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

Percentage Accurate: 95.9% → 99.9%

Time: 13.4s

Alternatives: 11

Speedup: 1.1×

• 21.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 95.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y + -1\right) \cdot \left(z \cdot x\right)\\ \mathbf{if}\;z \leq -4 \cdot 10^{+29}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 27000000000000:\\ \;\;\;\;\mathsf{fma}\left(\left(y + -1\right) \cdot z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (+ y -1.0) (* z x))))
   (if (<= z -4e+29)
     t_0
     (if (<= z 27000000000000.0) (fma (* (+ y -1.0) z) x x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (y + -1.0) * (z * x);
	double tmp;
	if (z <= -4e+29) {
		tmp = t_0;
	} else if (z <= 27000000000000.0) {
		tmp = fma(((y + -1.0) * z), x, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y + -1.0) * Float64(z * x))
	tmp = 0.0
	if (z <= -4e+29)
		tmp = t_0;
	elseif (z <= 27000000000000.0)
		tmp = fma(Float64(Float64(y + -1.0) * z), x, x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y + -1.0), $MachinePrecision] * N[(z * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -4e+29], t$95$0, If[LessEqual[z, 27000000000000.0], N[(N[(N[(y + -1.0), $MachinePrecision] * z), $MachinePrecision] * x + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.99999999999999966e29 or 2.7e13 < z

   1. Initial program 91.0%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   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 91.0

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

   5. Simplified 91.0%

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

      1. lift-*.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. neg-mul-1 N/A

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

      6. distribute-rgt-out N/A

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

      7. lift-+.f64 N/A

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

      8. *-lft-identity N/A

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

      9. rgt-mult-inverse N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l* N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

   7. Applied egg-rr 99.9%

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

   if -3.99999999999999966e29 < z < 2.7e13

   1. Initial program 99.9%

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

   4. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{+29}:\\ \;\;\;\;\left(y + -1\right) \cdot \left(z \cdot x\right)\\ \mathbf{elif}\;z \leq 27000000000000:\\ \;\;\;\;\mathsf{fma}\left(\left(y + -1\right) \cdot z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y + -1\right) \cdot \left(z \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(y \cdot z - z\right)\\ t_1 := z \cdot \left(1 - y\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;y \cdot \left(z \cdot x\right)\\ \mathbf{elif}\;t\_1 \leq -5000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 400000000:\\ \;\;\;\;\mathsf{fma}\left(-z, x, x\right)\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+304}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (- (* y z) z))) (t_1 (* z (- 1.0 y))))
   (if (<= t_1 (- INFINITY))
     (* y (* z x))
     (if (<= t_1 -5000000.0)
       t_0
       (if (<= t_1 400000000.0)
         (fma (- z) x x)
         (if (<= t_1 4e+304) t_0 (* z (* y x))))))))

C

double code(double x, double y, double z) {
	double t_0 = x * ((y * z) - z);
	double t_1 = z * (1.0 - y);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = y * (z * x);
	} else if (t_1 <= -5000000.0) {
		tmp = t_0;
	} else if (t_1 <= 400000000.0) {
		tmp = fma(-z, x, x);
	} else if (t_1 <= 4e+304) {
		tmp = t_0;
	} else {
		tmp = z * (y * x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(Float64(y * z) - z))
	t_1 = Float64(z * Float64(1.0 - y))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(y * Float64(z * x));
	elseif (t_1 <= -5000000.0)
		tmp = t_0;
	elseif (t_1 <= 400000000.0)
		tmp = fma(Float64(-z), x, x);
	elseif (t_1 <= 4e+304)
		tmp = t_0;
	else
		tmp = Float64(z * Float64(y * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(N[(y * z), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(z * N[(1.0 - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(y * N[(z * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -5000000.0], t$95$0, If[LessEqual[t$95$1, 400000000.0], N[((-z) * x + x), $MachinePrecision], If[LessEqual[t$95$1, 4e+304], t$95$0, N[(z * N[(y * x), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (-.f64 #s(literal 1 binary64) y) z) < -inf.0

   1. Initial program 76.2%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   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 99.8

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

   5. Simplified 99.8%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 100.0

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

   7. Applied egg-rr 100.0%

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

   if -inf.0 < (*.f64 (-.f64 #s(literal 1 binary64) y) z) < -5e6 or 4e8 < (*.f64 (-.f64 #s(literal 1 binary64) y) z) < 3.9999999999999998e304

   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 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 98.9

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

   5. Simplified 98.9%

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

   if -5e6 < (*.f64 (-.f64 #s(literal 1 binary64) y) z) < 4e8

   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. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 99.6

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

   5. Simplified 99.6%

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

      1. cancel-sign-sub-inv N/A

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

      2. +-commutative N/A

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

      3. neg-mul-1 N/A

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

      4. lower-fma.f64 N/A

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

      5. neg-mul-1 N/A

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

      6. lower-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 3.9999999999999998e304 < (*.f64 (-.f64 #s(literal 1 binary64) y) z)

   1. Initial program 58.6%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   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 99.8

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

   5. Simplified 99.8%

      \[\leadsto \color{blue}{z \cdot \left(y \cdot x\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot \left(1 - y\right) \leq -\infty:\\ \;\;\;\;y \cdot \left(z \cdot x\right)\\ \mathbf{elif}\;z \cdot \left(1 - y\right) \leq -5000000:\\ \;\;\;\;x \cdot \left(y \cdot z - z\right)\\ \mathbf{elif}\;z \cdot \left(1 - y\right) \leq 400000000:\\ \;\;\;\;\mathsf{fma}\left(-z, x, x\right)\\ \mathbf{elif}\;z \cdot \left(1 - y\right) \leq 4 \cdot 10^{+304}:\\ \;\;\;\;x \cdot \left(y \cdot z - z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y \cdot z, x, x\right)\\ \mathbf{if}\;1 - y \leq -0.5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;1 - y \leq 1.000000000001:\\ \;\;\;\;\mathsf{fma}\left(-z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (* y z) x x)))
   (if (<= (- 1.0 y) -0.5)
     t_0
     (if (<= (- 1.0 y) 1.000000000001) (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 ((1.0 - y) <= -0.5) {
		tmp = t_0;
	} else if ((1.0 - y) <= 1.000000000001) {
		tmp = fma(-z, x, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(Float64(y * z), x, x)
	tmp = 0.0
	if (Float64(1.0 - y) <= -0.5)
		tmp = t_0;
	elseif (Float64(1.0 - y) <= 1.000000000001)
		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[(y * z), $MachinePrecision] * x + x), $MachinePrecision]}, If[LessEqual[N[(1.0 - y), $MachinePrecision], -0.5], t$95$0, If[LessEqual[N[(1.0 - y), $MachinePrecision], 1.000000000001], N[((-z) * x + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) y) < -0.5 or 1.0000000000010001 < (-.f64 #s(literal 1 binary64) y)

   1. Initial program 90.7%

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

   4. Applied egg-rr 90.7%

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

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 88.3

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

   7. Simplified 88.3%

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

   if -0.5 < (-.f64 #s(literal 1 binary64) y) < 1.0000000000010001

   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. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.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. cancel-sign-sub-inv N/A

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

      2. +-commutative N/A

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

      3. neg-mul-1 N/A

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

      4. lower-fma.f64 N/A

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

      5. neg-mul-1 N/A

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

      6. lower-neg.f64 99.5

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

   7. Applied egg-rr 99.5%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - y \leq -0.5:\\ \;\;\;\;\mathsf{fma}\left(y \cdot z, x, x\right)\\ \mathbf{elif}\;1 - y \leq 1.000000000001:\\ \;\;\;\;\mathsf{fma}\left(-z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot z, x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(y \cdot x\right)\\ \mathbf{if}\;1 - y \leq -5 \cdot 10^{+30}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;1 - y \leq 2 \cdot 10^{+57}:\\ \;\;\;\;\mathsf{fma}\left(-z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (* y x))))
   (if (<= (- 1.0 y) -5e+30)
     t_0
     (if (<= (- 1.0 y) 2e+57) (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) <= -5e+30) {
		tmp = t_0;
	} else if ((1.0 - y) <= 2e+57) {
		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) <= -5e+30)
		tmp = t_0;
	elseif (Float64(1.0 - y) <= 2e+57)
		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], -5e+30], t$95$0, If[LessEqual[N[(1.0 - y), $MachinePrecision], 2e+57], N[((-z) * x + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) y) < -4.9999999999999998e30 or 2.0000000000000001e57 < (-.f64 #s(literal 1 binary64) y)

   1. Initial program 88.2%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   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 82.5

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

   5. Simplified 82.5%

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

   if -4.9999999999999998e30 < (-.f64 #s(literal 1 binary64) y) < 2.0000000000000001e57

   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. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 94.2

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

   5. Simplified 94.2%

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

      1. cancel-sign-sub-inv N/A

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

      2. +-commutative N/A

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

      3. neg-mul-1 N/A

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

      4. lower-fma.f64 N/A

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

      5. neg-mul-1 N/A

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

      6. lower-neg.f64 94.3

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

   7. Applied egg-rr 94.3%

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

Alternative 5: 83.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(y \cdot z\right)\\ \mathbf{if}\;1 - y \leq -5 \cdot 10^{+30}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;1 - y \leq 2 \cdot 10^{+57}:\\ \;\;\;\;\mathsf{fma}\left(-z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (* y z))))
   (if (<= (- 1.0 y) -5e+30)
     t_0
     (if (<= (- 1.0 y) 2e+57) (fma (- z) x x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * (y * z);
	double tmp;
	if ((1.0 - y) <= -5e+30) {
		tmp = t_0;
	} else if ((1.0 - y) <= 2e+57) {
		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 (Float64(1.0 - y) <= -5e+30)
		tmp = t_0;
	elseif (Float64(1.0 - y) <= 2e+57)
		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[N[(1.0 - y), $MachinePrecision], -5e+30], t$95$0, If[LessEqual[N[(1.0 - y), $MachinePrecision], 2e+57], N[((-z) * x + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) y) < -4.9999999999999998e30 or 2.0000000000000001e57 < (-.f64 #s(literal 1 binary64) y)

   1. Initial program 88.2%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   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. lower-*.f64 73.8

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

   5. Simplified 73.8%

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

   if -4.9999999999999998e30 < (-.f64 #s(literal 1 binary64) y) < 2.0000000000000001e57

   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. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 94.2

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

   5. Simplified 94.2%

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

      1. cancel-sign-sub-inv N/A

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

      2. +-commutative N/A

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

      3. neg-mul-1 N/A

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

      4. lower-fma.f64 N/A

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

      5. neg-mul-1 N/A

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

      6. lower-neg.f64 94.3

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

   7. Applied egg-rr 94.3%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - y \leq -5 \cdot 10^{+30}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;1 - y \leq 2 \cdot 10^{+57}:\\ \;\;\;\;\mathsf{fma}\left(-z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.8% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (+ y -1.0) (* z x))))
   (if (<= z -1.0) 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 = (y + -1.0) * (z * x);
	double tmp;
	if (z <= -1.0) {
		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(Float64(y + -1.0) * Float64(z * x))
	tmp = 0.0
	if (z <= -1.0)
		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[(N[(y + -1.0), $MachinePrecision] * N[(z * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.0], 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 or 1 < z

   1. Initial program 91.7%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   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 89.8

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

   5. Simplified 89.8%

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

      1. lift-*.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. neg-mul-1 N/A

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

      6. distribute-rgt-out N/A

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

      7. lift-+.f64 N/A

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

      8. *-lft-identity N/A

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

      9. rgt-mult-inverse N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l* N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

   7. Applied egg-rr 98.0%

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

   if -1 < z < 1

   1. Initial program 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 97.8

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

   7. Simplified 97.8%

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

4. Final simplification 97.9%

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

Alternative 7: 64.6% 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 91.7%

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

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

      4. *-commutative N/A

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

      5. lower-*.f64 58.4

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

   5. Simplified 58.4%

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

   6. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot z\right)} \]
   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. lower-*.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. lower-neg.f64 56.5

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

   8. Simplified 56.5%

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

   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 z around 0

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

   5. Simplified 75.5%

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

      1. *-rgt-identity 75.5

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

   7. Applied egg-rr 75.5%

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

Alternative 8: 97.9% 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 95.5%

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

4. Applied egg-rr 98.0%

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

Alternative 9: 65.6% 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 95.5%

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

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

   4. *-commutative N/A

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

   5. lower-*.f64 67.2

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

5. Simplified 67.2%

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

   1. cancel-sign-sub-inv N/A

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

   2. +-commutative N/A

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

   3. neg-mul-1 N/A

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

   4. lower-fma.f64 N/A

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

   5. neg-mul-1 N/A

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

   6. lower-neg.f64 67.2

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

7. Applied egg-rr 67.2%

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

Alternative 10: 65.6% 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 95.5%

   \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
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 67.2

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

5. Simplified 67.2%

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

Alternative 11: 37.8% 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 95.5%

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

3. Taylor expanded in z around 0

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

5. Simplified 36.4%

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

   1. *-rgt-identity 36.4

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

7. Applied egg-rr 36.4%

   \[\leadsto \color{blue}{x} \]
8. 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 2024207 
(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))))