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

Percentage Accurate: 96.1% → 97.6%

Time: 10.2s

Alternatives: 9

Speedup: 1.1×

• 20.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\left(y + -1\right) \cdot x, z, x\right)\\ \mathbf{if}\;z \leq -5.6 \cdot 10^{-55}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-232}:\\ \;\;\;\;\mathsf{fma}\left(y, z \cdot x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (* (+ y -1.0) x) z x)))
   (if (<= z -5.6e-55) t_0 (if (<= z 5e-232) (fma y (* z x) x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(((y + -1.0) * x), z, x);
	double tmp;
	if (z <= -5.6e-55) {
		tmp = t_0;
	} else if (z <= 5e-232) {
		tmp = fma(y, (z * x), x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(Float64(Float64(y + -1.0) * x), z, x)
	tmp = 0.0
	if (z <= -5.6e-55)
		tmp = t_0;
	elseif (z <= 5e-232)
		tmp = fma(y, Float64(z * x), x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(y + -1.0), $MachinePrecision] * x), $MachinePrecision] * z + x), $MachinePrecision]}, If[LessEqual[z, -5.6e-55], t$95$0, If[LessEqual[z, 5e-232], N[(y * N[(z * x), $MachinePrecision] + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.59999999999999968e-55 or 4.9999999999999999e-232 < z

   1. Initial program 91.2%

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

   4. Applied rewrites 99.4%

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

   if -5.59999999999999968e-55 < z < 4.9999999999999999e-232

   1. Initial program 99.9%

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

   4. Applied rewrites 99.4%

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

      1. lift-+.f64 N/A

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

      2. unpow1 N/A

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

      3. sqr-pow N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. metadata-eval N/A

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

      7. lower-pow.f64 N/A

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

      8. metadata-eval 52.0

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left({y}^{0.5}, {y}^{\color{blue}{0.5}}, -1\right), z \cdot x, x\right) \]

   6. Applied rewrites 52.0%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left({y}^{0.5}, {y}^{0.5}, -1\right)}, z \cdot x, x\right) \]

   7. Taylor expanded in y around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. rem-square-sqrt N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg 99.4

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

   9. Applied rewrites 99.4%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.6 \cdot 10^{-55}:\\ \;\;\;\;\mathsf{fma}\left(\left(y + -1\right) \cdot x, z, x\right)\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-232}:\\ \;\;\;\;\mathsf{fma}\left(y, z \cdot x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(y + -1\right) \cdot x, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y, z \cdot x, x\right)\\ \mathbf{if}\;1 - y \leq -1 \cdot 10^{+15}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;1 - y \leq 1.2:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma y (* z x) x)))
   (if (<= (- 1.0 y) -1e+15) t_0 (if (<= (- 1.0 y) 1.2) (* x (- 1.0 z)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(y, (z * x), x);
	double tmp;
	if ((1.0 - y) <= -1e+15) {
		tmp = t_0;
	} else if ((1.0 - y) <= 1.2) {
		tmp = x * (1.0 - z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(y, Float64(z * x), x)
	tmp = 0.0
	if (Float64(1.0 - y) <= -1e+15)
		tmp = t_0;
	elseif (Float64(1.0 - y) <= 1.2)
		tmp = Float64(x * Float64(1.0 - z));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(z * x), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[N[(1.0 - y), $MachinePrecision], -1e+15], t$95$0, If[LessEqual[N[(1.0 - y), $MachinePrecision], 1.2], N[(x * N[(1.0 - z), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) y) < -1e15 or 1.19999999999999996 < (-.f64 #s(literal 1 binary64) y)

   1. Initial program 87.4%

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

   4. Applied rewrites 98.3%

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

      1. lift-+.f64 N/A

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

      2. unpow1 N/A

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

      3. sqr-pow N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. metadata-eval N/A

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

      7. lower-pow.f64 N/A

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

      8. metadata-eval 41.6

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left({y}^{0.5}, {y}^{\color{blue}{0.5}}, -1\right), z \cdot x, x\right) \]

   6. Applied rewrites 41.6%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left({y}^{0.5}, {y}^{0.5}, -1\right)}, z \cdot x, x\right) \]

   7. Taylor expanded in y around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. rem-square-sqrt N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg 98.3

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

   9. Applied rewrites 98.3%

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

   if -1e15 < (-.f64 #s(literal 1 binary64) y) < 1.19999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 99.5

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

   5. Applied rewrites 99.5%

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

Alternative 3: 93.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y \cdot x, z, x\right)\\ \mathbf{if}\;1 - y \leq -1 \cdot 10^{+15}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;1 - y \leq 10^{+63}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (* y x) z x)))
   (if (<= (- 1.0 y) -1e+15)
     t_0
     (if (<= (- 1.0 y) 1e+63) (* x (- 1.0 z)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma((y * x), z, x);
	double tmp;
	if ((1.0 - y) <= -1e+15) {
		tmp = t_0;
	} else if ((1.0 - y) <= 1e+63) {
		tmp = x * (1.0 - z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(Float64(y * x), z, x)
	tmp = 0.0
	if (Float64(1.0 - y) <= -1e+15)
		tmp = t_0;
	elseif (Float64(1.0 - y) <= 1e+63)
		tmp = Float64(x * Float64(1.0 - z));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y * x), $MachinePrecision] * z + x), $MachinePrecision]}, If[LessEqual[N[(1.0 - y), $MachinePrecision], -1e+15], t$95$0, If[LessEqual[N[(1.0 - y), $MachinePrecision], 1e+63], N[(x * N[(1.0 - z), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) y) < -1e15 or 1.00000000000000006e63 < (-.f64 #s(literal 1 binary64) y)

   1. Initial program 86.5%

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

   4. Applied rewrites 98.2%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 91.0

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

   6. Applied rewrites 91.0%

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

   7. Taylor expanded in y around inf

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

      1. lower-*.f64 91.0

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

   9. Applied rewrites 91.0%

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

   if -1e15 < (-.f64 #s(literal 1 binary64) y) < 1.00000000000000006e63

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 98.8

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

   5. Applied rewrites 98.8%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - y \leq -1 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, z, x\right)\\ \mathbf{elif}\;1 - y \leq 10^{+63}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(z \cdot x\right)\\ \mathbf{if}\;1 - y \leq -1 \cdot 10^{+147}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;1 - y \leq 5 \cdot 10^{+64}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (* z x))))
   (if (<= (- 1.0 y) -1e+147)
     t_0
     (if (<= (- 1.0 y) 5e+64) (* x (- 1.0 z)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = y * (z * x);
	double tmp;
	if ((1.0 - y) <= -1e+147) {
		tmp = t_0;
	} else if ((1.0 - y) <= 5e+64) {
		tmp = x * (1.0 - 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 = y * (z * x)
    if ((1.0d0 - y) <= (-1d+147)) then
        tmp = t_0
    else if ((1.0d0 - y) <= 5d+64) then
        tmp = x * (1.0d0 - 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 = y * (z * x);
	double tmp;
	if ((1.0 - y) <= -1e+147) {
		tmp = t_0;
	} else if ((1.0 - y) <= 5e+64) {
		tmp = x * (1.0 - z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * (z * x)
	tmp = 0
	if (1.0 - y) <= -1e+147:
		tmp = t_0
	elif (1.0 - y) <= 5e+64:
		tmp = x * (1.0 - z)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(z * x))
	tmp = 0.0
	if (Float64(1.0 - y) <= -1e+147)
		tmp = t_0;
	elseif (Float64(1.0 - y) <= 5e+64)
		tmp = Float64(x * Float64(1.0 - z));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * (z * x);
	tmp = 0.0;
	if ((1.0 - y) <= -1e+147)
		tmp = t_0;
	elseif ((1.0 - y) <= 5e+64)
		tmp = x * (1.0 - z);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(z * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 - y), $MachinePrecision], -1e+147], t$95$0, If[LessEqual[N[(1.0 - y), $MachinePrecision], 5e+64], N[(x * N[(1.0 - z), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) y) < -9.9999999999999998e146 or 5e64 < (-.f64 #s(literal 1 binary64) y)

   1. Initial program 84.4%

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

   4. Applied rewrites 97.9%

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

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 82.2

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

   7. Applied rewrites 82.2%

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

   if -9.9999999999999998e146 < (-.f64 #s(literal 1 binary64) y) < 5e64

   1. Initial program 99.4%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 94.0

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

   5. Applied rewrites 94.0%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - y \leq -1 \cdot 10^{+147}:\\ \;\;\;\;y \cdot \left(z \cdot x\right)\\ \mathbf{elif}\;1 - y \leq 5 \cdot 10^{+64}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(y \cdot x\right)\\ \mathbf{if}\;1 - y \leq -1 \cdot 10^{+147}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;1 - y \leq 5 \cdot 10^{+64}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (* y x))))
   (if (<= (- 1.0 y) -1e+147)
     t_0
     (if (<= (- 1.0 y) 5e+64) (* x (- 1.0 z)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = z * (y * x);
	double tmp;
	if ((1.0 - y) <= -1e+147) {
		tmp = t_0;
	} else if ((1.0 - y) <= 5e+64) {
		tmp = x * (1.0 - 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 = z * (y * x)
    if ((1.0d0 - y) <= (-1d+147)) then
        tmp = t_0
    else if ((1.0d0 - y) <= 5d+64) then
        tmp = x * (1.0d0 - 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 = z * (y * x);
	double tmp;
	if ((1.0 - y) <= -1e+147) {
		tmp = t_0;
	} else if ((1.0 - y) <= 5e+64) {
		tmp = x * (1.0 - z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * (y * x)
	tmp = 0
	if (1.0 - y) <= -1e+147:
		tmp = t_0
	elif (1.0 - y) <= 5e+64:
		tmp = x * (1.0 - z)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(y * x))
	tmp = 0.0
	if (Float64(1.0 - y) <= -1e+147)
		tmp = t_0;
	elseif (Float64(1.0 - y) <= 5e+64)
		tmp = Float64(x * Float64(1.0 - z));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * (y * x);
	tmp = 0.0;
	if ((1.0 - y) <= -1e+147)
		tmp = t_0;
	elseif ((1.0 - y) <= 5e+64)
		tmp = x * (1.0 - z);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(y * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 - y), $MachinePrecision], -1e+147], t$95$0, If[LessEqual[N[(1.0 - y), $MachinePrecision], 5e+64], N[(x * N[(1.0 - z), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) y) < -9.9999999999999998e146 or 5e64 < (-.f64 #s(literal 1 binary64) y)

   1. Initial program 84.4%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 80.3

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

   5. Applied rewrites 80.3%

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

   if -9.9999999999999998e146 < (-.f64 #s(literal 1 binary64) y) < 5e64

   1. Initial program 99.4%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 94.0

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

   5. Applied rewrites 94.0%

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

Alternative 6: 64.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(-z\right)\\ \mathbf{if}\;z \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 0.0085:\\ \;\;\;\;x \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (- z))))
   (if (<= z -1.0) t_0 (if (<= z 0.0085) (* x 1.0) 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 <= 0.0085) {
		tmp = x * 1.0;
	} 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 <= 0.0085d0) then
        tmp = x * 1.0d0
    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 <= 0.0085) {
		tmp = x * 1.0;
	} 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 <= 0.0085:
		tmp = x * 1.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(-z))
	tmp = 0.0
	if (z <= -1.0)
		tmp = t_0;
	elseif (z <= 0.0085)
		tmp = Float64(x * 1.0);
	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 <= 0.0085)
		tmp = x * 1.0;
	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, 0.0085], N[(x * 1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 0.0085000000000000006 < z

   1. Initial program 88.0%

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

   3. Taylor expanded in z around inf

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

      1. distribute-rgt-out-- N/A

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

      2. *-lft-identity N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 87.1

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

   5. Applied rewrites 87.1%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 53.5%

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

   if -1 < z < 0.0085000000000000006

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 79.4%

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

Alternative 7: 98.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 93.7%

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

4. Applied rewrites 99.2%

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

Alternative 8: 66.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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 - z)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.7%

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

3. Taylor expanded in y around 0

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

   1. lower--.f64 66.7

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

5. Applied rewrites 66.7%

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

Alternative 9: 38.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ x \cdot 1 \end{array} \]

FPCore

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

C

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

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.7%

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

3. Taylor expanded in z around 0

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

5. Applied rewrites 39.8%

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

Developer Target 1: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(1 - \left(1 - y\right) \cdot z\right)\\ t_1 := x + \left(1 - y\right) \cdot \left(\left(-z\right) \cdot x\right)\\ \mathbf{if}\;t\_0 < -1.618195973607049 \cdot 10^{+50}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 < 3.892237649663903 \cdot 10^{+134}:\\ \;\;\;\;\left(x \cdot y\right) \cdot z - \left(x \cdot z - x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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 2024221 
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSV:hsv from colour-2.3.3, J"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< (* x (- 1 (* (- 1 y) z))) -161819597360704900000000000000000000000000000000000) (+ x (* (- 1 y) (* (- z) x))) (if (< (* x (- 1 (* (- 1 y) z))) 389223764966390300000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- (* (* x y) z) (- (* x z) x)) (+ x (* (- 1 y) (* (- z) x))))))

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