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

Percentage Accurate: 96.1% → 98.8%

Time: 7.0s

Alternatives: 10

Speedup: 0.8×

• 21.0% 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: 98.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8200000000 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;z \cdot \left(x \cdot \left(y + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + x \cdot \left(z \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -8200000000.0) (not (<= z 1.0)))
   (* z (* x (+ y -1.0)))
   (+ x (* x (* z y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -8200000000.0) || !(z <= 1.0)) {
		tmp = z * (x * (y + -1.0));
	} else {
		tmp = x + (x * (z * y));
	}
	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 <= (-8200000000.0d0)) .or. (.not. (z <= 1.0d0))) then
        tmp = z * (x * (y + (-1.0d0)))
    else
        tmp = x + (x * (z * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -8200000000.0) || !(z <= 1.0)) {
		tmp = z * (x * (y + -1.0));
	} else {
		tmp = x + (x * (z * y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -8200000000.0) or not (z <= 1.0):
		tmp = z * (x * (y + -1.0))
	else:
		tmp = x + (x * (z * y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -8200000000.0) || !(z <= 1.0))
		tmp = Float64(z * Float64(x * Float64(y + -1.0)));
	else
		tmp = Float64(x + Float64(x * Float64(z * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -8200000000.0) || ~((z <= 1.0)))
		tmp = z * (x * (y + -1.0));
	else
		tmp = x + (x * (z * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -8200000000.0], N[Not[LessEqual[z, 1.0]], $MachinePrecision]], N[(z * N[(x * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(x * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -8.2e9 or 1 < z

   1. Initial program 94.5%

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

   2. Taylor expanded in z around inf 93.9%

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

      1. *-commutative 93.9%

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

      2. associate-*r* 99.3%

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

      3. *-commutative 99.3%

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

      4. sub-neg 99.3%

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

      5. metadata-eval 99.3%

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

   4. Simplified 99.3%

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

   if -8.2e9 < z < 1

   1. Initial program 99.9%

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

   2. Taylor expanded in z around 0 99.9%

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

   3. Taylor expanded in y around inf 98.4%

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

      1. *-commutative 98.4%

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

   5. Simplified 98.4%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8200000000 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;z \cdot \left(x \cdot \left(y + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + x \cdot \left(z \cdot y\right)\\ \end{array} \]

Alternative 2: 98.0% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 97.0%

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

2. Taylor expanded in z around 0 97.0%

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

   1. +-commutative 97.0%

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

   2. associate-*r* 98.4%

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

   3. fma-def 98.4%

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

   4. sub-neg 98.4%

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

   5. metadata-eval 98.4%

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

4. Applied egg-rr 98.4%

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

5. Final simplification 98.4%

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

Alternative 3: 98.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + z \cdot \left(y + -1\right)\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{+195}:\\ \;\;\;\;\left(x \cdot z\right) \cdot \frac{1}{\frac{1}{y + -1}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* z (+ y -1.0)))))
   (if (<= t_0 -5e+195) (* (* x z) (/ 1.0 (/ 1.0 (+ y -1.0)))) (* x t_0))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 + (z * (y + -1.0));
	double tmp;
	if (t_0 <= -5e+195) {
		tmp = (x * z) * (1.0 / (1.0 / (y + -1.0)));
	} else {
		tmp = x * 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 = 1.0d0 + (z * (y + (-1.0d0)))
    if (t_0 <= (-5d+195)) then
        tmp = (x * z) * (1.0d0 / (1.0d0 / (y + (-1.0d0))))
    else
        tmp = x * t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = 1.0 + (z * (y + -1.0));
	double tmp;
	if (t_0 <= -5e+195) {
		tmp = (x * z) * (1.0 / (1.0 / (y + -1.0)));
	} else {
		tmp = x * t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 + (z * (y + -1.0))
	tmp = 0
	if t_0 <= -5e+195:
		tmp = (x * z) * (1.0 / (1.0 / (y + -1.0)))
	else:
		tmp = x * t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(1.0 + Float64(z * Float64(y + -1.0)))
	tmp = 0.0
	if (t_0 <= -5e+195)
		tmp = Float64(Float64(x * z) * Float64(1.0 / Float64(1.0 / Float64(y + -1.0))));
	else
		tmp = Float64(x * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 1.0 + (z * (y + -1.0));
	tmp = 0.0;
	if (t_0 <= -5e+195)
		tmp = (x * z) * (1.0 / (1.0 / (y + -1.0)));
	else
		tmp = x * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(1.0 + N[(z * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+195], N[(N[(x * z), $MachinePrecision] * N[(1.0 / N[(1.0 / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 1 (*.f64 (-.f64 1 y) z)) < -4.9999999999999998e195

   1. Initial program 89.8%

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

   2. Taylor expanded in z around inf 89.8%

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

      1. *-commutative 89.8%

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

      2. associate-*r* 99.9%

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

      3. *-commutative 99.9%

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

      4. sub-neg 99.9%

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

      5. metadata-eval 99.9%

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

   4. Simplified 99.9%

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

      1. associate-*r* 99.9%

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

      2. *-commutative 99.9%

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

      3. flip-+ 83.8%

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

      4. associate-*r/ 79.9%

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

      5. metadata-eval 79.9%

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

      6. fma-neg 79.9%

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

      7. metadata-eval 79.9%

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

      8. sub-neg 79.9%

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

      9. metadata-eval 79.9%

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

   6. Applied egg-rr 79.9%

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

      1. associate-/l* 83.8%

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

      2. div-inv 83.7%

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

      3. clear-num 83.8%

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

      4. metadata-eval 83.8%

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

      5. fma-neg 83.8%

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

      6. metadata-eval 83.8%

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

      7. flip-- 100.0%

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

      8. sub-neg 100.0%

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

      9. metadata-eval 100.0%

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

   8. Applied egg-rr 100.0%

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

   if -4.9999999999999998e195 < (-.f64 1 (*.f64 (-.f64 1 y) z))

   1. Initial program 98.6%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 + z \cdot \left(y + -1\right) \leq -5 \cdot 10^{+195}:\\ \;\;\;\;\left(x \cdot z\right) \cdot \frac{1}{\frac{1}{y + -1}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + z \cdot \left(y + -1\right)\right)\\ \end{array} \]

Alternative 4: 60.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(z \cdot y\right)\\ t_1 := x \cdot \left(-z\right)\\ \mathbf{if}\;z \leq -3.2 \cdot 10^{+77}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -1:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -6 \cdot 10^{-73}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -8 \cdot 10^{-118}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{-11}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 95000000000000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (* z y))) (t_1 (* x (- z))))
   (if (<= z -3.2e+77)
     t_0
     (if (<= z -1.0)
       t_1
       (if (<= z -6e-73)
         x
         (if (<= z -8e-118)
           t_0
           (if (<= z 4.4e-11) x (if (<= z 95000000000000.0) t_0 t_1))))))))

C

double code(double x, double y, double z) {
	double t_0 = x * (z * y);
	double t_1 = x * -z;
	double tmp;
	if (z <= -3.2e+77) {
		tmp = t_0;
	} else if (z <= -1.0) {
		tmp = t_1;
	} else if (z <= -6e-73) {
		tmp = x;
	} else if (z <= -8e-118) {
		tmp = t_0;
	} else if (z <= 4.4e-11) {
		tmp = x;
	} else if (z <= 95000000000000.0) {
		tmp = t_0;
	} 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 * (z * y)
    t_1 = x * -z
    if (z <= (-3.2d+77)) then
        tmp = t_0
    else if (z <= (-1.0d0)) then
        tmp = t_1
    else if (z <= (-6d-73)) then
        tmp = x
    else if (z <= (-8d-118)) then
        tmp = t_0
    else if (z <= 4.4d-11) then
        tmp = x
    else if (z <= 95000000000000.0d0) then
        tmp = t_0
    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 * (z * y);
	double t_1 = x * -z;
	double tmp;
	if (z <= -3.2e+77) {
		tmp = t_0;
	} else if (z <= -1.0) {
		tmp = t_1;
	} else if (z <= -6e-73) {
		tmp = x;
	} else if (z <= -8e-118) {
		tmp = t_0;
	} else if (z <= 4.4e-11) {
		tmp = x;
	} else if (z <= 95000000000000.0) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * (z * y)
	t_1 = x * -z
	tmp = 0
	if z <= -3.2e+77:
		tmp = t_0
	elif z <= -1.0:
		tmp = t_1
	elif z <= -6e-73:
		tmp = x
	elif z <= -8e-118:
		tmp = t_0
	elif z <= 4.4e-11:
		tmp = x
	elif z <= 95000000000000.0:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(z * y))
	t_1 = Float64(x * Float64(-z))
	tmp = 0.0
	if (z <= -3.2e+77)
		tmp = t_0;
	elseif (z <= -1.0)
		tmp = t_1;
	elseif (z <= -6e-73)
		tmp = x;
	elseif (z <= -8e-118)
		tmp = t_0;
	elseif (z <= 4.4e-11)
		tmp = x;
	elseif (z <= 95000000000000.0)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * (z * y);
	t_1 = x * -z;
	tmp = 0.0;
	if (z <= -3.2e+77)
		tmp = t_0;
	elseif (z <= -1.0)
		tmp = t_1;
	elseif (z <= -6e-73)
		tmp = x;
	elseif (z <= -8e-118)
		tmp = t_0;
	elseif (z <= 4.4e-11)
		tmp = x;
	elseif (z <= 95000000000000.0)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(z * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * (-z)), $MachinePrecision]}, If[LessEqual[z, -3.2e+77], t$95$0, If[LessEqual[z, -1.0], t$95$1, If[LessEqual[z, -6e-73], x, If[LessEqual[z, -8e-118], t$95$0, If[LessEqual[z, 4.4e-11], x, If[LessEqual[z, 95000000000000.0], t$95$0, t$95$1]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.2000000000000002e77 or -6e-73 < z < -7.99999999999999988e-118 or 4.4000000000000003e-11 < z < 9.5e13

   1. Initial program 95.0%

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

   2. Taylor expanded in y around inf 72.2%

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

      1. *-commutative 72.2%

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

   4. Simplified 72.2%

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

   if -3.2000000000000002e77 < z < -1 or 9.5e13 < z

   1. Initial program 94.9%

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

   2. Taylor expanded in z around inf 94.7%

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

      1. *-commutative 94.7%

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

      2. associate-*r* 99.7%

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

      3. *-commutative 99.7%

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

      4. sub-neg 99.7%

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

      5. metadata-eval 99.7%

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

   4. Simplified 99.7%

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

   5. Taylor expanded in y around 0 65.4%

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

      1. mul-1-neg 65.4%

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

   7. Simplified 65.4%

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

   if -1 < z < -6e-73 or -7.99999999999999988e-118 < z < 4.4000000000000003e-11

   1. Initial program 99.9%

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

   2. Taylor expanded in z around 0 81.4%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.2 \cdot 10^{+77}:\\ \;\;\;\;x \cdot \left(z \cdot y\right)\\ \mathbf{elif}\;z \leq -1:\\ \;\;\;\;x \cdot \left(-z\right)\\ \mathbf{elif}\;z \leq -6 \cdot 10^{-73}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -8 \cdot 10^{-118}:\\ \;\;\;\;x \cdot \left(z \cdot y\right)\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{-11}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 95000000000000:\\ \;\;\;\;x \cdot \left(z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-z\right)\\ \end{array} \]

Alternative 5: 85.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(x \cdot \left(y + -1\right)\right)\\ \mathbf{if}\;z \leq -0.00115:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -8.4 \cdot 10^{-72}:\\ \;\;\;\;x - x \cdot z\\ \mathbf{elif}\;z \leq -8 \cdot 10^{-118} \lor \neg \left(z \leq 1.15 \cdot 10^{-9}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (* x (+ y -1.0)))))
   (if (<= z -0.00115)
     t_0
     (if (<= z -8.4e-72)
       (- x (* x z))
       (if (or (<= z -8e-118) (not (<= z 1.15e-9))) t_0 (* x (- 1.0 z)))))))

C

double code(double x, double y, double z) {
	double t_0 = z * (x * (y + -1.0));
	double tmp;
	if (z <= -0.00115) {
		tmp = t_0;
	} else if (z <= -8.4e-72) {
		tmp = x - (x * z);
	} else if ((z <= -8e-118) || !(z <= 1.15e-9)) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = z * (x * (y + (-1.0d0)))
    if (z <= (-0.00115d0)) then
        tmp = t_0
    else if (z <= (-8.4d-72)) then
        tmp = x - (x * z)
    else if ((z <= (-8d-118)) .or. (.not. (z <= 1.15d-9))) then
        tmp = t_0
    else
        tmp = x * (1.0d0 - z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = z * (x * (y + -1.0));
	double tmp;
	if (z <= -0.00115) {
		tmp = t_0;
	} else if (z <= -8.4e-72) {
		tmp = x - (x * z);
	} else if ((z <= -8e-118) || !(z <= 1.15e-9)) {
		tmp = t_0;
	} else {
		tmp = x * (1.0 - z);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * (x * (y + -1.0))
	tmp = 0
	if z <= -0.00115:
		tmp = t_0
	elif z <= -8.4e-72:
		tmp = x - (x * z)
	elif (z <= -8e-118) or not (z <= 1.15e-9):
		tmp = t_0
	else:
		tmp = x * (1.0 - z)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(x * Float64(y + -1.0)))
	tmp = 0.0
	if (z <= -0.00115)
		tmp = t_0;
	elseif (z <= -8.4e-72)
		tmp = Float64(x - Float64(x * z));
	elseif ((z <= -8e-118) || !(z <= 1.15e-9))
		tmp = t_0;
	else
		tmp = Float64(x * Float64(1.0 - z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * (x * (y + -1.0));
	tmp = 0.0;
	if (z <= -0.00115)
		tmp = t_0;
	elseif (z <= -8.4e-72)
		tmp = x - (x * z);
	elseif ((z <= -8e-118) || ~((z <= 1.15e-9)))
		tmp = t_0;
	else
		tmp = x * (1.0 - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(x * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -0.00115], t$95$0, If[LessEqual[z, -8.4e-72], N[(x - N[(x * z), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[z, -8e-118], N[Not[LessEqual[z, 1.15e-9]], $MachinePrecision]], t$95$0, N[(x * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -0.00115 or -8.4e-72 < z < -7.99999999999999988e-118 or 1.15e-9 < z

   1. Initial program 95.0%

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

   2. Taylor expanded in z around inf 93.2%

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

      1. *-commutative 93.2%

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

      2. associate-*r* 98.1%

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

      3. *-commutative 98.1%

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

      4. sub-neg 98.1%

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

      5. metadata-eval 98.1%

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

   4. Simplified 98.1%

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

   if -0.00115 < z < -8.4e-72

   1. Initial program 99.7%

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

   2. Taylor expanded in z around 0 99.9%

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

      1. +-commutative 99.9%

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

      2. associate-*r* 99.9%

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

      3. fma-def 99.9%

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

      4. sub-neg 99.9%

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

      5. metadata-eval 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in y around 0 82.6%

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

      1. mul-1-neg 82.6%

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

      2. sub-neg 82.6%

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

   7. Simplified 82.6%

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

   if -7.99999999999999988e-118 < z < 1.15e-9

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 82.4%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.00115:\\ \;\;\;\;z \cdot \left(x \cdot \left(y + -1\right)\right)\\ \mathbf{elif}\;z \leq -8.4 \cdot 10^{-72}:\\ \;\;\;\;x - x \cdot z\\ \mathbf{elif}\;z \leq -8 \cdot 10^{-118} \lor \neg \left(z \leq 1.15 \cdot 10^{-9}\right):\\ \;\;\;\;z \cdot \left(x \cdot \left(y + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \end{array} \]

Alternative 6: 98.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot \left(1 - y\right) \leq 5 \cdot 10^{+191}:\\ \;\;\;\;x \cdot \left(1 + z \cdot \left(y + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot z\right) \cdot \left(y + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z (- 1.0 y)) 5e+191)
   (* x (+ 1.0 (* z (+ y -1.0))))
   (* (* x z) (+ y -1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * (1.0 - y)) <= 5e+191) {
		tmp = x * (1.0 + (z * (y + -1.0)));
	} else {
		tmp = (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 ((z * (1.0d0 - y)) <= 5d+191) then
        tmp = x * (1.0d0 + (z * (y + (-1.0d0))))
    else
        tmp = (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 ((z * (1.0 - y)) <= 5e+191) {
		tmp = x * (1.0 + (z * (y + -1.0)));
	} else {
		tmp = (x * z) * (y + -1.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * (1.0 - y)) <= 5e+191:
		tmp = x * (1.0 + (z * (y + -1.0)))
	else:
		tmp = (x * z) * (y + -1.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * Float64(1.0 - y)) <= 5e+191)
		tmp = Float64(x * Float64(1.0 + Float64(z * Float64(y + -1.0))));
	else
		tmp = Float64(Float64(x * z) * Float64(y + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * (1.0 - y)) <= 5e+191)
		tmp = x * (1.0 + (z * (y + -1.0)));
	else
		tmp = (x * z) * (y + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * N[(1.0 - y), $MachinePrecision]), $MachinePrecision], 5e+191], N[(x * N[(1.0 + N[(z * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * z), $MachinePrecision] * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 1 y) z) < 5.0000000000000002e191

   1. Initial program 98.6%

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

   if 5.0000000000000002e191 < (*.f64 (-.f64 1 y) z)

   1. Initial program 89.8%

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

   2. Taylor expanded in z around inf 89.8%

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

      1. *-commutative 89.8%

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

      2. associate-*r* 99.9%

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

      3. *-commutative 99.9%

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

      4. sub-neg 99.9%

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

      5. metadata-eval 99.9%

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

   4. Simplified 99.9%

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

      1. associate-*r* 99.9%

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

      2. *-commutative 99.9%

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

      3. flip-+ 83.8%

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

      4. associate-*r/ 79.9%

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

      5. metadata-eval 79.9%

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

      6. fma-neg 79.9%

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

      7. metadata-eval 79.9%

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

      8. sub-neg 79.9%

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

      9. metadata-eval 79.9%

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

   6. Applied egg-rr 79.9%

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

   7. Taylor expanded in y around 0 83.4%

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

      1. *-commutative 83.4%

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

      2. *-commutative 83.4%

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

      3. associate-*r* 93.6%

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

      4. distribute-lft-in 99.9%

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

   9. Simplified 99.9%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot \left(1 - y\right) \leq 5 \cdot 10^{+191}:\\ \;\;\;\;x \cdot \left(1 + z \cdot \left(y + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot z\right) \cdot \left(y + -1\right)\\ \end{array} \]

Alternative 7: 83.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.9 \cdot 10^{+96} \lor \neg \left(y \leq -9.2 \cdot 10^{+61} \lor \neg \left(y \leq -2.5 \cdot 10^{+38}\right) \land y \leq 1.3 \cdot 10^{+14}\right):\\ \;\;\;\;x \cdot \left(z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -3.9e+96)
         (not (or (<= y -9.2e+61) (and (not (<= y -2.5e+38)) (<= y 1.3e+14)))))
   (* x (* z y))
   (* x (- 1.0 z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -3.9e+96) || !((y <= -9.2e+61) || (!(y <= -2.5e+38) && (y <= 1.3e+14)))) {
		tmp = x * (z * y);
	} 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 <= (-3.9d+96)) .or. (.not. (y <= (-9.2d+61)) .or. (.not. (y <= (-2.5d+38))) .and. (y <= 1.3d+14))) then
        tmp = x * (z * y)
    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 <= -3.9e+96) || !((y <= -9.2e+61) || (!(y <= -2.5e+38) && (y <= 1.3e+14)))) {
		tmp = x * (z * y);
	} else {
		tmp = x * (1.0 - z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -3.9e+96) or not ((y <= -9.2e+61) or (not (y <= -2.5e+38) and (y <= 1.3e+14))):
		tmp = x * (z * y)
	else:
		tmp = x * (1.0 - z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -3.9e+96) || !((y <= -9.2e+61) || (!(y <= -2.5e+38) && (y <= 1.3e+14))))
		tmp = Float64(x * Float64(z * y));
	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 <= -3.9e+96) || ~(((y <= -9.2e+61) || (~((y <= -2.5e+38)) && (y <= 1.3e+14)))))
		tmp = x * (z * y);
	else
		tmp = x * (1.0 - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -3.9e+96], N[Not[Or[LessEqual[y, -9.2e+61], And[N[Not[LessEqual[y, -2.5e+38]], $MachinePrecision], LessEqual[y, 1.3e+14]]]], $MachinePrecision]], N[(x * N[(z * y), $MachinePrecision]), $MachinePrecision], N[(x * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.9e96 or -9.1999999999999998e61 < y < -2.49999999999999985e38 or 1.3e14 < y

   1. Initial program 92.3%

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

   2. Taylor expanded in y around inf 77.9%

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

      1. *-commutative 77.9%

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

   4. Simplified 77.9%

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

   if -3.9e96 < y < -9.1999999999999998e61 or -2.49999999999999985e38 < y < 1.3e14

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 94.3%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.9 \cdot 10^{+96} \lor \neg \left(y \leq -9.2 \cdot 10^{+61} \lor \neg \left(y \leq -2.5 \cdot 10^{+38}\right) \land y \leq 1.3 \cdot 10^{+14}\right):\\ \;\;\;\;x \cdot \left(z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \end{array} \]

Alternative 8: 84.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(x \cdot y\right)\\ t_1 := x \cdot \left(1 - z\right)\\ \mathbf{if}\;y \leq -9 \cdot 10^{+88}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1.1 \cdot 10^{+59}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -4.2 \cdot 10^{+39}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 3.9 \cdot 10^{+14}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (* x y))) (t_1 (* x (- 1.0 z))))
   (if (<= y -9e+88)
     t_0
     (if (<= y -1.1e+59)
       t_1
       (if (<= y -4.2e+39) t_0 (if (<= y 3.9e+14) t_1 (* x (* z y))))))))

C

double code(double x, double y, double z) {
	double t_0 = z * (x * y);
	double t_1 = x * (1.0 - z);
	double tmp;
	if (y <= -9e+88) {
		tmp = t_0;
	} else if (y <= -1.1e+59) {
		tmp = t_1;
	} else if (y <= -4.2e+39) {
		tmp = t_0;
	} else if (y <= 3.9e+14) {
		tmp = t_1;
	} else {
		tmp = x * (z * y);
	}
	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 = z * (x * y)
    t_1 = x * (1.0d0 - z)
    if (y <= (-9d+88)) then
        tmp = t_0
    else if (y <= (-1.1d+59)) then
        tmp = t_1
    else if (y <= (-4.2d+39)) then
        tmp = t_0
    else if (y <= 3.9d+14) then
        tmp = t_1
    else
        tmp = x * (z * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = z * (x * y);
	double t_1 = x * (1.0 - z);
	double tmp;
	if (y <= -9e+88) {
		tmp = t_0;
	} else if (y <= -1.1e+59) {
		tmp = t_1;
	} else if (y <= -4.2e+39) {
		tmp = t_0;
	} else if (y <= 3.9e+14) {
		tmp = t_1;
	} else {
		tmp = x * (z * y);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * (x * y)
	t_1 = x * (1.0 - z)
	tmp = 0
	if y <= -9e+88:
		tmp = t_0
	elif y <= -1.1e+59:
		tmp = t_1
	elif y <= -4.2e+39:
		tmp = t_0
	elif y <= 3.9e+14:
		tmp = t_1
	else:
		tmp = x * (z * y)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(x * y))
	t_1 = Float64(x * Float64(1.0 - z))
	tmp = 0.0
	if (y <= -9e+88)
		tmp = t_0;
	elseif (y <= -1.1e+59)
		tmp = t_1;
	elseif (y <= -4.2e+39)
		tmp = t_0;
	elseif (y <= 3.9e+14)
		tmp = t_1;
	else
		tmp = Float64(x * Float64(z * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * (x * y);
	t_1 = x * (1.0 - z);
	tmp = 0.0;
	if (y <= -9e+88)
		tmp = t_0;
	elseif (y <= -1.1e+59)
		tmp = t_1;
	elseif (y <= -4.2e+39)
		tmp = t_0;
	elseif (y <= 3.9e+14)
		tmp = t_1;
	else
		tmp = x * (z * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(x * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -9e+88], t$95$0, If[LessEqual[y, -1.1e+59], t$95$1, If[LessEqual[y, -4.2e+39], t$95$0, If[LessEqual[y, 3.9e+14], t$95$1, N[(x * N[(z * y), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -9e88 or -1.1e59 < y < -4.1999999999999997e39

   1. Initial program 88.8%

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

   2. Taylor expanded in y around inf 80.3%

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

      1. associate-*r* 88.7%

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

      2. *-commutative 88.7%

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

   4. Simplified 88.7%

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

   if -9e88 < y < -1.1e59 or -4.1999999999999997e39 < y < 3.9e14

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 94.3%

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

   if 3.9e14 < y

   1. Initial program 96.0%

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

   2. Taylor expanded in y around inf 75.4%

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

      1. *-commutative 75.4%

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

   4. Simplified 75.4%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{+88}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;y \leq -1.1 \cdot 10^{+59}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{elif}\;y \leq -4.2 \cdot 10^{+39}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;y \leq 3.9 \cdot 10^{+14}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z \cdot y\right)\\ \end{array} \]

Alternative 9: 64.2% accurate, 1.1× 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 94.5%

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

   2. Taylor expanded in z around inf 93.9%

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

      1. *-commutative 93.9%

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

      2. associate-*r* 99.3%

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

      3. *-commutative 99.3%

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

      4. sub-neg 99.3%

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

      5. metadata-eval 99.3%

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

   4. Simplified 99.3%

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

   5. Taylor expanded in y around 0 60.0%

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

      1. mul-1-neg 60.0%

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

   7. Simplified 60.0%

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

   if -1 < z < 1

   1. Initial program 99.9%

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

   2. Taylor expanded in z around 0 73.7%

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

4. Final simplification 66.2%

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

Alternative 10: 37.7% 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 97.0%

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

2. Taylor expanded in z around 0 35.2%

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

3. Final simplification 35.2%

   \[\leadsto x \]

Developer target: 99.7% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

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 2023332 
(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))))