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

Percentage Accurate: 96.1% → 97.8%

Time: 8.5s

Alternatives: 9

Speedup: 0.6×

• 20.6% 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: 97.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot \left(1 - y\right) \leq -4 \cdot 10^{+151}:\\ \;\;\;\;\left(\left(y - 1\right) \cdot x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(y - 1\right) \cdot z, x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z (- 1.0 y)) -4e+151)
   (* (* (- y 1.0) x) z)
   (fma (* (- y 1.0) z) x x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * (1.0 - y)) <= -4e+151) {
		tmp = ((y - 1.0) * x) * z;
	} else {
		tmp = fma(((y - 1.0) * z), x, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * Float64(1.0 - y)) <= -4e+151)
		tmp = Float64(Float64(Float64(y - 1.0) * x) * z);
	else
		tmp = fma(Float64(Float64(y - 1.0) * z), x, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * N[(1.0 - y), $MachinePrecision]), $MachinePrecision], -4e+151], N[(N[(N[(y - 1.0), $MachinePrecision] * x), $MachinePrecision] * z), $MachinePrecision], N[(N[(N[(y - 1.0), $MachinePrecision] * z), $MachinePrecision] * x + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 #s(literal 1 binary64) y) z) < -4.00000000000000007e151

   1. Initial program 84.0%

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

   3. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-commutative N/A

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

      7. cancel-sign-sub N/A

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

      8. mul-1-neg N/A

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

      9. *-rgt-identity N/A

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

      10. distribute-lft-out-- N/A

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

      11. mul-1-neg N/A

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

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

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

      13. *-commutative N/A

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

      14. associate-*r* N/A

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

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

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

   5. Applied rewrites 99.9%

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

   if -4.00000000000000007e151 < (*.f64 (-.f64 #s(literal 1 binary64) y) z)

   1. Initial program 98.5%

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

   4. Applied rewrites 98.5%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot \left(1 - y\right) \leq -4 \cdot 10^{+151}:\\ \;\;\;\;\left(\left(y - 1\right) \cdot x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(y - 1\right) \cdot z, x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(1 - y\right)\\ t_1 := \left(\left(y - 1\right) \cdot x\right) \cdot z\\ \mathbf{if}\;t\_0 \leq -200000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(-z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (- 1.0 y))) (t_1 (* (* (- y 1.0) x) z)))
   (if (<= t_0 -200000000.0) t_1 (if (<= t_0 1e+15) (fma (- z) x x) t_1))))

C

double code(double x, double y, double z) {
	double t_0 = z * (1.0 - y);
	double t_1 = ((y - 1.0) * x) * z;
	double tmp;
	if (t_0 <= -200000000.0) {
		tmp = t_1;
	} else if (t_0 <= 1e+15) {
		tmp = fma(-z, x, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(1.0 - y))
	t_1 = Float64(Float64(Float64(y - 1.0) * x) * z)
	tmp = 0.0
	if (t_0 <= -200000000.0)
		tmp = t_1;
	elseif (t_0 <= 1e+15)
		tmp = fma(Float64(-z), x, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(1.0 - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(y - 1.0), $MachinePrecision] * x), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[t$95$0, -200000000.0], t$95$1, If[LessEqual[t$95$0, 1e+15], N[((-z) * x + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 #s(literal 1 binary64) y) z) < -2e8 or 1e15 < (*.f64 (-.f64 #s(literal 1 binary64) y) z)

   1. Initial program 92.2%

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

   3. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-commutative N/A

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

      7. cancel-sign-sub N/A

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

      8. mul-1-neg N/A

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

      9. *-rgt-identity N/A

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

      10. distribute-lft-out-- N/A

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

      11. mul-1-neg N/A

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

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

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

      13. *-commutative N/A

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

      14. associate-*r* N/A

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

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

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

   5. Applied rewrites 98.7%

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

   if -2e8 < (*.f64 (-.f64 #s(literal 1 binary64) y) z) < 1e15

   1. Initial program 100.0%

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 97.7

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

   7. Applied rewrites 97.7%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot \left(1 - y\right) \leq -200000000:\\ \;\;\;\;\left(\left(y - 1\right) \cdot x\right) \cdot z\\ \mathbf{elif}\;z \cdot \left(1 - y\right) \leq 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(-z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y - 1\right) \cdot x\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(z \cdot y, x, x\right)\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.02:\\ \;\;\;\;\mathsf{fma}\left(-z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (* z y) x x)))
   (if (<= y -1.0) t_0 (if (<= y 0.02) (fma (- z) x x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma((z * y), x, x);
	double tmp;
	if (y <= -1.0) {
		tmp = t_0;
	} else if (y <= 0.02) {
		tmp = fma(-z, x, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(Float64(z * y), x, x)
	tmp = 0.0
	if (y <= -1.0)
		tmp = t_0;
	elseif (y <= 0.02)
		tmp = fma(Float64(-z), x, x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(z * y), $MachinePrecision] * x + x), $MachinePrecision]}, If[LessEqual[y, -1.0], t$95$0, If[LessEqual[y, 0.02], N[((-z) * x + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 0.0200000000000000004 < y

   1. Initial program 90.1%

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

   4. Applied rewrites 90.1%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 29.4

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

   7. Applied rewrites 29.4%

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

   8. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 88.8

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

   10. Applied rewrites 88.8%

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

   if -1 < y < 0.0200000000000000004

   1. Initial program 100.0%

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 100.0

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

   7. Applied rewrites 100.0%

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

Alternative 4: 85.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot y\right) \cdot z\\ \mathbf{if}\;y \leq -19500:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 320:\\ \;\;\;\;\mathsf{fma}\left(-z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (* x y) z)))
   (if (<= y -19500.0) t_0 (if (<= y 320.0) (fma (- z) x x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (x * y) * z;
	double tmp;
	if (y <= -19500.0) {
		tmp = t_0;
	} else if (y <= 320.0) {
		tmp = fma(-z, x, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x * y) * z)
	tmp = 0.0
	if (y <= -19500.0)
		tmp = t_0;
	elseif (y <= 320.0)
		tmp = fma(Float64(-z), x, x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x * y), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[y, -19500.0], t$95$0, If[LessEqual[y, 320.0], N[((-z) * x + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -19500 or 320 < y

   1. Initial program 89.9%

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

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

      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 70.4

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

   5. Applied rewrites 70.4%

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

   7. Applied rewrites 75.7%

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

   if -19500 < y < 320

   1. Initial program 100.0%

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 100.0

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

   7. Applied rewrites 100.0%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -19500:\\ \;\;\;\;\left(x \cdot y\right) \cdot z\\ \mathbf{elif}\;y \leq 320:\\ \;\;\;\;\mathsf{fma}\left(-z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot z\right) \cdot y\\ \mathbf{if}\;y \leq -1.35 \cdot 10^{+137}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 320:\\ \;\;\;\;\mathsf{fma}\left(-z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (* x z) y)))
   (if (<= y -1.35e+137) t_0 (if (<= y 320.0) (fma (- z) x x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (x * z) * y;
	double tmp;
	if (y <= -1.35e+137) {
		tmp = t_0;
	} else if (y <= 320.0) {
		tmp = fma(-z, x, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x * z) * y)
	tmp = 0.0
	if (y <= -1.35e+137)
		tmp = t_0;
	elseif (y <= 320.0)
		tmp = fma(Float64(-z), x, x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x * z), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[y, -1.35e+137], t$95$0, If[LessEqual[y, 320.0], N[((-z) * x + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.35000000000000009e137 or 320 < y

   1. Initial program 88.0%

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

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

      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 75.3

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

   5. Applied rewrites 75.3%

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

   if -1.35000000000000009e137 < y < 320

   1. Initial program 99.4%

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 92.0

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

   7. Applied rewrites 92.0%

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

Alternative 6: 65.5% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- x) z))) (if (<= z -1.0) t_0 (if (<= z 1.0) (* 1.0 x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = -x * z;
	double tmp;
	if (z <= -1.0) {
		tmp = t_0;
	} else if (z <= 1.0) {
		tmp = 1.0 * x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -x * z
    if (z <= (-1.0d0)) then
        tmp = t_0
    else if (z <= 1.0d0) then
        tmp = 1.0d0 * x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

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

Python

def code(x, y, z):
	t_0 = -x * z
	tmp = 0
	if z <= -1.0:
		tmp = t_0
	elif z <= 1.0:
		tmp = 1.0 * x
	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 <= 1.0)
		tmp = Float64(1.0 * x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = -x * z;
	tmp = 0.0;
	if (z <= -1.0)
		tmp = t_0;
	elseif (z <= 1.0)
		tmp = 1.0 * x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if z < -1 or 1 < z

   1. Initial program 90.7%

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

   3. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-commutative N/A

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

      7. cancel-sign-sub N/A

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

      8. mul-1-neg N/A

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

      9. *-rgt-identity N/A

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

      10. distribute-lft-out-- N/A

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

      11. mul-1-neg N/A

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

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

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

      13. *-commutative N/A

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

      14. associate-*r* N/A

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

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

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

   5. Applied rewrites 98.8%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 55.5%

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

   if -1 < z < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 75.0

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

   5. Applied rewrites 75.0%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 72.2%

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

4. Final simplification 63.6%

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

Alternative 7: 66.6% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (x y z) :precision binary64 (fma (- z) x x))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 95.1%

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

4. Applied rewrites 95.2%

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

5. Taylor expanded in y around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 65.5

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

7. Applied rewrites 65.5%

   \[\leadsto \mathsf{fma}\left(\color{blue}{-z}, x, x\right) \]
8. Add Preprocessing

Alternative 8: 66.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (1.0 - z) * 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 = (1.0d0 - z) * x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.1%

   \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
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 65.5

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

5. Applied rewrites 65.5%

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

6. Final simplification 65.5%

   \[\leadsto \left(1 - z\right) \cdot x \]
7. Add Preprocessing

Alternative 9: 38.6% accurate, 2.8× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return 1.0 * 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 = 1.0d0 * x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.1%

   \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
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 65.5

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

5. Applied rewrites 65.5%

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

6. Taylor expanded in z around 0

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

8. Applied rewrites 36.6%

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

9. Final simplification 36.6%

   \[\leadsto 1 \cdot x \]
10. Add Preprocessing

Developer Target 1: 99.8% 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 2024294 
(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))))