\ \;\;\;\;x - x \cdot z\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -3.5e+69)
   (* y (* x z))
   (if (<= y -5.8e+48)
     x
     (if (<= y -4000000.0)
       (* x (* z (+ y -1.0)))
       (if (<= y 1.45e+79) (- x (* x z)) (* z (* y x)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.5e+69) {
		tmp = y * (x * z);
	} else if (y <= -5.8e+48) {
		tmp = x;
	} else if (y <= -4000000.0) {
		tmp = x * (z * (y + -1.0));
	} else if (y <= 1.45e+79) {
		tmp = x - (x * z);
	} else {
		tmp = z * (y * 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 (y <= (-3.5d+69)) then
        tmp = y * (x * z)
    else if (y <= (-5.8d+48)) then
        tmp = x
    else if (y <= (-4000000.0d0)) then
        tmp = x * (z * (y + (-1.0d0)))
    else if (y <= 1.45d+79) then
        tmp = x - (x * z)
    else
        tmp = z * (y * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.5e+69) {
		tmp = y * (x * z);
	} else if (y <= -5.8e+48) {
		tmp = x;
	} else if (y <= -4000000.0) {
		tmp = x * (z * (y + -1.0));
	} else if (y <= 1.45e+79) {
		tmp = x - (x * z);
	} else {
		tmp = z * (y * x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -3.5e+69:
		tmp = y * (x * z)
	elif y <= -5.8e+48:
		tmp = x
	elif y <= -4000000.0:
		tmp = x * (z * (y + -1.0))
	elif y <= 1.45e+79:
		tmp = x - (x * z)
	else:
		tmp = z * (y * x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -3.5e+69)
		tmp = Float64(y * Float64(x * z));
	elseif (y <= -5.8e+48)
		tmp = x;
	elseif (y <= -4000000.0)
		tmp = Float64(x * Float64(z * Float64(y + -1.0)));
	elseif (y <= 1.45e+79)
		tmp = Float64(x - Float64(x * z));
	else
		tmp = Float64(z * Float64(y * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -3.5e+69)
		tmp = y * (x * z);
	elseif (y <= -5.8e+48)
		tmp = x;
	elseif (y <= -4000000.0)
		tmp = x * (z * (y + -1.0));
	elseif (y <= 1.45e+79)
		tmp = x - (x * z);
	else
		tmp = z * (y * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -3.5e+69], N[(y * N[(x * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -5.8e+48], x, If[LessEqual[y, -4000000.0], N[(x * N[(z * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.45e+79], N[(x - N[(x * z), $MachinePrecision]), $MachinePrecision], N[(z * N[(y * x), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if y < -3.49999999999999987e69

   1. Initial program 77.8%

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

   2. Taylor expanded in y around inf 85.8%

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

   if -3.49999999999999987e69 < y < -5.7999999999999998e48

   1. Initial program 100.0%

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

   2. Taylor expanded in z around 0 100.0%

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

   if -5.7999999999999998e48 < y < -4e6

   1. Initial program 99.6%

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

   2. Taylor expanded in z around inf 79.9%

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

   if -4e6 < y < 1.44999999999999996e79

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 94.1%

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

      1. sub-neg 94.1%

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

      2. +-commutative 94.1%

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

      3. distribute-rgt1-in 94.1%

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

      4. cancel-sign-sub-inv 94.1%

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

   4. Simplified 94.1%

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

   if 1.44999999999999996e79 < y

   1. Initial program 95.4%

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

   2. Taylor expanded in y around inf 71.3%

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

      1. associate-*r* 73.5%

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

      2. *-commutative 73.5%

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

      3. associate-*l* 75.7%

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

   4. Simplified 75.7%

      \[\leadsto \color{blue}{z \cdot \left(y \cdot x\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{+69}:\\ \;\;\;\;y \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;y \leq -5.8 \cdot 10^{+48}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -4000000:\\ \;\;\;\;x \cdot \left(z \cdot \left(y + -1\right)\right)\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{+79}:\\ \;\;\;\;x - x \cdot z\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot x\right)\\ \end{array} \]

Alternative 5: 63.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -x \cdot z\\ \mathbf{if}\;z \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-21}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 2.25 \cdot 10^{+213}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (* x z))))
   (if (<= z -1.0)
     t_0
     (if (<= z 4.2e-21) x (if (<= z 2.25e+213) (* 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 <= 4.2e-21) {
		tmp = x;
	} else if (z <= 2.25e+213) {
		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 <= 4.2d-21) then
        tmp = x
    else if (z <= 2.25d+213) 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 <= 4.2e-21) {
		tmp = x;
	} else if (z <= 2.25e+213) {
		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 <= 4.2e-21:
		tmp = x
	elif z <= 2.25e+213:
		tmp = x * (y * z)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(-Float64(x * z))
	tmp = 0.0
	if (z <= -1.0)
		tmp = t_0;
	elseif (z <= 4.2e-21)
		tmp = x;
	elseif (z <= 2.25e+213)
		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 <= 4.2e-21)
		tmp = x;
	elseif (z <= 2.25e+213)
		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, 4.2e-21], x, If[LessEqual[z, 2.25e+213], N[(x * N[(y * z), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1 or 2.2500000000000001e213 < z

   1. Initial program 90.1%

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

   2. Taylor expanded in z around inf 87.9%

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

   3. Taylor expanded in y around 0 63.2%

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

      1. mul-1-neg 63.2%

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

      2. distribute-rgt-neg-in 63.2%

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

   5. Simplified 63.2%

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

   if -1 < z < 4.20000000000000025e-21

   1. Initial program 99.9%

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

   2. Taylor expanded in z around 0 80.3%

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

   if 4.20000000000000025e-21 < z < 2.2500000000000001e213

   1. Initial program 93.2%

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

   2. Taylor expanded in y around inf 64.1%

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

      1. *-commutative 64.1%

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

   4. Simplified 64.1%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;-x \cdot z\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-21}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 2.25 \cdot 10^{+213}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;-x \cdot z\\ \end{array} \]

Alternative 6: 65.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -x \cdot z\\ \mathbf{if}\;z \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{-21}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{+217}:\\ \;\;\;\;y \cdot \left(x \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (* x z))))
   (if (<= z -1.0)
     t_0
     (if (<= z 2.9e-21) x (if (<= z 4.2e+217) (* y (* x 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 <= 2.9e-21) {
		tmp = x;
	} else if (z <= 4.2e+217) {
		tmp = y * (x * 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 <= 2.9d-21) then
        tmp = x
    else if (z <= 4.2d+217) then
        tmp = y * (x * 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 <= 2.9e-21) {
		tmp = x;
	} else if (z <= 4.2e+217) {
		tmp = y * (x * 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 <= 2.9e-21:
		tmp = x
	elif z <= 4.2e+217:
		tmp = y * (x * z)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(-Float64(x * z))
	tmp = 0.0
	if (z <= -1.0)
		tmp = t_0;
	elseif (z <= 2.9e-21)
		tmp = x;
	elseif (z <= 4.2e+217)
		tmp = Float64(y * Float64(x * 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 <= 2.9e-21)
		tmp = x;
	elseif (z <= 4.2e+217)
		tmp = y * (x * 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, 2.9e-21], x, If[LessEqual[z, 4.2e+217], N[(y * N[(x * z), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1 or 4.2000000000000002e217 < z

   1. Initial program 90.1%

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

   2. Taylor expanded in z around inf 87.9%

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

   3. Taylor expanded in y around 0 63.2%

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

      1. mul-1-neg 63.2%

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

      2. distribute-rgt-neg-in 63.2%

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

   5. Simplified 63.2%

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

   if -1 < z < 2.9e-21

   1. Initial program 99.9%

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

   2. Taylor expanded in z around 0 80.3%

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

   if 2.9e-21 < z < 4.2000000000000002e217

   1. Initial program 93.2%

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

   2. Taylor expanded in y around inf 72.9%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;-x \cdot z\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{-21}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{+217}:\\ \;\;\;\;y \cdot \left(x \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;-x \cdot z\\ \end{array} \]

Alternative 7: 96.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 5.2 \cdot 10^{-29}\right):\\ \;\;\;\;x + y \cdot \left(x \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x - x \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 5.2e-29))) (+ x (* y (* x z))) (- x (* x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.0) || !(y <= 5.2e-29)) {
		tmp = x + (y * (x * z));
	} 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 ((y <= (-1.0d0)) .or. (.not. (y <= 5.2d-29))) then
        tmp = x + (y * (x * z))
    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 ((y <= -1.0) || !(y <= 5.2e-29)) {
		tmp = x + (y * (x * z));
	} else {
		tmp = x - (x * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1.0) or not (y <= 5.2e-29):
		tmp = x + (y * (x * z))
	else:
		tmp = x - (x * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 5.2e-29))
		tmp = Float64(x + Float64(y * Float64(x * z)));
	else
		tmp = Float64(x - Float64(x * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -1.0) || ~((y <= 5.2e-29)))
		tmp = x + (y * (x * z));
	else
		tmp = x - (x * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 5.2e-29]], $MachinePrecision]], N[(x + N[(y * N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(x * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 5.2000000000000004e-29 < y

   1. Initial program 91.1%

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

      1. flip-- 67.4%

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

      2. associate-*l/ 63.0%

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

      3. metadata-eval 63.0%

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

   3. Applied egg-rr 63.0%

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

   4. Taylor expanded in y around inf 90.4%

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

      1. associate-*r* 90.4%

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

      2. neg-mul-1 90.4%

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

   6. Simplified 90.4%

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

   7. Taylor expanded in y around 0 96.0%

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

   if -1 < y < 5.2000000000000004e-29

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-rgt1-in 100.0%

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

      4. cancel-sign-sub-inv 100.0%

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

   4. Simplified 100.0%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 5.2 \cdot 10^{-29}\right):\\ \;\;\;\;x + y \cdot \left(x \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x - x \cdot z\\ \end{array} \]

Alternative 8: 95.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x + y \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{-29}:\\ \;\;\;\;x - x \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.0)
   (+ x (* y (* x z)))
   (if (<= y 5.2e-29) (- x (* x z)) (+ x (* z (* y x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.0) {
		tmp = x + (y * (x * z));
	} else if (y <= 5.2e-29) {
		tmp = x - (x * z);
	} else {
		tmp = x + (z * (y * 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 (y <= (-1.0d0)) then
        tmp = x + (y * (x * z))
    else if (y <= 5.2d-29) then
        tmp = x - (x * z)
    else
        tmp = x + (z * (y * x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.0) {
		tmp = x + (y * (x * z));
	} else if (y <= 5.2e-29) {
		tmp = x - (x * z);
	} else {
		tmp = x + (z * (y * x));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.0:
		tmp = x + (y * (x * z))
	elif y <= 5.2e-29:
		tmp = x - (x * z)
	else:
		tmp = x + (z * (y * x))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.0)
		tmp = Float64(x + Float64(y * Float64(x * z)));
	elseif (y <= 5.2e-29)
		tmp = Float64(x - Float64(x * z));
	else
		tmp = Float64(x + Float64(z * Float64(y * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.0)
		tmp = x + (y * (x * z));
	elseif (y <= 5.2e-29)
		tmp = x - (x * z);
	else
		tmp = x + (z * (y * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.0], N[(x + N[(y * N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.2e-29], N[(x - N[(x * z), $MachinePrecision]), $MachinePrecision], N[(x + N[(z * N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1

   1. Initial program 84.3%

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

      1. flip-- 65.6%

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

      2. associate-*l/ 64.0%

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

      3. metadata-eval 64.0%

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

   3. Applied egg-rr 64.0%

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

   4. Taylor expanded in y around inf 83.6%

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

      1. associate-*r* 83.6%

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

      2. neg-mul-1 83.6%

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

   6. Simplified 83.6%

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

   7. Taylor expanded in y around 0 97.8%

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

   if -1 < y < 5.2000000000000004e-29

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-rgt1-in 100.0%

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

      4. cancel-sign-sub-inv 100.0%

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

   4. Simplified 100.0%

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

   if 5.2000000000000004e-29 < y

   1. Initial program 97.1%

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

      1. distribute-rgt-out-- 97.2%

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

      2. *-lft-identity 97.2%

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

      3. cancel-sign-sub-inv 97.2%

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

      4. +-commutative 97.2%

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

      5. distribute-lft-neg-in 97.2%

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

      6. associate-*l* 95.1%

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

      7. fma-def 95.1%

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

      8. neg-sub0 95.1%

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

      9. associate--r- 95.1%

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

      10. metadata-eval 95.1%

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

      11. +-commutative 95.1%

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

      12. *-commutative 95.1%

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

   3. Simplified 95.1%

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

      1. fma-udef 95.1%

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

      2. associate-*r* 95.7%

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

   5. Applied egg-rr 95.7%

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

   6. Taylor expanded in y around inf 95.0%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x + y \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{-29}:\\ \;\;\;\;x - x \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(y \cdot x\right)\\ \end{array} \]

Alternative 9: 95.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x + y \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{-29}:\\ \;\;\;\;x - x \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + y \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.0)
   (+ x (* y (* x z)))
   (if (<= y 5.2e-29) (- x (* x z)) (* x (+ 1.0 (* y z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.0) {
		tmp = x + (y * (x * z));
	} else if (y <= 5.2e-29) {
		tmp = x - (x * z);
	} else {
		tmp = x * (1.0 + (y * 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 <= (-1.0d0)) then
        tmp = x + (y * (x * z))
    else if (y <= 5.2d-29) then
        tmp = x - (x * z)
    else
        tmp = x * (1.0d0 + (y * z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.0) {
		tmp = x + (y * (x * z));
	} else if (y <= 5.2e-29) {
		tmp = x - (x * z);
	} else {
		tmp = x * (1.0 + (y * z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.0:
		tmp = x + (y * (x * z))
	elif y <= 5.2e-29:
		tmp = x - (x * z)
	else:
		tmp = x * (1.0 + (y * z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.0)
		tmp = Float64(x + Float64(y * Float64(x * z)));
	elseif (y <= 5.2e-29)
		tmp = Float64(x - Float64(x * z));
	else
		tmp = Float64(x * Float64(1.0 + Float64(y * z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.0)
		tmp = x + (y * (x * z));
	elseif (y <= 5.2e-29)
		tmp = x - (x * z);
	else
		tmp = x * (1.0 + (y * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.0], N[(x + N[(y * N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.2e-29], N[(x - N[(x * z), $MachinePrecision]), $MachinePrecision], N[(x * N[(1.0 + N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1

   1. Initial program 84.3%

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

      1. flip-- 65.6%

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

      2. associate-*l/ 64.0%

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

      3. metadata-eval 64.0%

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

   3. Applied egg-rr 64.0%

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

   4. Taylor expanded in y around inf 83.6%

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

      1. associate-*r* 83.6%

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

      2. neg-mul-1 83.6%

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

   6. Simplified 83.6%

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

   7. Taylor expanded in y around 0 97.8%

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

   if -1 < y < 5.2000000000000004e-29

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-rgt1-in 100.0%

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

      4. cancel-sign-sub-inv 100.0%

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

   4. Simplified 100.0%

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

   if 5.2000000000000004e-29 < y

   1. Initial program 97.1%

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

   2. Taylor expanded in y around inf 96.5%

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

      1. mul-1-neg 96.5%

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

      2. distribute-lft-neg-out 96.5%

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

      3. *-commutative 96.5%

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

   4. Simplified 96.5%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x + y \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{-29}:\\ \;\;\;\;x - x \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + y \cdot z\right)\\ \end{array} \]

Alternative 10: 64.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 1.42 \cdot 10^{-12}\right):\\ \;\;\;\;-x \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.0) (not (<= z 1.42e-12))) (- (* x z)) x))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.0) || !(z <= 1.42e-12)) {
		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 <= 1.42d-12))) 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 <= 1.42e-12)) {
		tmp = -(x * z);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.0) or not (z <= 1.42e-12):
		tmp = -(x * z)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.0) || !(z <= 1.42e-12))
		tmp = Float64(-Float64(x * z));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.0) || ~((z <= 1.42e-12)))
		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, 1.42e-12]], $MachinePrecision]], (-N[(x * z), $MachinePrecision]), x]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 1.42e-12 < z

   1. Initial program 91.0%

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

   2. Taylor expanded in z around inf 89.0%

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

   3. Taylor expanded in y around 0 53.5%

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

      1. mul-1-neg 53.5%

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

      2. distribute-rgt-neg-in 53.5%

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

   5. Simplified 53.5%

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

   if -1 < z < 1.42e-12

   1. Initial program 99.9%

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

   2. Taylor expanded in z around 0 79.6%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 1.42 \cdot 10^{-12}\right):\\ \;\;\;\;-x \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 11: 38.7% 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 95.5%

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

2. Taylor expanded in z around 0 41.5%

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

3. Final simplification 41.5%

   \[\leadsto x \]

Developer target: 99.6% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

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 2023257 
(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))))