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

Percentage Accurate: 96.1% → 98.8%

Time: 11.0s

Alternatives: 12

Speedup: 1.1×

• 21.2% 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, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(y \cdot x - x\right)\\ \mathbf{if}\;z \leq -1.06:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\mathsf{fma}\left(y \cdot z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (- (* y x) x))))
   (if (<= z -1.06) t_0 (if (<= z 1.0) (fma (* y z) x x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = z * ((y * x) - x);
	double tmp;
	if (z <= -1.06) {
		tmp = t_0;
	} else if (z <= 1.0) {
		tmp = fma((y * z), x, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(Float64(y * x) - x))
	tmp = 0.0
	if (z <= -1.06)
		tmp = t_0;
	elseif (z <= 1.0)
		tmp = fma(Float64(y * z), x, x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(N[(y * x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.06], t$95$0, If[LessEqual[z, 1.0], N[(N[(y * z), $MachinePrecision] * x + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.0600000000000001 or 1 < z

   1. Initial program 92.7%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   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. *-commutative N/A

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

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

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

      14. associate-*r* N/A

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

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

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

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

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

      17. *-lowering-*.f64 N/A

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

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

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

      19. *-lft-identity N/A

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

      20. unsub-neg N/A

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

      21. +-commutative N/A

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

      22. distribute-neg-in N/A

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

      23. remove-double-neg N/A

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

      24. unsub-neg N/A

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

      25. --lowering--.f64 N/A

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

      26. *-lowering-*.f64 98.4

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

   5. Simplified 98.4%

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

   if -1.0600000000000001 < z < 1

   1. Initial program 99.8%

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

   4. Applied egg-rr 95.0%

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

      1. *-commutative N/A

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 99.7

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

   9. Simplified 99.7%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.06:\\ \;\;\;\;z \cdot \left(y \cdot x - x\right)\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\mathsf{fma}\left(y \cdot z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot x - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.0% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 88.6%

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

   4. Applied egg-rr 97.0%

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

   5. Taylor expanded in y around inf

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

   7. Simplified 95.8%

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

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

   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

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

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

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

      2. *-rgt-identity N/A

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

      3. --lowering--.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 99.5

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

   5. Simplified 99.5%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. neg-lowering-neg.f64 99.6

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

   7. Applied egg-rr 99.6%

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

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

   1. Initial program 96.9%

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

   4. Applied egg-rr 93.2%

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

      1. *-commutative N/A

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

   6. Applied egg-rr 96.9%

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

   7. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 94.0

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

   9. Simplified 94.0%

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

4. Final simplification 97.1%

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

Alternative 3: 85.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - y \leq -1.5 \cdot 10^{+17}:\\ \;\;\;\;y \cdot \left(z \cdot x\right)\\ \mathbf{elif}\;1 - y \leq 5 \cdot 10^{+36}:\\ \;\;\;\;\mathsf{fma}\left(-z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (- 1.0 y) -1.5e+17)
   (* y (* z x))
   (if (<= (- 1.0 y) 5e+36) (fma (- z) x x) (* z (* y x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((1.0 - y) <= -1.5e+17) {
		tmp = y * (z * x);
	} else if ((1.0 - y) <= 5e+36) {
		tmp = fma(-z, x, x);
	} else {
		tmp = z * (y * x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(1.0 - y) <= -1.5e+17)
		tmp = Float64(y * Float64(z * x));
	elseif (Float64(1.0 - y) <= 5e+36)
		tmp = fma(Float64(-z), x, x);
	else
		tmp = Float64(z * Float64(y * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(1.0 - y), $MachinePrecision], -1.5e+17], N[(y * N[(z * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 - y), $MachinePrecision], 5e+36], N[((-z) * x + x), $MachinePrecision], N[(z * N[(y * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 #s(literal 1 binary64) y) < -1.5e17

   1. Initial program 88.3%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 61.3

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

   5. Simplified 61.3%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 69.8

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

   7. Applied egg-rr 69.8%

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

   if -1.5e17 < (-.f64 #s(literal 1 binary64) y) < 4.99999999999999977e36

   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

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

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

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

      2. *-rgt-identity N/A

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

      3. --lowering--.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 96.2

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

   5. Simplified 96.2%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. neg-lowering-neg.f64 96.2

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

   7. Applied egg-rr 96.2%

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

   if 4.99999999999999977e36 < (-.f64 #s(literal 1 binary64) y)

   1. Initial program 96.5%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   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. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 81.1

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

   5. Simplified 81.1%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - y \leq -1.5 \cdot 10^{+17}:\\ \;\;\;\;y \cdot \left(z \cdot x\right)\\ \mathbf{elif}\;1 - y \leq 5 \cdot 10^{+36}:\\ \;\;\;\;\mathsf{fma}\left(-z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(y \cdot x\right)\\ \mathbf{if}\;1 - y \leq -1.5 \cdot 10^{+17}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;1 - y \leq 5 \cdot 10^{+36}:\\ \;\;\;\;\mathsf{fma}\left(-z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (* y x))))
   (if (<= (- 1.0 y) -1.5e+17)
     t_0
     (if (<= (- 1.0 y) 5e+36) (fma (- z) x x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = z * (y * x);
	double tmp;
	if ((1.0 - y) <= -1.5e+17) {
		tmp = t_0;
	} else if ((1.0 - y) <= 5e+36) {
		tmp = fma(-z, x, x);
	} 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) <= -1.5e+17)
		tmp = t_0;
	elseif (Float64(1.0 - y) <= 5e+36)
		tmp = fma(Float64(-z), x, x);
	else
		tmp = t_0;
	end
	return 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], -1.5e+17], t$95$0, If[LessEqual[N[(1.0 - y), $MachinePrecision], 5e+36], N[((-z) * x + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) y) < -1.5e17 or 4.99999999999999977e36 < (-.f64 #s(literal 1 binary64) y)

   1. Initial program 92.1%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   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. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 73.2

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

   5. Simplified 73.2%

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

   if -1.5e17 < (-.f64 #s(literal 1 binary64) y) < 4.99999999999999977e36

   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

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

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

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

      2. *-rgt-identity N/A

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

      3. --lowering--.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 96.2

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

   5. Simplified 96.2%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. neg-lowering-neg.f64 96.2

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

   7. Applied egg-rr 96.2%

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

Alternative 5: 97.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 92.7%

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

   4. Applied egg-rr 95.2%

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

   5. Taylor expanded in y around inf

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

   7. Simplified 93.1%

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

   if -1 < y < 1

   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

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

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

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

      2. *-rgt-identity N/A

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

      3. --lowering--.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 99.5

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

   5. Simplified 99.5%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. neg-lowering-neg.f64 99.6

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

   7. Applied egg-rr 99.6%

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

Alternative 6: 96.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - y \leq -1 \cdot 10^{+37}:\\ \;\;\;\;\mathsf{fma}\left(y, z \cdot x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(y + -1\right) \cdot z, x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (- 1.0 y) -1e+37) (fma y (* z x) x) (fma (* (+ y -1.0) z) x x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((1.0 - y) <= -1e+37) {
		tmp = fma(y, (z * x), x);
	} else {
		tmp = fma(((y + -1.0) * z), x, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(1.0 - y) <= -1e+37)
		tmp = fma(y, Float64(z * x), x);
	else
		tmp = fma(Float64(Float64(y + -1.0) * z), x, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(1.0 - y), $MachinePrecision], -1e+37], N[(y * N[(z * x), $MachinePrecision] + x), $MachinePrecision], N[(N[(N[(y + -1.0), $MachinePrecision] * z), $MachinePrecision] * x + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) y) < -9.99999999999999954e36

   1. Initial program 87.8%

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

   4. Applied egg-rr 96.8%

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

   5. Taylor expanded in y around inf

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

   7. Simplified 96.8%

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

   if -9.99999999999999954e36 < (-.f64 #s(literal 1 binary64) y)

   1. Initial program 98.9%

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

   4. Applied egg-rr 98.9%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - y \leq -1 \cdot 10^{+37}:\\ \;\;\;\;\mathsf{fma}\left(y, z \cdot x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(y + -1\right) \cdot z, x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 83.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(y \cdot z\right)\\ \mathbf{if}\;y \leq -2.8 \cdot 10^{+67}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+17}:\\ \;\;\;\;\mathsf{fma}\left(-z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (* y z))))
   (if (<= y -2.8e+67) t_0 (if (<= y 1.5e+17) (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 <= -2.8e+67) {
		tmp = t_0;
	} else if (y <= 1.5e+17) {
		tmp = fma(-z, x, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(y * z))
	tmp = 0.0
	if (y <= -2.8e+67)
		tmp = t_0;
	elseif (y <= 1.5e+17)
		tmp = fma(Float64(-z), x, x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(y * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.8e+67], t$95$0, If[LessEqual[y, 1.5e+17], N[((-z) * x + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.7999999999999998e67 or 1.5e17 < y

   1. Initial program 91.7%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 71.3

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

   5. Simplified 71.3%

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

   if -2.7999999999999998e67 < y < 1.5e17

   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

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

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

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

      2. *-rgt-identity N/A

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

      3. --lowering--.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 94.4

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

   5. Simplified 94.4%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. neg-lowering-neg.f64 94.4

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

   7. Applied egg-rr 94.4%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{+67}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+17}:\\ \;\;\;\;\mathsf{fma}\left(-z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 65.6% accurate, 0.8× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	double t_0 = z * -x;
	double tmp;
	if (z <= -1.0) {
		tmp = t_0;
	} else if (z <= 1.0) {
		tmp = 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 = z * -x
    if (z <= (-1.0d0)) then
        tmp = t_0
    else if (z <= 1.0d0) then
        tmp = 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 = z * -x;
	double tmp;
	if (z <= -1.0) {
		tmp = t_0;
	} else if (z <= 1.0) {
		tmp = x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if z < -1 or 1 < z

   1. Initial program 92.7%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   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. *-commutative N/A

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

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

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

      14. associate-*r* N/A

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

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

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

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

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

      17. *-lowering-*.f64 N/A

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

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

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

      19. *-lft-identity N/A

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

      20. unsub-neg N/A

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

      21. +-commutative N/A

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

      22. distribute-neg-in N/A

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

      23. remove-double-neg N/A

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

      24. unsub-neg N/A

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

      25. --lowering--.f64 N/A

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

      26. *-lowering-*.f64 98.4

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

   5. Simplified 98.4%

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

   6. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. neg-lowering-neg.f64 57.2

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

   8. Simplified 57.2%

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

   if -1 < z < 1

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0

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

   5. Simplified 69.0%

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

Alternative 9: 97.8% 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 96.2%

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

4. Applied egg-rr 97.5%

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

Alternative 10: 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 96.2%

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

3. Taylor expanded in y around 0

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

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

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

   2. *-rgt-identity N/A

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

   3. --lowering--.f64 N/A

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

   4. *-commutative N/A

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

   5. *-lowering-*.f64 63.8

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

5. Simplified 63.8%

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

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

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

   4. accelerator-lowering-fma.f64 N/A

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

   5. neg-lowering-neg.f64 63.8

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

7. Applied egg-rr 63.8%

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

Alternative 11: 66.6% 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 96.2%

   \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
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. --lowering--.f64 63.8

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

5. Simplified 63.8%

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

Alternative 12: 38.5% accurate, 17.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x
end function

Java

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

Python

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

Julia

function code(x, y, z)
	return x
end

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 96.2%

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

3. Taylor expanded in z around 0

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

5. Simplified 35.7%

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

Developer Target 1: 99.6% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x * (1.0d0 - ((1.0d0 - y) * z))
    t_1 = x + ((1.0d0 - y) * (-z * x))
    if (t_0 < (-1.618195973607049d+50)) then
        tmp = t_1
    else if (t_0 < 3.892237649663903d+134) then
        tmp = ((x * y) * z) - ((x * z) - x)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

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

Python

def code(x, y, z):
	t_0 = x * (1.0 - ((1.0 - y) * z))
	t_1 = x + ((1.0 - y) * (-z * x))
	tmp = 0
	if t_0 < -1.618195973607049e+50:
		tmp = t_1
	elif t_0 < 3.892237649663903e+134:
		tmp = ((x * y) * z) - ((x * z) - x)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(1.0 - Float64(Float64(1.0 - y) * z)))
	t_1 = Float64(x + Float64(Float64(1.0 - y) * Float64(Float64(-z) * x)))
	tmp = 0.0
	if (t_0 < -1.618195973607049e+50)
		tmp = t_1;
	elseif (t_0 < 3.892237649663903e+134)
		tmp = Float64(Float64(Float64(x * y) * z) - Float64(Float64(x * z) - x));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * (1.0 - ((1.0 - y) * z));
	t_1 = x + ((1.0 - y) * (-z * x));
	tmp = 0.0;
	if (t_0 < -1.618195973607049e+50)
		tmp = t_1;
	elseif (t_0 < 3.892237649663903e+134)
		tmp = ((x * y) * z) - ((x * z) - x);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(1.0 - N[(N[(1.0 - y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x + N[(N[(1.0 - y), $MachinePrecision] * N[((-z) * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[t$95$0, -1.618195973607049e+50], t$95$1, If[Less[t$95$0, 3.892237649663903e+134], N[(N[(N[(x * y), $MachinePrecision] * z), $MachinePrecision] - N[(N[(x * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Reproduce

herbie shell --seed 2024198 
(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))))