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

Percentage Accurate: 95.9% → 99.7%

Time: 18.2s

Alternatives: 10

Speedup: 0.6×

• 20.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 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: 99.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2 \cdot 10^{-51}:\\ \;\;\;\;z \cdot \left(x\_m \cdot \left(y + -1\right) + \frac{x\_m}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \left(1 + z \cdot \left(y + -1\right)\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= x_m 2e-51)
    (* z (+ (* x_m (+ y -1.0)) (/ x_m z)))
    (* x_m (+ 1.0 (* z (+ y -1.0)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (x_m <= 2e-51) {
		tmp = z * ((x_m * (y + -1.0)) + (x_m / z));
	} else {
		tmp = x_m * (1.0 + (z * (y + -1.0)));
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x_m <= 2d-51) then
        tmp = z * ((x_m * (y + (-1.0d0))) + (x_m / z))
    else
        tmp = x_m * (1.0d0 + (z * (y + (-1.0d0))))
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (x_m <= 2e-51) {
		tmp = z * ((x_m * (y + -1.0)) + (x_m / z));
	} else {
		tmp = x_m * (1.0 + (z * (y + -1.0)));
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if x_m <= 2e-51:
		tmp = z * ((x_m * (y + -1.0)) + (x_m / z))
	else:
		tmp = x_m * (1.0 + (z * (y + -1.0)))
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (x_m <= 2e-51)
		tmp = Float64(z * Float64(Float64(x_m * Float64(y + -1.0)) + Float64(x_m / z)));
	else
		tmp = Float64(x_m * Float64(1.0 + Float64(z * Float64(y + -1.0))));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if (x_m <= 2e-51)
		tmp = z * ((x_m * (y + -1.0)) + (x_m / z));
	else
		tmp = x_m * (1.0 + (z * (y + -1.0)));
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[x$95$m, 2e-51], N[(z * N[(N[(x$95$m * N[(y + -1.0), $MachinePrecision]), $MachinePrecision] + N[(x$95$m / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(1.0 + N[(z * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2e-51

   1. Initial program 95.6%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \left(z \cdot \color{blue}{\left(1 - y\right)}\right)\right)\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \left(z \cdot \left(1 + \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right)\right)\right) \]

      3. distribute-lft-in N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \left(z \cdot 1 + \color{blue}{z \cdot \left(\mathsf{neg}\left(y\right)\right)}\right)\right)\right) \]

      4. fma-define N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \left(\mathsf{fma}\left(z, \color{blue}{1}, z \cdot \left(\mathsf{neg}\left(y\right)\right)\right)\right)\right)\right) \]

      5. distribute-rgt-neg-out N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \left(\mathsf{fma}\left(z, 1, \mathsf{neg}\left(z \cdot y\right)\right)\right)\right)\right) \]

      6. fmm-undef N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \left(z \cdot 1 - \color{blue}{z \cdot y}\right)\right)\right) \]

      7. *-rgt-identity N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \left(z - \color{blue}{z} \cdot y\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{\_.f64}\left(z, \color{blue}{\left(z \cdot y\right)}\right)\right)\right) \]

      9. *-lowering-*.f64 95.6%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{\_.f64}\left(z, \mathsf{*.f64}\left(z, \color{blue}{y}\right)\right)\right)\right) \]

   4. Applied egg-rr 95.6%

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

   5. Taylor expanded in z around inf

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

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

         \[\leadsto \mathsf{*.f64}\left(z, \color{blue}{\left(x \cdot \left(y - 1\right) + \frac{x}{z}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(x \cdot \left(y - 1\right)\right), \color{blue}{\left(\frac{x}{z}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(y - 1\right)\right), \left(\frac{\color{blue}{x}}{z}\right)\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(y + \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(\frac{x}{z}\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(y + -1\right)\right), \left(\frac{x}{z}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(y, -1\right)\right), \left(\frac{x}{z}\right)\right)\right) \]

      7. /-lowering-/.f64 89.7%

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(y, -1\right)\right), \mathsf{/.f64}\left(x, \color{blue}{z}\right)\right)\right) \]

   7. Simplified 89.7%

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

   if 2e-51 < x

   1. Initial program 100.0%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-51}:\\ \;\;\;\;z \cdot \left(x \cdot \left(y + -1\right) + \frac{x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + z \cdot \left(y + -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot \left(1 - y\right) \leq 10^{+308}:\\ \;\;\;\;x\_m \cdot \left(1 + \left(z \cdot y - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(x\_m \cdot y\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= (* z (- 1.0 y)) 1e+308)
    (* x_m (+ 1.0 (- (* z y) z)))
    (* z (* x_m y)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((z * (1.0 - y)) <= 1e+308) {
		tmp = x_m * (1.0 + ((z * y) - z));
	} else {
		tmp = z * (x_m * y);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * (1.0d0 - y)) <= 1d+308) then
        tmp = x_m * (1.0d0 + ((z * y) - z))
    else
        tmp = z * (x_m * y)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((z * (1.0 - y)) <= 1e+308) {
		tmp = x_m * (1.0 + ((z * y) - z));
	} else {
		tmp = z * (x_m * y);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if (z * (1.0 - y)) <= 1e+308:
		tmp = x_m * (1.0 + ((z * y) - z))
	else:
		tmp = z * (x_m * y)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(z * Float64(1.0 - y)) <= 1e+308)
		tmp = Float64(x_m * Float64(1.0 + Float64(Float64(z * y) - z)));
	else
		tmp = Float64(z * Float64(x_m * y));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if ((z * (1.0 - y)) <= 1e+308)
		tmp = x_m * (1.0 + ((z * y) - z));
	else
		tmp = z * (x_m * y);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(z * N[(1.0 - y), $MachinePrecision]), $MachinePrecision], 1e+308], N[(x$95$m * N[(1.0 + N[(N[(z * y), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[(x$95$m * y), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 #s(literal 1 binary64) y) z) < 1e308

   1. Initial program 98.8%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \left(z \cdot \color{blue}{\left(1 - y\right)}\right)\right)\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \left(z \cdot \left(1 + \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right)\right)\right) \]

      3. distribute-lft-in N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \left(z \cdot 1 + \color{blue}{z \cdot \left(\mathsf{neg}\left(y\right)\right)}\right)\right)\right) \]

      4. fma-define N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \left(\mathsf{fma}\left(z, \color{blue}{1}, z \cdot \left(\mathsf{neg}\left(y\right)\right)\right)\right)\right)\right) \]

      5. distribute-rgt-neg-out N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \left(\mathsf{fma}\left(z, 1, \mathsf{neg}\left(z \cdot y\right)\right)\right)\right)\right) \]

      6. fmm-undef N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \left(z \cdot 1 - \color{blue}{z \cdot y}\right)\right)\right) \]

      7. *-rgt-identity N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \left(z - \color{blue}{z} \cdot y\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{\_.f64}\left(z, \color{blue}{\left(z \cdot y\right)}\right)\right)\right) \]

      9. *-lowering-*.f64 98.8%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{\_.f64}\left(z, \mathsf{*.f64}\left(z, \color{blue}{y}\right)\right)\right)\right) \]

   4. Applied egg-rr 98.8%

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

   if 1e308 < (*.f64 (-.f64 #s(literal 1 binary64) y) z)

   1. Initial program 68.4%

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

   3. Taylor expanded in y around inf

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

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

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(y \cdot z\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(z \cdot \color{blue}{y}\right)\right) \]

      3. *-lowering-*.f64 68.4%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(z, \color{blue}{y}\right)\right) \]

   5. Simplified 68.4%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(y \cdot x\right), \color{blue}{z}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(x \cdot y\right), z\right) \]

      6. *-lowering-*.f64 99.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right) \]

   7. Applied egg-rr 99.8%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot \left(1 - y\right) \leq 10^{+308}:\\ \;\;\;\;x \cdot \left(1 + \left(z \cdot y - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot \left(1 - y\right) \leq 10^{+308}:\\ \;\;\;\;x\_m \cdot \left(1 + z \cdot \left(y + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(x\_m \cdot y\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= (* z (- 1.0 y)) 1e+308)
    (* x_m (+ 1.0 (* z (+ y -1.0))))
    (* z (* x_m y)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((z * (1.0 - y)) <= 1e+308) {
		tmp = x_m * (1.0 + (z * (y + -1.0)));
	} else {
		tmp = z * (x_m * y);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * (1.0d0 - y)) <= 1d+308) then
        tmp = x_m * (1.0d0 + (z * (y + (-1.0d0))))
    else
        tmp = z * (x_m * y)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((z * (1.0 - y)) <= 1e+308) {
		tmp = x_m * (1.0 + (z * (y + -1.0)));
	} else {
		tmp = z * (x_m * y);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if (z * (1.0 - y)) <= 1e+308:
		tmp = x_m * (1.0 + (z * (y + -1.0)))
	else:
		tmp = z * (x_m * y)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(z * Float64(1.0 - y)) <= 1e+308)
		tmp = Float64(x_m * Float64(1.0 + Float64(z * Float64(y + -1.0))));
	else
		tmp = Float64(z * Float64(x_m * y));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if ((z * (1.0 - y)) <= 1e+308)
		tmp = x_m * (1.0 + (z * (y + -1.0)));
	else
		tmp = z * (x_m * y);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(z * N[(1.0 - y), $MachinePrecision]), $MachinePrecision], 1e+308], N[(x$95$m * N[(1.0 + N[(z * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[(x$95$m * y), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 #s(literal 1 binary64) y) z) < 1e308

   1. Initial program 98.8%

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

   if 1e308 < (*.f64 (-.f64 #s(literal 1 binary64) y) z)

   1. Initial program 68.4%

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

   3. Taylor expanded in y around inf

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

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

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(y \cdot z\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(z \cdot \color{blue}{y}\right)\right) \]

      3. *-lowering-*.f64 68.4%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(z, \color{blue}{y}\right)\right) \]

   5. Simplified 68.4%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(y \cdot x\right), \color{blue}{z}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(x \cdot y\right), z\right) \]

      6. *-lowering-*.f64 99.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right) \]

   7. Applied egg-rr 99.8%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot \left(1 - y\right) \leq 10^{+308}:\\ \;\;\;\;x \cdot \left(1 + z \cdot \left(y + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \left(y + -1\right) \cdot \left(x\_m \cdot z\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -0.026:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 0.000105:\\ \;\;\;\;x\_m \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (* (+ y -1.0) (* x_m z))))
   (* x_s (if (<= z -0.026) t_0 (if (<= z 0.000105) (* x_m (- 1.0 z)) t_0)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double t_0 = (y + -1.0) * (x_m * z);
	double tmp;
	if (z <= -0.026) {
		tmp = t_0;
	} else if (z <= 0.000105) {
		tmp = x_m * (1.0 - z);
	} else {
		tmp = t_0;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (y + (-1.0d0)) * (x_m * z)
    if (z <= (-0.026d0)) then
        tmp = t_0
    else if (z <= 0.000105d0) then
        tmp = x_m * (1.0d0 - z)
    else
        tmp = t_0
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double t_0 = (y + -1.0) * (x_m * z);
	double tmp;
	if (z <= -0.026) {
		tmp = t_0;
	} else if (z <= 0.000105) {
		tmp = x_m * (1.0 - z);
	} else {
		tmp = t_0;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	t_0 = (y + -1.0) * (x_m * z)
	tmp = 0
	if z <= -0.026:
		tmp = t_0
	elif z <= 0.000105:
		tmp = x_m * (1.0 - z)
	else:
		tmp = t_0
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	t_0 = Float64(Float64(y + -1.0) * Float64(x_m * z))
	tmp = 0.0
	if (z <= -0.026)
		tmp = t_0;
	elseif (z <= 0.000105)
		tmp = Float64(x_m * Float64(1.0 - z));
	else
		tmp = t_0;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	t_0 = (y + -1.0) * (x_m * z);
	tmp = 0.0;
	if (z <= -0.026)
		tmp = t_0;
	elseif (z <= 0.000105)
		tmp = x_m * (1.0 - z);
	else
		tmp = t_0;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := Block[{t$95$0 = N[(N[(y + -1.0), $MachinePrecision] * N[(x$95$m * z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[z, -0.026], t$95$0, If[LessEqual[z, 0.000105], N[(x$95$m * N[(1.0 - z), $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -0.0259999999999999988 or 1.05e-4 < z

   1. Initial program 93.5%

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

   3. Taylor expanded in z around inf

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

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

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(z \cdot \left(y - 1\right)\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(z, \color{blue}{\left(y - 1\right)}\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(z, \left(y + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(z, \left(y + -1\right)\right)\right) \]

      5. +-lowering-+.f64 91.7%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(y, \color{blue}{-1}\right)\right)\right) \]

   5. Simplified 91.7%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(y + -1\right), \color{blue}{\left(x \cdot z\right)}\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(y, -1\right), \left(\color{blue}{x} \cdot z\right)\right) \]

      5. *-lowering-*.f64 98.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(y, -1\right), \mathsf{*.f64}\left(x, \color{blue}{z}\right)\right) \]

   7. Applied egg-rr 98.2%

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

   if -0.0259999999999999988 < z < 1.05e-4

   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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 - z\right)}\right) \]

      2. --lowering--.f64 82.6%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \color{blue}{z}\right)\right) \]

   5. Simplified 82.6%

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

Alternative 5: 83.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{+120}:\\ \;\;\;\;y \cdot \left(x\_m \cdot z\right)\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{+49}:\\ \;\;\;\;x\_m \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \left(z \cdot \left(y + -1\right)\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= y -6e+120)
    (* y (* x_m z))
    (if (<= y 4.2e+49) (* x_m (- 1.0 z)) (* x_m (* z (+ y -1.0)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (y <= -6e+120) {
		tmp = y * (x_m * z);
	} else if (y <= 4.2e+49) {
		tmp = x_m * (1.0 - z);
	} else {
		tmp = x_m * (z * (y + -1.0));
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-6d+120)) then
        tmp = y * (x_m * z)
    else if (y <= 4.2d+49) then
        tmp = x_m * (1.0d0 - z)
    else
        tmp = x_m * (z * (y + (-1.0d0)))
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (y <= -6e+120) {
		tmp = y * (x_m * z);
	} else if (y <= 4.2e+49) {
		tmp = x_m * (1.0 - z);
	} else {
		tmp = x_m * (z * (y + -1.0));
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if y <= -6e+120:
		tmp = y * (x_m * z)
	elif y <= 4.2e+49:
		tmp = x_m * (1.0 - z)
	else:
		tmp = x_m * (z * (y + -1.0))
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (y <= -6e+120)
		tmp = Float64(y * Float64(x_m * z));
	elseif (y <= 4.2e+49)
		tmp = Float64(x_m * Float64(1.0 - z));
	else
		tmp = Float64(x_m * Float64(z * Float64(y + -1.0)));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if (y <= -6e+120)
		tmp = y * (x_m * z);
	elseif (y <= 4.2e+49)
		tmp = x_m * (1.0 - z);
	else
		tmp = x_m * (z * (y + -1.0));
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[y, -6e+120], N[(y * N[(x$95$m * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.2e+49], N[(x$95$m * N[(1.0 - z), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(z * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -6e120

   1. Initial program 88.0%

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

   3. Taylor expanded in y around inf

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

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

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(y \cdot z\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(z \cdot \color{blue}{y}\right)\right) \]

      3. *-lowering-*.f64 81.0%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(z, \color{blue}{y}\right)\right) \]

   5. Simplified 81.0%

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

      1. associate-*r* N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(x \cdot z\right), \color{blue}{y}\right) \]

      3. *-lowering-*.f64 85.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, z\right), y\right) \]

   7. Applied egg-rr 85.6%

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

   if -6e120 < y < 4.20000000000000022e49

   1. Initial program 98.4%

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

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 - z\right)}\right) \]

      2. --lowering--.f64 91.9%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \color{blue}{z}\right)\right) \]

   5. Simplified 91.9%

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

   if 4.20000000000000022e49 < y

   1. Initial program 97.4%

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

   3. Taylor expanded in z around inf

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

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

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(z \cdot \left(y - 1\right)\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(z, \color{blue}{\left(y - 1\right)}\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(z, \left(y + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(z, \left(y + -1\right)\right)\right) \]

      5. +-lowering-+.f64 75.1%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(y, \color{blue}{-1}\right)\right)\right) \]

   5. Simplified 75.1%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{+120}:\\ \;\;\;\;y \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{+49}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z \cdot \left(y + -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -5.1 \cdot 10^{+120}:\\ \;\;\;\;y \cdot \left(x\_m \cdot z\right)\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+49}:\\ \;\;\;\;x\_m \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \left(z \cdot y\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= y -5.1e+120)
    (* y (* x_m z))
    (if (<= y 6.5e+49) (* x_m (- 1.0 z)) (* x_m (* z y))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (y <= -5.1e+120) {
		tmp = y * (x_m * z);
	} else if (y <= 6.5e+49) {
		tmp = x_m * (1.0 - z);
	} else {
		tmp = x_m * (z * y);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-5.1d+120)) then
        tmp = y * (x_m * z)
    else if (y <= 6.5d+49) then
        tmp = x_m * (1.0d0 - z)
    else
        tmp = x_m * (z * y)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (y <= -5.1e+120) {
		tmp = y * (x_m * z);
	} else if (y <= 6.5e+49) {
		tmp = x_m * (1.0 - z);
	} else {
		tmp = x_m * (z * y);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if y <= -5.1e+120:
		tmp = y * (x_m * z)
	elif y <= 6.5e+49:
		tmp = x_m * (1.0 - z)
	else:
		tmp = x_m * (z * y)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (y <= -5.1e+120)
		tmp = Float64(y * Float64(x_m * z));
	elseif (y <= 6.5e+49)
		tmp = Float64(x_m * Float64(1.0 - z));
	else
		tmp = Float64(x_m * Float64(z * y));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if (y <= -5.1e+120)
		tmp = y * (x_m * z);
	elseif (y <= 6.5e+49)
		tmp = x_m * (1.0 - z);
	else
		tmp = x_m * (z * y);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[y, -5.1e+120], N[(y * N[(x$95$m * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6.5e+49], N[(x$95$m * N[(1.0 - z), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(z * y), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -5.10000000000000027e120

   1. Initial program 88.0%

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

   3. Taylor expanded in y around inf

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

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

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(y \cdot z\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(z \cdot \color{blue}{y}\right)\right) \]

      3. *-lowering-*.f64 81.0%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(z, \color{blue}{y}\right)\right) \]

   5. Simplified 81.0%

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

      1. associate-*r* N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(x \cdot z\right), \color{blue}{y}\right) \]

      3. *-lowering-*.f64 85.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, z\right), y\right) \]

   7. Applied egg-rr 85.6%

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

   if -5.10000000000000027e120 < y < 6.5000000000000005e49

   1. Initial program 98.4%

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

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 - z\right)}\right) \]

      2. --lowering--.f64 91.9%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \color{blue}{z}\right)\right) \]

   5. Simplified 91.9%

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

   if 6.5000000000000005e49 < y

   1. Initial program 97.4%

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

   3. Taylor expanded in y around inf

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

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

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(y \cdot z\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(z \cdot \color{blue}{y}\right)\right) \]

      3. *-lowering-*.f64 75.1%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(z, \color{blue}{y}\right)\right) \]

   5. Simplified 75.1%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.1 \cdot 10^{+120}:\\ \;\;\;\;y \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+49}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 83.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -4.6 \cdot 10^{+120}:\\ \;\;\;\;z \cdot \left(x\_m \cdot y\right)\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{+52}:\\ \;\;\;\;x\_m \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \left(z \cdot y\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= y -4.6e+120)
    (* z (* x_m y))
    (if (<= y 2.7e+52) (* x_m (- 1.0 z)) (* x_m (* z y))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (y <= -4.6e+120) {
		tmp = z * (x_m * y);
	} else if (y <= 2.7e+52) {
		tmp = x_m * (1.0 - z);
	} else {
		tmp = x_m * (z * y);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-4.6d+120)) then
        tmp = z * (x_m * y)
    else if (y <= 2.7d+52) then
        tmp = x_m * (1.0d0 - z)
    else
        tmp = x_m * (z * y)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (y <= -4.6e+120) {
		tmp = z * (x_m * y);
	} else if (y <= 2.7e+52) {
		tmp = x_m * (1.0 - z);
	} else {
		tmp = x_m * (z * y);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if y <= -4.6e+120:
		tmp = z * (x_m * y)
	elif y <= 2.7e+52:
		tmp = x_m * (1.0 - z)
	else:
		tmp = x_m * (z * y)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (y <= -4.6e+120)
		tmp = Float64(z * Float64(x_m * y));
	elseif (y <= 2.7e+52)
		tmp = Float64(x_m * Float64(1.0 - z));
	else
		tmp = Float64(x_m * Float64(z * y));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if (y <= -4.6e+120)
		tmp = z * (x_m * y);
	elseif (y <= 2.7e+52)
		tmp = x_m * (1.0 - z);
	else
		tmp = x_m * (z * y);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[y, -4.6e+120], N[(z * N[(x$95$m * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.7e+52], N[(x$95$m * N[(1.0 - z), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(z * y), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -4.59999999999999985e120

   1. Initial program 88.0%

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

   3. Taylor expanded in y around inf

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

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

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(y \cdot z\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(z \cdot \color{blue}{y}\right)\right) \]

      3. *-lowering-*.f64 81.0%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(z, \color{blue}{y}\right)\right) \]

   5. Simplified 81.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(y \cdot x\right), \color{blue}{z}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(x \cdot y\right), z\right) \]

      6. *-lowering-*.f64 81.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right) \]

   7. Applied egg-rr 81.3%

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

   if -4.59999999999999985e120 < y < 2.7e52

   1. Initial program 98.4%

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

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 - z\right)}\right) \]

      2. --lowering--.f64 91.9%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \color{blue}{z}\right)\right) \]

   5. Simplified 91.9%

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

   if 2.7e52 < y

   1. Initial program 97.4%

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

   3. Taylor expanded in y around inf

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

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

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(y \cdot z\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(z \cdot \color{blue}{y}\right)\right) \]

      3. *-lowering-*.f64 75.1%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(z, \color{blue}{y}\right)\right) \]

   5. Simplified 75.1%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.6 \cdot 10^{+120}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{+52}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 82.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := x\_m \cdot \left(z \cdot y\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{+120}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{+49}:\\ \;\;\;\;x\_m \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (* x_m (* z y))))
   (* x_s (if (<= y -6e+120) t_0 (if (<= y 2.9e+49) (* x_m (- 1.0 z)) t_0)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double t_0 = x_m * (z * y);
	double tmp;
	if (y <= -6e+120) {
		tmp = t_0;
	} else if (y <= 2.9e+49) {
		tmp = x_m * (1.0 - z);
	} else {
		tmp = t_0;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x_m * (z * y)
    if (y <= (-6d+120)) then
        tmp = t_0
    else if (y <= 2.9d+49) then
        tmp = x_m * (1.0d0 - z)
    else
        tmp = t_0
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double t_0 = x_m * (z * y);
	double tmp;
	if (y <= -6e+120) {
		tmp = t_0;
	} else if (y <= 2.9e+49) {
		tmp = x_m * (1.0 - z);
	} else {
		tmp = t_0;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	t_0 = x_m * (z * y)
	tmp = 0
	if y <= -6e+120:
		tmp = t_0
	elif y <= 2.9e+49:
		tmp = x_m * (1.0 - z)
	else:
		tmp = t_0
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	t_0 = Float64(x_m * Float64(z * y))
	tmp = 0.0
	if (y <= -6e+120)
		tmp = t_0;
	elseif (y <= 2.9e+49)
		tmp = Float64(x_m * Float64(1.0 - z));
	else
		tmp = t_0;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	t_0 = x_m * (z * y);
	tmp = 0.0;
	if (y <= -6e+120)
		tmp = t_0;
	elseif (y <= 2.9e+49)
		tmp = x_m * (1.0 - z);
	else
		tmp = t_0;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := Block[{t$95$0 = N[(x$95$m * N[(z * y), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[y, -6e+120], t$95$0, If[LessEqual[y, 2.9e+49], N[(x$95$m * N[(1.0 - z), $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6e120 or 2.9e49 < y

   1. Initial program 92.6%

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

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(y \cdot z\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(z \cdot \color{blue}{y}\right)\right) \]

      3. *-lowering-*.f64 78.1%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(z, \color{blue}{y}\right)\right) \]

   5. Simplified 78.1%

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

   if -6e120 < y < 2.9e49

   1. Initial program 98.4%

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

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 - z\right)}\right) \]

      2. --lowering--.f64 91.9%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \color{blue}{z}\right)\right) \]

   5. Simplified 91.9%

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

Alternative 9: 59.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := x\_m \cdot \left(z \cdot y\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -0.00035:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 10^{-5}:\\ \;\;\;\;x\_m\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (* x_m (* z y))))
   (* x_s (if (<= z -0.00035) t_0 (if (<= z 1e-5) x_m t_0)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double t_0 = x_m * (z * y);
	double tmp;
	if (z <= -0.00035) {
		tmp = t_0;
	} else if (z <= 1e-5) {
		tmp = x_m;
	} else {
		tmp = t_0;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x_m * (z * y)
    if (z <= (-0.00035d0)) then
        tmp = t_0
    else if (z <= 1d-5) then
        tmp = x_m
    else
        tmp = t_0
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double t_0 = x_m * (z * y);
	double tmp;
	if (z <= -0.00035) {
		tmp = t_0;
	} else if (z <= 1e-5) {
		tmp = x_m;
	} else {
		tmp = t_0;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	t_0 = x_m * (z * y)
	tmp = 0
	if z <= -0.00035:
		tmp = t_0
	elif z <= 1e-5:
		tmp = x_m
	else:
		tmp = t_0
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	t_0 = Float64(x_m * Float64(z * y))
	tmp = 0.0
	if (z <= -0.00035)
		tmp = t_0;
	elseif (z <= 1e-5)
		tmp = x_m;
	else
		tmp = t_0;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	t_0 = x_m * (z * y);
	tmp = 0.0;
	if (z <= -0.00035)
		tmp = t_0;
	elseif (z <= 1e-5)
		tmp = x_m;
	else
		tmp = t_0;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := Block[{t$95$0 = N[(x$95$m * N[(z * y), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[z, -0.00035], t$95$0, If[LessEqual[z, 1e-5], x$95$m, t$95$0]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.49999999999999996e-4 or 1.00000000000000008e-5 < z

   1. Initial program 93.5%

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

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(y \cdot z\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(z \cdot \color{blue}{y}\right)\right) \]

      3. *-lowering-*.f64 42.2%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(z, \color{blue}{y}\right)\right) \]

   5. Simplified 42.2%

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

   if -3.49999999999999996e-4 < z < 1.00000000000000008e-5

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

   5. Simplified 81.7%

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

Alternative 10: 38.7% accurate, 9.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot x\_m \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z) :precision binary64 (* x_s x_m))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	return x_s * x_m;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x_s * x_m
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	return x_s * x_m;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	return x_s * x_m

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	return Float64(x_s * x_m)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m, y, z)
	tmp = x_s * x_m;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * x$95$m), $MachinePrecision]

Derivation

1. Initial program 96.6%

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

3. Taylor expanded in z around 0

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

5. Simplified 41.4%

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

Developer Target 1: 99.6% accurate, 0.2× 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 2024141 
(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))))