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

Percentage Accurate: 96.2% → 99.9%

Time: 7.1s

Alternatives: 13

Speedup: 0.8×

• 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: 96.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \begin{array}{l} \mathbf{if}\;x_m \leq 7 \cdot 10^{-60}:\\ \;\;\;\;x_m - z \cdot \left(x_m \cdot \left(1 - y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x_m + \left(-1 + y\right) \cdot \left(x_m \cdot z\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= x_m 7e-60)
    (- x_m (* z (* x_m (- 1.0 y))))
    (+ x_m (* (+ -1.0 y) (* x_m z))))))

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 <= 7e-60) {
		tmp = x_m - (z * (x_m * (1.0 - y)));
	} else {
		tmp = x_m + ((-1.0 + y) * (x_m * z));
	}
	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 <= 7d-60) then
        tmp = x_m - (z * (x_m * (1.0d0 - y)))
    else
        tmp = x_m + (((-1.0d0) + y) * (x_m * z))
    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 <= 7e-60) {
		tmp = x_m - (z * (x_m * (1.0 - y)));
	} else {
		tmp = x_m + ((-1.0 + y) * (x_m * z));
	}
	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 <= 7e-60:
		tmp = x_m - (z * (x_m * (1.0 - y)))
	else:
		tmp = x_m + ((-1.0 + y) * (x_m * z))
	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 <= 7e-60)
		tmp = Float64(x_m - Float64(z * Float64(x_m * Float64(1.0 - y))));
	else
		tmp = Float64(x_m + Float64(Float64(-1.0 + y) * Float64(x_m * z)));
	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 <= 7e-60)
		tmp = x_m - (z * (x_m * (1.0 - y)));
	else
		tmp = x_m + ((-1.0 + y) * (x_m * z));
	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, 7e-60], N[(x$95$m - N[(z * N[(x$95$m * N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m + N[(N[(-1.0 + y), $MachinePrecision] * N[(x$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 6.99999999999999952e-60

   1. Initial program 95.9%

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

   3. Taylor expanded in z around 0 95.9%

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

   4. Taylor expanded in y around 0 90.5%

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

      1. +-commutative 90.5%

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

      2. *-commutative 90.5%

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

      3. associate-*r* 88.8%

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

      4. *-commutative 88.8%

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

      5. distribute-lft-in 95.8%

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

      6. *-commutative 95.8%

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

      7. associate-*l* 99.4%

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

      8. +-commutative 99.4%

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

   6. Simplified 99.4%

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

   if 6.99999999999999952e-60 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0 99.9%

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

   4. Taylor expanded in y around 0 92.7%

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

      1. +-commutative 92.7%

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

      2. *-commutative 92.7%

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

      3. associate-*r* 88.3%

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

      4. *-commutative 88.3%

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

      5. distribute-lft-in 99.9%

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

      6. *-commutative 99.9%

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

      7. associate-*l* 90.4%

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

      8. +-commutative 90.4%

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

   6. Simplified 90.4%

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

   7. Taylor expanded in z around 0 99.9%

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

      1. associate-*r* 99.9%

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

      2. sub-neg 99.9%

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

      3. metadata-eval 99.9%

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

   9. Simplified 99.9%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7 \cdot 10^{-60}:\\ \;\;\;\;x - z \cdot \left(x \cdot \left(1 - y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(-1 + y\right) \cdot \left(x \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot \left(1 - y\right) \leq 5 \cdot 10^{+286}:\\ \;\;\;\;x_m \cdot \left(1 + z \cdot \left(-1 + y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x_m + y \cdot \left(x_m \cdot z\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= (* z (- 1.0 y)) 5e+286)
    (* x_m (+ 1.0 (* z (+ -1.0 y))))
    (+ x_m (* y (* x_m z))))))

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)) <= 5e+286) {
		tmp = x_m * (1.0 + (z * (-1.0 + y)));
	} else {
		tmp = x_m + (y * (x_m * z));
	}
	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)) <= 5d+286) then
        tmp = x_m * (1.0d0 + (z * ((-1.0d0) + y)))
    else
        tmp = x_m + (y * (x_m * z))
    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)) <= 5e+286) {
		tmp = x_m * (1.0 + (z * (-1.0 + y)));
	} else {
		tmp = x_m + (y * (x_m * z));
	}
	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)) <= 5e+286:
		tmp = x_m * (1.0 + (z * (-1.0 + y)))
	else:
		tmp = x_m + (y * (x_m * z))
	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)) <= 5e+286)
		tmp = Float64(x_m * Float64(1.0 + Float64(z * Float64(-1.0 + y))));
	else
		tmp = Float64(x_m + Float64(y * Float64(x_m * z)));
	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)) <= 5e+286)
		tmp = x_m * (1.0 + (z * (-1.0 + y)));
	else
		tmp = x_m + (y * (x_m * z));
	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], 5e+286], N[(x$95$m * N[(1.0 + N[(z * N[(-1.0 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m + N[(y * N[(x$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 1 y) z) < 5.0000000000000004e286

   1. Initial program 98.7%

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

   if 5.0000000000000004e286 < (*.f64 (-.f64 1 y) z)

   1. Initial program 76.6%

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

   3. Taylor expanded in z around 0 76.6%

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

   4. Taylor expanded in y around 0 51.6%

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

      1. +-commutative 51.6%

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

      2. *-commutative 51.6%

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

      3. associate-*r* 74.8%

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

      4. *-commutative 74.8%

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

      5. distribute-lft-in 99.8%

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

      6. *-commutative 99.8%

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

      7. associate-*l* 99.7%

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

      8. +-commutative 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in z around 0 76.6%

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

      1. associate-*r* 99.8%

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

      2. sub-neg 99.8%

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

      3. metadata-eval 99.8%

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

   9. Simplified 99.8%

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

   10. Taylor expanded in y around inf 76.6%

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

       1. associate-*r* 99.7%

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

       2. *-commutative 99.7%

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

       3. associate-*r* 99.8%

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

   12. Simplified 99.8%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot \left(1 - y\right) \leq 5 \cdot 10^{+286}:\\ \;\;\;\;x \cdot \left(1 + z \cdot \left(-1 + y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(x \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot \left(1 - y\right) \leq 5 \cdot 10^{+286}:\\ \;\;\;\;x_m + x_m \cdot \left(z \cdot \left(-1 + y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x_m + y \cdot \left(x_m \cdot z\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= (* z (- 1.0 y)) 5e+286)
    (+ x_m (* x_m (* z (+ -1.0 y))))
    (+ x_m (* y (* x_m z))))))

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)) <= 5e+286) {
		tmp = x_m + (x_m * (z * (-1.0 + y)));
	} else {
		tmp = x_m + (y * (x_m * z));
	}
	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)) <= 5d+286) then
        tmp = x_m + (x_m * (z * ((-1.0d0) + y)))
    else
        tmp = x_m + (y * (x_m * z))
    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)) <= 5e+286) {
		tmp = x_m + (x_m * (z * (-1.0 + y)));
	} else {
		tmp = x_m + (y * (x_m * z));
	}
	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)) <= 5e+286:
		tmp = x_m + (x_m * (z * (-1.0 + y)))
	else:
		tmp = x_m + (y * (x_m * z))
	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)) <= 5e+286)
		tmp = Float64(x_m + Float64(x_m * Float64(z * Float64(-1.0 + y))));
	else
		tmp = Float64(x_m + Float64(y * Float64(x_m * z)));
	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)) <= 5e+286)
		tmp = x_m + (x_m * (z * (-1.0 + y)));
	else
		tmp = x_m + (y * (x_m * z));
	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], 5e+286], N[(x$95$m + N[(x$95$m * N[(z * N[(-1.0 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m + N[(y * N[(x$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 1 y) z) < 5.0000000000000004e286

   1. Initial program 98.7%

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

   3. Taylor expanded in z around 0 98.7%

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

   if 5.0000000000000004e286 < (*.f64 (-.f64 1 y) z)

   1. Initial program 76.6%

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

   3. Taylor expanded in z around 0 76.6%

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

   4. Taylor expanded in y around 0 51.6%

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

      1. +-commutative 51.6%

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

      2. *-commutative 51.6%

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

      3. associate-*r* 74.8%

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

      4. *-commutative 74.8%

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

      5. distribute-lft-in 99.8%

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

      6. *-commutative 99.8%

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

      7. associate-*l* 99.7%

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

      8. +-commutative 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in z around 0 76.6%

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

      1. associate-*r* 99.8%

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

      2. sub-neg 99.8%

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

      3. metadata-eval 99.8%

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

   9. Simplified 99.8%

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

   10. Taylor expanded in y around inf 76.6%

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

       1. associate-*r* 99.7%

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

       2. *-commutative 99.7%

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

       3. associate-*r* 99.8%

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

   12. Simplified 99.8%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot \left(1 - y\right) \leq 5 \cdot 10^{+286}:\\ \;\;\;\;x + x \cdot \left(z \cdot \left(-1 + y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(x \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 59.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := x_m \cdot \left(z \cdot y\right)\\ x_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -13500000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1.05 \cdot 10^{-245}:\\ \;\;\;\;x_m\\ \mathbf{elif}\;y \leq 7:\\ \;\;\;\;x_m \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (* x_m (* z y))))
   (*
    x_s
    (if (<= y -13500000.0)
      t_0
      (if (<= y -1.05e-245) x_m (if (<= y 7.0) (* x_m (- 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 <= -13500000.0) {
		tmp = t_0;
	} else if (y <= -1.05e-245) {
		tmp = x_m;
	} else if (y <= 7.0) {
		tmp = x_m * -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 <= (-13500000.0d0)) then
        tmp = t_0
    else if (y <= (-1.05d-245)) then
        tmp = x_m
    else if (y <= 7.0d0) then
        tmp = x_m * -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 <= -13500000.0) {
		tmp = t_0;
	} else if (y <= -1.05e-245) {
		tmp = x_m;
	} else if (y <= 7.0) {
		tmp = x_m * -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 <= -13500000.0:
		tmp = t_0
	elif y <= -1.05e-245:
		tmp = x_m
	elif y <= 7.0:
		tmp = x_m * -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 <= -13500000.0)
		tmp = t_0;
	elseif (y <= -1.05e-245)
		tmp = x_m;
	elseif (y <= 7.0)
		tmp = Float64(x_m * Float64(-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 <= -13500000.0)
		tmp = t_0;
	elseif (y <= -1.05e-245)
		tmp = x_m;
	elseif (y <= 7.0)
		tmp = x_m * -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, -13500000.0], t$95$0, If[LessEqual[y, -1.05e-245], x$95$m, If[LessEqual[y, 7.0], N[(x$95$m * (-z)), $MachinePrecision], t$95$0]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if y < -1.35e7 or 7 < y

   1. Initial program 94.3%

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

   3. Taylor expanded in y around inf 71.6%

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

      1. *-commutative 71.6%

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

   5. Simplified 71.6%

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

   if -1.35e7 < y < -1.05000000000000005e-245

   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 68.0%

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

   if -1.05000000000000005e-245 < y < 7

   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 inf 59.5%

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

   4. Taylor expanded in y around 0 59.6%

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

      1. mul-1-neg 59.6%

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

      2. *-commutative 59.6%

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

      3. distribute-rgt-neg-in 59.6%

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

   6. Simplified 59.6%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -13500000:\\ \;\;\;\;x \cdot \left(z \cdot y\right)\\ \mathbf{elif}\;y \leq -1.05 \cdot 10^{-245}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 7:\\ \;\;\;\;x \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 2.8 \cdot 10^{-8}\right):\\ \;\;\;\;x_m + x_m \cdot \left(z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x_m - x_m \cdot z\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (or (<= y -1.0) (not (<= y 2.8e-8)))
    (+ x_m (* x_m (* z y)))
    (- x_m (* x_m z)))))

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 <= -1.0) || !(y <= 2.8e-8)) {
		tmp = x_m + (x_m * (z * y));
	} else {
		tmp = x_m - (x_m * z);
	}
	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 <= (-1.0d0)) .or. (.not. (y <= 2.8d-8))) then
        tmp = x_m + (x_m * (z * y))
    else
        tmp = x_m - (x_m * z)
    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 <= -1.0) || !(y <= 2.8e-8)) {
		tmp = x_m + (x_m * (z * y));
	} else {
		tmp = x_m - (x_m * z);
	}
	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 <= -1.0) or not (y <= 2.8e-8):
		tmp = x_m + (x_m * (z * y))
	else:
		tmp = x_m - (x_m * z)
	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 <= -1.0) || !(y <= 2.8e-8))
		tmp = Float64(x_m + Float64(x_m * Float64(z * y)));
	else
		tmp = Float64(x_m - Float64(x_m * z));
	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 <= -1.0) || ~((y <= 2.8e-8)))
		tmp = x_m + (x_m * (z * y));
	else
		tmp = x_m - (x_m * z);
	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[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 2.8e-8]], $MachinePrecision]], N[(x$95$m + N[(x$95$m * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m - N[(x$95$m * z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 2.7999999999999999e-8 < y

   1. Initial program 94.4%

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

   3. Taylor expanded in z around 0 94.5%

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

   4. Taylor expanded in y around inf 93.4%

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

      1. *-commutative 93.4%

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

   6. Simplified 93.4%

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

   if -1 < y < 2.7999999999999999e-8

   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 0 100.0%

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

   4. Taylor expanded in y around 0 95.7%

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

      1. +-commutative 95.7%

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

      2. *-commutative 95.7%

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

      3. associate-*r* 90.5%

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

      4. *-commutative 90.5%

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

      5. distribute-lft-in 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*l* 100.0%

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

      8. +-commutative 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in z around 0 100.0%

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

      1. associate-*r* 100.0%

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

      2. sub-neg 100.0%

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

      3. metadata-eval 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in y around 0 100.0%

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

       1. neg-mul-1 100.0%

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

       2. unsub-neg 100.0%

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

   12. Simplified 100.0%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 2.8 \cdot 10^{-8}\right):\\ \;\;\;\;x + x \cdot \left(z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x - x \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -5.9 \cdot 10^{+46}:\\ \;\;\;\;z \cdot \left(x_m \cdot y\right)\\ \mathbf{elif}\;y \leq 2200:\\ \;\;\;\;x_m - x_m \cdot z\\ \mathbf{else}:\\ \;\;\;\;x_m \cdot \left(z \cdot \left(-1 + y\right)\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= y -5.9e+46)
    (* z (* x_m y))
    (if (<= y 2200.0) (- x_m (* x_m z)) (* x_m (* z (+ -1.0 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 <= -5.9e+46) {
		tmp = z * (x_m * y);
	} else if (y <= 2200.0) {
		tmp = x_m - (x_m * z);
	} else {
		tmp = x_m * (z * (-1.0 + 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 <= (-5.9d+46)) then
        tmp = z * (x_m * y)
    else if (y <= 2200.0d0) then
        tmp = x_m - (x_m * z)
    else
        tmp = x_m * (z * ((-1.0d0) + 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 <= -5.9e+46) {
		tmp = z * (x_m * y);
	} else if (y <= 2200.0) {
		tmp = x_m - (x_m * z);
	} else {
		tmp = x_m * (z * (-1.0 + 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 <= -5.9e+46:
		tmp = z * (x_m * y)
	elif y <= 2200.0:
		tmp = x_m - (x_m * z)
	else:
		tmp = x_m * (z * (-1.0 + 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 <= -5.9e+46)
		tmp = Float64(z * Float64(x_m * y));
	elseif (y <= 2200.0)
		tmp = Float64(x_m - Float64(x_m * z));
	else
		tmp = Float64(x_m * Float64(z * Float64(-1.0 + 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 <= -5.9e+46)
		tmp = z * (x_m * y);
	elseif (y <= 2200.0)
		tmp = x_m - (x_m * z);
	else
		tmp = x_m * (z * (-1.0 + 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, -5.9e+46], N[(z * N[(x$95$m * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2200.0], N[(x$95$m - N[(x$95$m * z), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(z * N[(-1.0 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -5.8999999999999999e46

   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 71.0%

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

      1. associate-*r* 77.8%

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

      2. *-commutative 77.8%

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

   5. Simplified 77.8%

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

   if -5.8999999999999999e46 < y < 2200

   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 0 100.0%

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

   4. Taylor expanded in y around 0 96.2%

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

      1. +-commutative 96.2%

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

      2. *-commutative 96.2%

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

      3. associate-*r* 91.6%

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

      4. *-commutative 91.6%

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

      5. distribute-lft-in 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*l* 100.0%

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

      8. +-commutative 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in z around 0 100.0%

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

      1. associate-*r* 100.0%

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

      2. sub-neg 100.0%

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

      3. metadata-eval 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in y around 0 95.8%

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

       1. neg-mul-1 95.8%

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

       2. unsub-neg 95.8%

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

   12. Simplified 95.8%

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

   if 2200 < y

   1. Initial program 96.9%

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

   3. Taylor expanded in z around inf 76.9%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.9 \cdot 10^{+46}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;y \leq 2200:\\ \;\;\;\;x - x \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z \cdot \left(-1 + y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 95.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x_m + y \cdot \left(x_m \cdot z\right)\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{-8}:\\ \;\;\;\;x_m - x_m \cdot z\\ \mathbf{else}:\\ \;\;\;\;x_m + x_m \cdot \left(z \cdot y\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= y -1.0)
    (+ x_m (* y (* x_m z)))
    (if (<= y 2.8e-8) (- x_m (* x_m z)) (+ x_m (* 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 <= -1.0) {
		tmp = x_m + (y * (x_m * z));
	} else if (y <= 2.8e-8) {
		tmp = x_m - (x_m * z);
	} else {
		tmp = x_m + (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 <= (-1.0d0)) then
        tmp = x_m + (y * (x_m * z))
    else if (y <= 2.8d-8) then
        tmp = x_m - (x_m * z)
    else
        tmp = x_m + (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 <= -1.0) {
		tmp = x_m + (y * (x_m * z));
	} else if (y <= 2.8e-8) {
		tmp = x_m - (x_m * z);
	} else {
		tmp = x_m + (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 <= -1.0:
		tmp = x_m + (y * (x_m * z))
	elif y <= 2.8e-8:
		tmp = x_m - (x_m * z)
	else:
		tmp = x_m + (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 <= -1.0)
		tmp = Float64(x_m + Float64(y * Float64(x_m * z)));
	elseif (y <= 2.8e-8)
		tmp = Float64(x_m - Float64(x_m * z));
	else
		tmp = Float64(x_m + 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 <= -1.0)
		tmp = x_m + (y * (x_m * z));
	elseif (y <= 2.8e-8)
		tmp = x_m - (x_m * z);
	else
		tmp = x_m + (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, -1.0], N[(x$95$m + N[(y * N[(x$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.8e-8], N[(x$95$m - N[(x$95$m * z), $MachinePrecision]), $MachinePrecision], N[(x$95$m + N[(x$95$m * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -1

   1. Initial program 91.7%

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

   3. Taylor expanded in z around 0 91.8%

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

   4. Taylor expanded in y around 0 91.7%

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

      1. +-commutative 91.7%

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

      2. *-commutative 91.7%

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

      3. associate-*r* 96.2%

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

      4. *-commutative 96.2%

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

      5. distribute-lft-in 96.2%

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

      6. *-commutative 96.2%

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

      7. associate-*l* 94.3%

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

      8. +-commutative 94.3%

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

   6. Simplified 94.3%

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

   7. Taylor expanded in z around 0 91.8%

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

      1. associate-*r* 96.2%

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

      2. sub-neg 96.2%

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

      3. metadata-eval 96.2%

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

   9. Simplified 96.2%

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

   10. Taylor expanded in y around inf 90.5%

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

       1. associate-*r* 93.1%

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

       2. *-commutative 93.1%

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

       3. associate-*r* 95.0%

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

   12. Simplified 95.0%

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

   if -1 < y < 2.7999999999999999e-8

   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 0 100.0%

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

   4. Taylor expanded in y around 0 95.7%

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

      1. +-commutative 95.7%

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

      2. *-commutative 95.7%

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

      3. associate-*r* 90.5%

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

      4. *-commutative 90.5%

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

      5. distribute-lft-in 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*l* 100.0%

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

      8. +-commutative 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in z around 0 100.0%

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

      1. associate-*r* 100.0%

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

      2. sub-neg 100.0%

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

      3. metadata-eval 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in y around 0 100.0%

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

       1. neg-mul-1 100.0%

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

       2. unsub-neg 100.0%

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

   12. Simplified 100.0%

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

   if 2.7999999999999999e-8 < y

   1. Initial program 97.0%

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

   3. Taylor expanded in z around 0 97.1%

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

   4. Taylor expanded in y around inf 96.1%

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

      1. *-commutative 96.1%

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

   6. Simplified 96.1%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x + y \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{-8}:\\ \;\;\;\;x - x \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + x \cdot \left(z \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \begin{array}{l} \mathbf{if}\;x_m \leq 10^{-25}:\\ \;\;\;\;x_m - z \cdot \left(x_m \cdot \left(1 - y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x_m + x_m \cdot \left(z \cdot \left(-1 + y\right)\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= x_m 1e-25)
    (- x_m (* z (* x_m (- 1.0 y))))
    (+ x_m (* x_m (* z (+ -1.0 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 (x_m <= 1e-25) {
		tmp = x_m - (z * (x_m * (1.0 - y)));
	} else {
		tmp = x_m + (x_m * (z * (-1.0 + 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 (x_m <= 1d-25) then
        tmp = x_m - (z * (x_m * (1.0d0 - y)))
    else
        tmp = x_m + (x_m * (z * ((-1.0d0) + 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 (x_m <= 1e-25) {
		tmp = x_m - (z * (x_m * (1.0 - y)));
	} else {
		tmp = x_m + (x_m * (z * (-1.0 + 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 x_m <= 1e-25:
		tmp = x_m - (z * (x_m * (1.0 - y)))
	else:
		tmp = x_m + (x_m * (z * (-1.0 + 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 (x_m <= 1e-25)
		tmp = Float64(x_m - Float64(z * Float64(x_m * Float64(1.0 - y))));
	else
		tmp = Float64(x_m + Float64(x_m * Float64(z * Float64(-1.0 + 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 (x_m <= 1e-25)
		tmp = x_m - (z * (x_m * (1.0 - y)));
	else
		tmp = x_m + (x_m * (z * (-1.0 + 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[x$95$m, 1e-25], N[(x$95$m - N[(z * N[(x$95$m * N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m + N[(x$95$m * N[(z * N[(-1.0 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.00000000000000004e-25

   1. Initial program 96.0%

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

   3. Taylor expanded in z around 0 96.0%

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

   4. Taylor expanded in y around 0 90.8%

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

      1. +-commutative 90.8%

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

      2. *-commutative 90.8%

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

      3. associate-*r* 89.2%

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

      4. *-commutative 89.2%

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

      5. distribute-lft-in 95.9%

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

      6. *-commutative 95.9%

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

      7. associate-*l* 99.4%

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

      8. +-commutative 99.4%

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

   6. Simplified 99.4%

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

   if 1.00000000000000004e-25 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0 99.9%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{-25}:\\ \;\;\;\;x - z \cdot \left(x \cdot \left(1 - y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + x \cdot \left(z \cdot \left(-1 + y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 83.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{+47} \lor \neg \left(y \leq 6.2 \cdot 10^{+16}\right):\\ \;\;\;\;x_m \cdot \left(z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x_m \cdot \left(1 - z\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (or (<= y -2.5e+47) (not (<= y 6.2e+16)))
    (* x_m (* z y))
    (* x_m (- 1.0 z)))))

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 <= -2.5e+47) || !(y <= 6.2e+16)) {
		tmp = x_m * (z * y);
	} else {
		tmp = x_m * (1.0 - z);
	}
	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 <= (-2.5d+47)) .or. (.not. (y <= 6.2d+16))) then
        tmp = x_m * (z * y)
    else
        tmp = x_m * (1.0d0 - z)
    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 <= -2.5e+47) || !(y <= 6.2e+16)) {
		tmp = x_m * (z * y);
	} else {
		tmp = x_m * (1.0 - z);
	}
	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 <= -2.5e+47) or not (y <= 6.2e+16):
		tmp = x_m * (z * y)
	else:
		tmp = x_m * (1.0 - z)
	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 <= -2.5e+47) || !(y <= 6.2e+16))
		tmp = Float64(x_m * Float64(z * y));
	else
		tmp = Float64(x_m * Float64(1.0 - z));
	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 <= -2.5e+47) || ~((y <= 6.2e+16)))
		tmp = x_m * (z * y);
	else
		tmp = x_m * (1.0 - z);
	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[Or[LessEqual[y, -2.5e+47], N[Not[LessEqual[y, 6.2e+16]], $MachinePrecision]], N[(x$95$m * N[(z * y), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < -2.50000000000000011e47 or 6.2e16 < y

   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 75.0%

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

      1. *-commutative 75.0%

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

   5. Simplified 75.0%

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

   if -2.50000000000000011e47 < y < 6.2e16

   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 94.5%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{+47} \lor \neg \left(y \leq 6.2 \cdot 10^{+16}\right):\\ \;\;\;\;x \cdot \left(z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 84.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{+44}:\\ \;\;\;\;z \cdot \left(x_m \cdot y\right)\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{+17}:\\ \;\;\;\;x_m \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;x_m \cdot \left(z \cdot y\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= y -1.8e+44)
    (* z (* x_m y))
    (if (<= y 1.8e+17) (* 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 <= -1.8e+44) {
		tmp = z * (x_m * y);
	} else if (y <= 1.8e+17) {
		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 <= (-1.8d+44)) then
        tmp = z * (x_m * y)
    else if (y <= 1.8d+17) 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 <= -1.8e+44) {
		tmp = z * (x_m * y);
	} else if (y <= 1.8e+17) {
		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 <= -1.8e+44:
		tmp = z * (x_m * y)
	elif y <= 1.8e+17:
		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 <= -1.8e+44)
		tmp = Float64(z * Float64(x_m * y));
	elseif (y <= 1.8e+17)
		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 <= -1.8e+44)
		tmp = z * (x_m * y);
	elseif (y <= 1.8e+17)
		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, -1.8e+44], N[(z * N[(x$95$m * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.8e+17], 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 < -1.8e44

   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 71.0%

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

      1. associate-*r* 77.8%

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

      2. *-commutative 77.8%

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

   5. Simplified 77.8%

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

   if -1.8e44 < y < 1.8e17

   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 94.5%

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

   if 1.8e17 < y

   1. Initial program 96.8%

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

   3. Taylor expanded in y around inf 78.4%

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

      1. *-commutative 78.4%

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

   5. Simplified 78.4%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{+44}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{+17}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 84.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{+47}:\\ \;\;\;\;z \cdot \left(x_m \cdot y\right)\\ \mathbf{elif}\;y \leq 5.4 \cdot 10^{+16}:\\ \;\;\;\;x_m - x_m \cdot z\\ \mathbf{else}:\\ \;\;\;\;x_m \cdot \left(z \cdot y\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= y -5.2e+47)
    (* z (* x_m y))
    (if (<= y 5.4e+16) (- x_m (* x_m 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 <= -5.2e+47) {
		tmp = z * (x_m * y);
	} else if (y <= 5.4e+16) {
		tmp = x_m - (x_m * 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 <= (-5.2d+47)) then
        tmp = z * (x_m * y)
    else if (y <= 5.4d+16) then
        tmp = x_m - (x_m * 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 <= -5.2e+47) {
		tmp = z * (x_m * y);
	} else if (y <= 5.4e+16) {
		tmp = x_m - (x_m * 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 <= -5.2e+47:
		tmp = z * (x_m * y)
	elif y <= 5.4e+16:
		tmp = x_m - (x_m * 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 <= -5.2e+47)
		tmp = Float64(z * Float64(x_m * y));
	elseif (y <= 5.4e+16)
		tmp = Float64(x_m - Float64(x_m * 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 <= -5.2e+47)
		tmp = z * (x_m * y);
	elseif (y <= 5.4e+16)
		tmp = x_m - (x_m * 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, -5.2e+47], N[(z * N[(x$95$m * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.4e+16], N[(x$95$m - N[(x$95$m * z), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(z * y), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -5.20000000000000007e47

   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 71.0%

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

      1. associate-*r* 77.8%

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

      2. *-commutative 77.8%

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

   5. Simplified 77.8%

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

   if -5.20000000000000007e47 < y < 5.4e16

   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 0 100.0%

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

   4. Taylor expanded in y around 0 96.3%

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

      1. +-commutative 96.3%

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

      2. *-commutative 96.3%

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

      3. associate-*r* 91.9%

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

      4. *-commutative 91.9%

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

      5. distribute-lft-in 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*l* 100.0%

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

      8. +-commutative 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in z around 0 100.0%

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

      1. associate-*r* 100.0%

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

      2. sub-neg 100.0%

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

      3. metadata-eval 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in y around 0 94.5%

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

       1. neg-mul-1 94.5%

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

       2. unsub-neg 94.5%

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

   12. Simplified 94.5%

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

   if 5.4e16 < y

   1. Initial program 96.8%

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

   3. Taylor expanded in y around inf 78.4%

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

      1. *-commutative 78.4%

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

   5. Simplified 78.4%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{+47}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;y \leq 5.4 \cdot 10^{+16}:\\ \;\;\;\;x - x \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 63.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -210000000 \lor \neg \left(z \leq 1.45 \cdot 10^{-5}\right):\\ \;\;\;\;x_m \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;x_m\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z)
 :precision binary64
 (* x_s (if (or (<= z -210000000.0) (not (<= z 1.45e-5))) (* x_m (- z)) 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) {
	double tmp;
	if ((z <= -210000000.0) || !(z <= 1.45e-5)) {
		tmp = x_m * -z;
	} else {
		tmp = x_m;
	}
	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 <= (-210000000.0d0)) .or. (.not. (z <= 1.45d-5))) then
        tmp = x_m * -z
    else
        tmp = x_m
    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 <= -210000000.0) || !(z <= 1.45e-5)) {
		tmp = x_m * -z;
	} else {
		tmp = x_m;
	}
	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 <= -210000000.0) or not (z <= 1.45e-5):
		tmp = x_m * -z
	else:
		tmp = x_m
	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 ((z <= -210000000.0) || !(z <= 1.45e-5))
		tmp = Float64(x_m * Float64(-z));
	else
		tmp = x_m;
	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 <= -210000000.0) || ~((z <= 1.45e-5)))
		tmp = x_m * -z;
	else
		tmp = x_m;
	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[Or[LessEqual[z, -210000000.0], N[Not[LessEqual[z, 1.45e-5]], $MachinePrecision]], N[(x$95$m * (-z)), $MachinePrecision], x$95$m]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -2.1e8 or 1.45e-5 < z

   1. Initial program 94.0%

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

   3. Taylor expanded in z around inf 93.3%

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

   4. Taylor expanded in y around 0 51.8%

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

      1. mul-1-neg 51.8%

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

      2. *-commutative 51.8%

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

      3. distribute-rgt-neg-in 51.8%

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

   6. Simplified 51.8%

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

   if -2.1e8 < z < 1.45e-5

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0 66.1%

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

4. Final simplification 59.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -210000000 \lor \neg \left(z \leq 1.45 \cdot 10^{-5}\right):\\ \;\;\;\;x \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 37.9% 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 1 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 96.9%

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

3. Taylor expanded in z around 0 35.2%

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

4. Final simplification 35.2%

   \[\leadsto x \]
5. Add Preprocessing

Developer target: 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 2024010 
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSV:hsv from colour-2.3.3, J"
  :precision binary64

  :herbie-target
  (if (< (* x (- 1.0 (* (- 1.0 y) z))) -1.618195973607049e+50) (+ x (* (- 1.0 y) (* (- z) x))) (if (< (* x (- 1.0 (* (- 1.0 y) z))) 3.892237649663903e+134) (- (* (* x y) z) (- (* x z) x)) (+ x (* (- 1.0 y) (* (- z) x)))))

  (* x (- 1.0 (* (- 1.0 y) z))))