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

Percentage Accurate: 96.1% → 99.7%

Time: 8.6s

Alternatives: 8

Speedup: 0.4×

• 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: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \left(1 - y\right) \cdot z\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(y \cdot x\right) \cdot z\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+119}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(-1 + y, z, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y - 1\right) \cdot x\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (* (- 1.0 y) z))))
   (if (<= t_0 (- INFINITY))
     (* (* y x) z)
     (if (<= t_0 4e+119) (* x (fma (+ -1.0 y) z 1.0)) (* (* (- y 1.0) x) z)))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 - ((1.0 - y) * z);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (y * x) * z;
	} else if (t_0 <= 4e+119) {
		tmp = x * fma((-1.0 + y), z, 1.0);
	} else {
		tmp = ((y - 1.0) * x) * z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(1.0 - Float64(Float64(1.0 - y) * z))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(y * x) * z);
	elseif (t_0 <= 4e+119)
		tmp = Float64(x * fma(Float64(-1.0 + y), z, 1.0));
	else
		tmp = Float64(Float64(Float64(y - 1.0) * x) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(1.0 - N[(N[(1.0 - y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(y * x), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[t$95$0, 4e+119], N[(x * N[(N[(-1.0 + y), $MachinePrecision] * z + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y - 1.0), $MachinePrecision] * x), $MachinePrecision] * z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 59.0%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 99.8

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

   5. Applied rewrites 99.8%

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

   7. Applied rewrites 99.9%

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

   if -inf.0 < (-.f64 #s(literal 1 binary64) (*.f64 (-.f64 #s(literal 1 binary64) y) z)) < 3.99999999999999978e119

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

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

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

      2. mul-1-neg N/A

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

      3. associate-*r* N/A

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

      4. associate--l- N/A

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

      5. associate-*r* N/A

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

      6. distribute-lft1-in N/A

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

      7. +-commutative N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

      9. metadata-eval N/A

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

      10. *-lft-identity N/A

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

      11. fp-cancel-sub-sign-inv N/A

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

      12. lft-mult-inverse N/A

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

      13. mul-1-neg N/A

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

      14. distribute-rgt-in N/A

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

      15. +-commutative N/A

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

      16. distribute-rgt-in N/A

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

      17. lft-mult-inverse N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 100.0%

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

   if 3.99999999999999978e119 < (-.f64 #s(literal 1 binary64) (*.f64 (-.f64 #s(literal 1 binary64) y) z))

   1. Initial program 91.6%

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

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 66.8

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

   5. Applied rewrites 66.8%

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

   7. Applied rewrites 61.3%

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

   8. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 99.9

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

   10. Applied rewrites 99.9%

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

   11. Taylor expanded in z around inf

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. lower-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. lower--.f64 99.9

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

   13. Applied rewrites 99.9%

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

Alternative 2: 96.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \left(1 - y\right) \cdot z\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+36} \lor \neg \left(t\_0 \leq 50000\right):\\ \;\;\;\;\left(\left(y - 1\right) \cdot x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (* (- 1.0 y) z))))
   (if (or (<= t_0 -5e+36) (not (<= t_0 50000.0)))
     (* (* (- y 1.0) x) z)
     (* x (- 1.0 z)))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 - ((1.0 - y) * z);
	double tmp;
	if ((t_0 <= -5e+36) || !(t_0 <= 50000.0)) {
		tmp = ((y - 1.0) * x) * z;
	} else {
		tmp = x * (1.0 - z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - ((1.0d0 - y) * z)
    if ((t_0 <= (-5d+36)) .or. (.not. (t_0 <= 50000.0d0))) then
        tmp = ((y - 1.0d0) * x) * z
    else
        tmp = x * (1.0d0 - z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = 1.0 - ((1.0 - y) * z);
	double tmp;
	if ((t_0 <= -5e+36) || !(t_0 <= 50000.0)) {
		tmp = ((y - 1.0) * x) * z;
	} else {
		tmp = x * (1.0 - z);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 - ((1.0 - y) * z)
	tmp = 0
	if (t_0 <= -5e+36) or not (t_0 <= 50000.0):
		tmp = ((y - 1.0) * x) * z
	else:
		tmp = x * (1.0 - z)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(1.0 - Float64(Float64(1.0 - y) * z))
	tmp = 0.0
	if ((t_0 <= -5e+36) || !(t_0 <= 50000.0))
		tmp = Float64(Float64(Float64(y - 1.0) * x) * z);
	else
		tmp = Float64(x * Float64(1.0 - z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 1.0 - ((1.0 - y) * z);
	tmp = 0.0;
	if ((t_0 <= -5e+36) || ~((t_0 <= 50000.0)))
		tmp = ((y - 1.0) * x) * z;
	else
		tmp = x * (1.0 - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(1.0 - N[(N[(1.0 - y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -5e+36], N[Not[LessEqual[t$95$0, 50000.0]], $MachinePrecision]], N[(N[(N[(y - 1.0), $MachinePrecision] * x), $MachinePrecision] * z), $MachinePrecision], N[(x * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (*.f64 (-.f64 #s(literal 1 binary64) y) z)) < -4.99999999999999977e36 or 5e4 < (-.f64 #s(literal 1 binary64) (*.f64 (-.f64 #s(literal 1 binary64) y) z))

   1. Initial program 91.9%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 63.7

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

   5. Applied rewrites 63.7%

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

   7. Applied rewrites 60.2%

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

   8. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 99.2

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

   10. Applied rewrites 99.2%

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

   11. Taylor expanded in z around inf

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. lower-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. lower--.f64 99.2

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

   13. Applied rewrites 99.2%

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

   if -4.99999999999999977e36 < (-.f64 #s(literal 1 binary64) (*.f64 (-.f64 #s(literal 1 binary64) y) z)) < 5e4

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 99.1

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

   5. Applied rewrites 99.1%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - \left(1 - y\right) \cdot z \leq -5 \cdot 10^{+36} \lor \neg \left(1 - \left(1 - y\right) \cdot z \leq 50000\right):\\ \;\;\;\;\left(\left(y - 1\right) \cdot x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-35}:\\ \;\;\;\;\mathsf{fma}\left(y - 1, x, \frac{x}{z}\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(-1 + y, z, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 2e-35)
   (* (fma (- y 1.0) x (/ x z)) z)
   (* x (fma (+ -1.0 y) z 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 2e-35) {
		tmp = fma((y - 1.0), x, (x / z)) * z;
	} else {
		tmp = x * fma((-1.0 + y), z, 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 2e-35)
		tmp = Float64(fma(Float64(y - 1.0), x, Float64(x / z)) * z);
	else
		tmp = Float64(x * fma(Float64(-1.0 + y), z, 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 2e-35], N[(N[(N[(y - 1.0), $MachinePrecision] * x + N[(x / z), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], N[(x * N[(N[(-1.0 + y), $MachinePrecision] * z + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.00000000000000002e-35

   1. Initial program 93.2%

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

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 37.5

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

   5. Applied rewrites 37.5%

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

   7. Applied rewrites 38.2%

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

   8. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 90.1

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

   10. Applied rewrites 90.1%

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

   if 2.00000000000000002e-35 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

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

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

      2. mul-1-neg N/A

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

      3. associate-*r* N/A

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

      4. associate--l- N/A

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

      5. associate-*r* N/A

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

      6. distribute-lft1-in N/A

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

      7. +-commutative N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

      9. metadata-eval N/A

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

      10. *-lft-identity N/A

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

      11. fp-cancel-sub-sign-inv N/A

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

      12. lft-mult-inverse N/A

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

      13. mul-1-neg N/A

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

      14. distribute-rgt-in N/A

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

      15. +-commutative N/A

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

      16. distribute-rgt-in N/A

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

      17. lft-mult-inverse N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 99.9%

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

Alternative 4: 85.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{+30} \lor \neg \left(y \leq 1.8 \cdot 10^{+23}\right):\\ \;\;\;\;\left(y \cdot x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -7e+30) (not (<= y 1.8e+23))) (* (* y x) z) (* x (- 1.0 z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -7e+30) || !(y <= 1.8e+23)) {
		tmp = (y * x) * z;
	} else {
		tmp = x * (1.0 - z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-7d+30)) .or. (.not. (y <= 1.8d+23))) then
        tmp = (y * x) * z
    else
        tmp = x * (1.0d0 - z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -7e+30) || !(y <= 1.8e+23)) {
		tmp = (y * x) * z;
	} else {
		tmp = x * (1.0 - z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -7e+30) or not (y <= 1.8e+23):
		tmp = (y * x) * z
	else:
		tmp = x * (1.0 - z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -7e+30) || !(y <= 1.8e+23))
		tmp = Float64(Float64(y * x) * z);
	else
		tmp = Float64(x * Float64(1.0 - z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -7e+30) || ~((y <= 1.8e+23)))
		tmp = (y * x) * z;
	else
		tmp = x * (1.0 - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -7e+30], N[Not[LessEqual[y, 1.8e+23]], $MachinePrecision]], N[(N[(y * x), $MachinePrecision] * z), $MachinePrecision], N[(x * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -7.00000000000000042e30 or 1.7999999999999999e23 < 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

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 78.7

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

   5. Applied rewrites 78.7%

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

   7. Applied rewrites 79.8%

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

   if -7.00000000000000042e30 < y < 1.7999999999999999e23

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 95.6

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

   5. Applied rewrites 95.6%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{+30} \lor \neg \left(y \leq 1.8 \cdot 10^{+23}\right):\\ \;\;\;\;\left(y \cdot x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{+30}:\\ \;\;\;\;\left(y \cdot x\right) \cdot z\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{+23}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -7e+30)
   (* (* y x) z)
   (if (<= y 1.8e+23) (* x (- 1.0 z)) (* (* z x) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -7e+30) {
		tmp = (y * x) * z;
	} else if (y <= 1.8e+23) {
		tmp = x * (1.0 - z);
	} else {
		tmp = (z * x) * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-7d+30)) then
        tmp = (y * x) * z
    else if (y <= 1.8d+23) then
        tmp = x * (1.0d0 - z)
    else
        tmp = (z * x) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -7e+30) {
		tmp = (y * x) * z;
	} else if (y <= 1.8e+23) {
		tmp = x * (1.0 - z);
	} else {
		tmp = (z * x) * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -7e+30:
		tmp = (y * x) * z
	elif y <= 1.8e+23:
		tmp = x * (1.0 - z)
	else:
		tmp = (z * x) * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -7e+30)
		tmp = Float64(Float64(y * x) * z);
	elseif (y <= 1.8e+23)
		tmp = Float64(x * Float64(1.0 - z));
	else
		tmp = Float64(Float64(z * x) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -7e+30)
		tmp = (y * x) * z;
	elseif (y <= 1.8e+23)
		tmp = x * (1.0 - z);
	else
		tmp = (z * x) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -7e+30], N[(N[(y * x), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[y, 1.8e+23], N[(x * N[(1.0 - z), $MachinePrecision]), $MachinePrecision], N[(N[(z * x), $MachinePrecision] * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.00000000000000042e30

   1. Initial program 86.5%

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

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 77.8

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

   5. Applied rewrites 77.8%

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

   7. Applied rewrites 80.3%

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

   if -7.00000000000000042e30 < y < 1.7999999999999999e23

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 95.6

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

   5. Applied rewrites 95.6%

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

   if 1.7999999999999999e23 < y

   1. Initial program 89.6%

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

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 79.4

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

   5. Applied rewrites 79.4%

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

Alternative 6: 65.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.9 \cdot 10^{-7} \lor \neg \left(z \leq 1.25 \cdot 10^{+22}\right):\\ \;\;\;\;x \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -3.9e-7) (not (<= z 1.25e+22))) (* x (- z)) (* x 1.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -3.9e-7) || !(z <= 1.25e+22)) {
		tmp = x * -z;
	} else {
		tmp = x * 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-3.9d-7)) .or. (.not. (z <= 1.25d+22))) then
        tmp = x * -z
    else
        tmp = x * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -3.9e-7) || !(z <= 1.25e+22)) {
		tmp = x * -z;
	} else {
		tmp = x * 1.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -3.9e-7) or not (z <= 1.25e+22):
		tmp = x * -z
	else:
		tmp = x * 1.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -3.9e-7) || !(z <= 1.25e+22))
		tmp = Float64(x * Float64(-z));
	else
		tmp = Float64(x * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -3.9e-7) || ~((z <= 1.25e+22)))
		tmp = x * -z;
	else
		tmp = x * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -3.9e-7], N[Not[LessEqual[z, 1.25e+22]], $MachinePrecision]], N[(x * (-z)), $MachinePrecision], N[(x * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.90000000000000025e-7 or 1.2499999999999999e22 < z

   1. Initial program 90.9%

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

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

      3. fp-cancel-sign-sub-inv N/A

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

      4. distribute-rgt-in N/A

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

      5. remove-double-neg N/A

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

      6. mul-1-neg N/A

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

      7. metadata-eval N/A

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

      8. distribute-neg-in N/A

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

      9. +-commutative N/A

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

      10. fp-cancel-sign-sub-inv N/A

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

      11. metadata-eval N/A

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

      12. *-lft-identity N/A

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

      13. mul-1-neg N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. mul-1-neg N/A

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

      17. *-lft-identity N/A

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

      18. metadata-eval N/A

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

      19. fp-cancel-sign-sub-inv N/A

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

      20. distribute-neg-in N/A

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

      21. metadata-eval N/A

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

      22. mul-1-neg N/A

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

      23. remove-double-neg N/A

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

      24. lower-+.f64 90.4

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

   5. Applied rewrites 90.4%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 56.1%

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

   if -3.90000000000000025e-7 < z < 1.2499999999999999e22

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

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

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

      2. mul-1-neg N/A

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

      3. associate-*r* N/A

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

      4. associate--l- N/A

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

      5. associate-*r* N/A

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

      6. distribute-lft1-in N/A

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

      7. +-commutative N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

      9. metadata-eval N/A

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

      10. *-lft-identity N/A

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

      11. fp-cancel-sub-sign-inv N/A

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

      12. lft-mult-inverse N/A

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

      13. mul-1-neg N/A

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

      14. distribute-rgt-in N/A

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

      15. +-commutative N/A

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

      16. distribute-rgt-in N/A

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

      17. lft-mult-inverse N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 81.3%

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

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.9 \cdot 10^{-7} \lor \neg \left(z \leq 1.25 \cdot 10^{+22}\right):\\ \;\;\;\;x \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.2% 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 95.1%

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

3. Taylor expanded in y around 0

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

   1. lower--.f64 69.1

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

5. Applied rewrites 69.1%

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

Alternative 8: 39.4% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.1%

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

3. Taylor expanded in y around 0

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

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

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

   2. mul-1-neg N/A

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

   3. associate-*r* N/A

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

   4. associate--l- N/A

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

   5. associate-*r* N/A

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

   6. distribute-lft1-in N/A

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

   7. +-commutative N/A

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

   8. fp-cancel-sign-sub-inv N/A

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

   9. metadata-eval N/A

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

   10. *-lft-identity N/A

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

   11. fp-cancel-sub-sign-inv N/A

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

   12. lft-mult-inverse N/A

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

   13. mul-1-neg N/A

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

   14. distribute-rgt-in N/A

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

   15. +-commutative N/A

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

   16. distribute-rgt-in N/A

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

   17. lft-mult-inverse N/A

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

   18. lower-fma.f64 N/A

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

5. Applied rewrites 95.1%

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

6. Taylor expanded in z around 0

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

8. Applied rewrites 40.2%

   \[\leadsto x \cdot \color{blue}{1} \]
9. 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 2024338 
(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))))