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

Percentage Accurate: 95.7% → 99.6%

Time: 8.0s

Alternatives: 10

Speedup: 0.6×

• 20.7% 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: 95.7% 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: 99.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 3.55 \cdot 10^{-65}:\\ \;\;\;\;x\_m + z \cdot \left(x\_m \cdot \left(y + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \left(1 + z \cdot \left(y + -1\right)\right)\\ \end{array} \end{array} \]

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 3.55e-65)
    (+ x_m (* z (* x_m (+ y -1.0))))
    (* 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 <= 3.55e-65) {
		tmp = x_m + (z * (x_m * (y + -1.0)));
	} 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 <= 3.55d-65) then
        tmp = x_m + (z * (x_m * (y + (-1.0d0))))
    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 <= 3.55e-65) {
		tmp = x_m + (z * (x_m * (y + -1.0)));
	} 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 <= 3.55e-65:
		tmp = x_m + (z * (x_m * (y + -1.0)))
	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 <= 3.55e-65)
		tmp = Float64(x_m + Float64(z * Float64(x_m * Float64(y + -1.0))));
	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 <= 3.55e-65)
		tmp = x_m + (z * (x_m * (y + -1.0)));
	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, 3.55e-65], N[(x$95$m + N[(z * N[(x$95$m * N[(y + -1.0), $MachinePrecision]), $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 < 3.55000000000000014e-65

   1. Initial program 93.8%

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

   3. Taylor expanded in z around 0 93.8%

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

      1. +-commutative 93.8%

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

      2. associate-*r* 97.0%

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

      3. *-commutative 97.0%

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

      4. associate-*l* 95.4%

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

      5. sub-neg 95.4%

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

      6. metadata-eval 95.4%

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

   5. Applied egg-rr 95.4%

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

   if 3.55000000000000014e-65 < x

   1. Initial program 99.9%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.55 \cdot 10^{-65}:\\ \;\;\;\;x + z \cdot \left(x \cdot \left(y + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + z \cdot \left(y + -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 62.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := x\_m \cdot \left(z \cdot y\right)\\ t_1 := x\_m \cdot \left(-z\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -2.45 \cdot 10^{+182}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -2.85 \cdot 10^{+149}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -60000000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;x\_m\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{+69} \lor \neg \left(z \leq 8.2 \cdot 10^{+185}\right):\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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))) (t_1 (* x_m (- z))))
   (*
    x_s
    (if (<= z -2.45e+182)
      t_1
      (if (<= z -2.85e+149)
        t_0
        (if (<= z -60000000000000.0)
          t_1
          (if (<= z 1.0)
            x_m
            (if (or (<= z 1.55e+69) (not (<= z 8.2e+185))) t_1 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 t_1 = x_m * -z;
	double tmp;
	if (z <= -2.45e+182) {
		tmp = t_1;
	} else if (z <= -2.85e+149) {
		tmp = t_0;
	} else if (z <= -60000000000000.0) {
		tmp = t_1;
	} else if (z <= 1.0) {
		tmp = x_m;
	} else if ((z <= 1.55e+69) || !(z <= 8.2e+185)) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_0 = x_m * (z * y)
    t_1 = x_m * -z
    if (z <= (-2.45d+182)) then
        tmp = t_1
    else if (z <= (-2.85d+149)) then
        tmp = t_0
    else if (z <= (-60000000000000.0d0)) then
        tmp = t_1
    else if (z <= 1.0d0) then
        tmp = x_m
    else if ((z <= 1.55d+69) .or. (.not. (z <= 8.2d+185))) then
        tmp = t_1
    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 t_1 = x_m * -z;
	double tmp;
	if (z <= -2.45e+182) {
		tmp = t_1;
	} else if (z <= -2.85e+149) {
		tmp = t_0;
	} else if (z <= -60000000000000.0) {
		tmp = t_1;
	} else if (z <= 1.0) {
		tmp = x_m;
	} else if ((z <= 1.55e+69) || !(z <= 8.2e+185)) {
		tmp = t_1;
	} 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)
	t_1 = x_m * -z
	tmp = 0
	if z <= -2.45e+182:
		tmp = t_1
	elif z <= -2.85e+149:
		tmp = t_0
	elif z <= -60000000000000.0:
		tmp = t_1
	elif z <= 1.0:
		tmp = x_m
	elif (z <= 1.55e+69) or not (z <= 8.2e+185):
		tmp = t_1
	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))
	t_1 = Float64(x_m * Float64(-z))
	tmp = 0.0
	if (z <= -2.45e+182)
		tmp = t_1;
	elseif (z <= -2.85e+149)
		tmp = t_0;
	elseif (z <= -60000000000000.0)
		tmp = t_1;
	elseif (z <= 1.0)
		tmp = x_m;
	elseif ((z <= 1.55e+69) || !(z <= 8.2e+185))
		tmp = t_1;
	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);
	t_1 = x_m * -z;
	tmp = 0.0;
	if (z <= -2.45e+182)
		tmp = t_1;
	elseif (z <= -2.85e+149)
		tmp = t_0;
	elseif (z <= -60000000000000.0)
		tmp = t_1;
	elseif (z <= 1.0)
		tmp = x_m;
	elseif ((z <= 1.55e+69) || ~((z <= 8.2e+185)))
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(x$95$m * (-z)), $MachinePrecision]}, N[(x$95$s * If[LessEqual[z, -2.45e+182], t$95$1, If[LessEqual[z, -2.85e+149], t$95$0, If[LessEqual[z, -60000000000000.0], t$95$1, If[LessEqual[z, 1.0], x$95$m, If[Or[LessEqual[z, 1.55e+69], N[Not[LessEqual[z, 8.2e+185]], $MachinePrecision]], t$95$1, t$95$0]]]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.45e182 or -2.84999999999999983e149 < z < -6e13 or 1 < z < 1.5499999999999999e69 or 8.2e185 < z

   1. Initial program 94.5%

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

   3. Taylor expanded in z around inf 93.2%

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

   4. Taylor expanded in y around 0 67.6%

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

      1. mul-1-neg 67.6%

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

      2. distribute-rgt-neg-in 67.6%

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

   6. Simplified 67.6%

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

   if -2.45e182 < z < -2.84999999999999983e149 or 1.5499999999999999e69 < z < 8.2e185

   1. Initial program 86.0%

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

   3. Taylor expanded in y around inf 69.0%

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

      1. *-commutative 69.0%

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

   5. Simplified 69.0%

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

   if -6e13 < z < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0 74.8%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.45 \cdot 10^{+182}:\\ \;\;\;\;x \cdot \left(-z\right)\\ \mathbf{elif}\;z \leq -2.85 \cdot 10^{+149}:\\ \;\;\;\;x \cdot \left(z \cdot y\right)\\ \mathbf{elif}\;z \leq -60000000000000:\\ \;\;\;\;x \cdot \left(-z\right)\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{+69} \lor \neg \left(z \leq 8.2 \cdot 10^{+185}\right):\\ \;\;\;\;x \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.8% accurate, 0.5× speedup

Math

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

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 (<= (* z (- 1.0 y)) (- INFINITY))
    (* z (* x_m y))
    (* 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 ((z * (1.0 - y)) <= -((double) INFINITY)) {
		tmp = z * (x_m * y);
	} else {
		tmp = x_m * (1.0 + (z * (y + -1.0)));
	}
	return x_s * tmp;
}

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)) <= -Double.POSITIVE_INFINITY) {
		tmp = z * (x_m * y);
	} 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 (z * (1.0 - y)) <= -math.inf:
		tmp = z * (x_m * y)
	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 (Float64(z * Float64(1.0 - y)) <= Float64(-Inf))
		tmp = Float64(z * Float64(x_m * y));
	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 ((z * (1.0 - y)) <= -Inf)
		tmp = z * (x_m * y);
	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[N[(z * N[(1.0 - y), $MachinePrecision]), $MachinePrecision], (-Infinity)], N[(z * N[(x$95$m * y), $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 (*.f64 (-.f64 #s(literal 1 binary64) y) z) < -inf.0

   1. Initial program 56.1%

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

   3. Taylor expanded in y around inf 93.8%

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

      1. distribute-rgt-in 93.8%

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

      2. associate-*r* 50.2%

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

      3. *-commutative 50.2%

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

      4. associate-*r* 93.9%

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

      5. *-commutative 93.9%

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

      6. associate-/l* 93.9%

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

      7. associate-*r* 93.9%

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

      8. *-commutative 93.9%

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

      9. distribute-lft-out 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in y around inf 99.8%

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

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

   1. Initial program 98.7%

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

4. Final simplification 98.8%

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

Alternative 4: 98.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -2.15 \cdot 10^{-11} \lor \neg \left(z \leq 1\right):\\ \;\;\;\;\left(y + -1\right) \cdot \left(x\_m \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m + x\_m \cdot \left(z \cdot y\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (or (<= z -2.15e-11) (not (<= z 1.0)))
    (* (+ y -1.0) (* 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 ((z <= -2.15e-11) || !(z <= 1.0)) {
		tmp = (y + -1.0) * (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 ((z <= (-2.15d-11)) .or. (.not. (z <= 1.0d0))) then
        tmp = (y + (-1.0d0)) * (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 ((z <= -2.15e-11) || !(z <= 1.0)) {
		tmp = (y + -1.0) * (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 (z <= -2.15e-11) or not (z <= 1.0):
		tmp = (y + -1.0) * (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 ((z <= -2.15e-11) || !(z <= 1.0))
		tmp = Float64(Float64(y + -1.0) * 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 ((z <= -2.15e-11) || ~((z <= 1.0)))
		tmp = (y + -1.0) * (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[Or[LessEqual[z, -2.15e-11], N[Not[LessEqual[z, 1.0]], $MachinePrecision]], N[(N[(y + -1.0), $MachinePrecision] * 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 2 regimes

2. if z < -2.15000000000000001e-11 or 1 < z

   1. Initial program 92.6%

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

   3. Taylor expanded in z around inf 91.6%

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

      1. associate-*r* 98.9%

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

      2. sub-neg 98.9%

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

      3. metadata-eval 98.9%

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

   5. Simplified 98.9%

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

   if -2.15000000000000001e-11 < z < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0 99.9%

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

   4. Taylor expanded in y around inf 99.0%

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

      1. *-commutative 99.0%

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

   6. Simplified 99.0%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.15 \cdot 10^{-11} \lor \neg \left(z \leq 1\right):\\ \;\;\;\;\left(y + -1\right) \cdot \left(x \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x + x \cdot \left(z \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.65 \cdot 10^{-81} \lor \neg \left(z \leq 2.4 \cdot 10^{+14}\right):\\ \;\;\;\;\left(y + -1\right) \cdot \left(x\_m \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \left(1 - z\right)\\ \end{array} \end{array} \]

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 (or (<= z -1.65e-81) (not (<= z 2.4e+14)))
    (* (+ y -1.0) (* x_m z))
    (* 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 ((z <= -1.65e-81) || !(z <= 2.4e+14)) {
		tmp = (y + -1.0) * (x_m * z);
	} 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 ((z <= (-1.65d-81)) .or. (.not. (z <= 2.4d+14))) then
        tmp = (y + (-1.0d0)) * (x_m * z)
    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 ((z <= -1.65e-81) || !(z <= 2.4e+14)) {
		tmp = (y + -1.0) * (x_m * z);
	} 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 (z <= -1.65e-81) or not (z <= 2.4e+14):
		tmp = (y + -1.0) * (x_m * z)
	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 ((z <= -1.65e-81) || !(z <= 2.4e+14))
		tmp = Float64(Float64(y + -1.0) * Float64(x_m * z));
	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 ((z <= -1.65e-81) || ~((z <= 2.4e+14)))
		tmp = (y + -1.0) * (x_m * z);
	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[z, -1.65e-81], N[Not[LessEqual[z, 2.4e+14]], $MachinePrecision]], N[(N[(y + -1.0), $MachinePrecision] * N[(x$95$m * z), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -1.64999999999999994e-81 or 2.4e14 < z

   1. Initial program 92.8%

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

   3. Taylor expanded in z around inf 89.7%

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

      1. associate-*r* 96.8%

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

      2. sub-neg 96.8%

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

      3. metadata-eval 96.8%

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

   5. Simplified 96.8%

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

   if -1.64999999999999994e-81 < z < 2.4e14

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 80.0%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.65 \cdot 10^{-81} \lor \neg \left(z \leq 2.4 \cdot 10^{+14}\right):\\ \;\;\;\;\left(y + -1\right) \cdot \left(x \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -3.9 \cdot 10^{+52} \lor \neg \left(y \leq 6.5 \cdot 10^{+66}\right):\\ \;\;\;\;z \cdot \left(x\_m \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \left(1 - z\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (or (<= y -3.9e+52) (not (<= y 6.5e+66)))
    (* z (* x_m 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 <= -3.9e+52) || !(y <= 6.5e+66)) {
		tmp = z * (x_m * 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 <= (-3.9d+52)) .or. (.not. (y <= 6.5d+66))) then
        tmp = z * (x_m * 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 <= -3.9e+52) || !(y <= 6.5e+66)) {
		tmp = z * (x_m * 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 <= -3.9e+52) or not (y <= 6.5e+66):
		tmp = z * (x_m * 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 <= -3.9e+52) || !(y <= 6.5e+66))
		tmp = Float64(z * Float64(x_m * 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 <= -3.9e+52) || ~((y <= 6.5e+66)))
		tmp = z * (x_m * 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, -3.9e+52], N[Not[LessEqual[y, 6.5e+66]], $MachinePrecision]], N[(z * N[(x$95$m * y), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < -3.9e52 or 6.5000000000000001e66 < y

   1. Initial program 90.2%

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

   3. Taylor expanded in y around inf 79.8%

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

      1. distribute-rgt-in 79.8%

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

      2. associate-*r* 74.6%

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

      3. *-commutative 74.6%

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

      4. associate-*r* 75.4%

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

      5. *-commutative 75.4%

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

      6. associate-/l* 75.4%

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

      7. associate-*r* 76.8%

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

      8. *-commutative 76.8%

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

      9. distribute-lft-out 91.0%

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

   5. Simplified 91.0%

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

   6. Taylor expanded in y around inf 72.0%

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

   if -3.9e52 < y < 6.5000000000000001e66

   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 93.1%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.9 \cdot 10^{+52} \lor \neg \left(y \leq 6.5 \cdot 10^{+66}\right):\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 83.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -3.1 \cdot 10^{+52} \lor \neg \left(y \leq 1.75 \cdot 10^{+67}\right):\\ \;\;\;\;x\_m \cdot \left(z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \left(1 - z\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (or (<= y -3.1e+52) (not (<= y 1.75e+67)))
    (* 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 <= -3.1e+52) || !(y <= 1.75e+67)) {
		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 <= (-3.1d+52)) .or. (.not. (y <= 1.75d+67))) 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 <= -3.1e+52) || !(y <= 1.75e+67)) {
		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 <= -3.1e+52) or not (y <= 1.75e+67):
		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 <= -3.1e+52) || !(y <= 1.75e+67))
		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 <= -3.1e+52) || ~((y <= 1.75e+67)))
		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, -3.1e+52], N[Not[LessEqual[y, 1.75e+67]], $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 < -3.1e52 or 1.75e67 < y

   1. Initial program 90.2%

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

   3. Taylor expanded in y around inf 68.0%

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

      1. *-commutative 68.0%

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

   5. Simplified 68.0%

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

   if -3.1e52 < y < 1.75e67

   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 93.1%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.1 \cdot 10^{+52} \lor \neg \left(y \leq 1.75 \cdot 10^{+67}\right):\\ \;\;\;\;x \cdot \left(z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -3.1 \cdot 10^{+52}:\\ \;\;\;\;z \cdot \left(x\_m \cdot y\right)\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{+66}:\\ \;\;\;\;x\_m \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x\_m \cdot z\right)\\ \end{array} \end{array} \]

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 -3.1e+52)
    (* z (* x_m y))
    (if (<= y 2.4e+66) (* x_m (- 1.0 z)) (* 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 (y <= -3.1e+52) {
		tmp = z * (x_m * y);
	} else if (y <= 2.4e+66) {
		tmp = x_m * (1.0 - z);
	} else {
		tmp = 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 (y <= (-3.1d+52)) then
        tmp = z * (x_m * y)
    else if (y <= 2.4d+66) then
        tmp = x_m * (1.0d0 - z)
    else
        tmp = 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 (y <= -3.1e+52) {
		tmp = z * (x_m * y);
	} else if (y <= 2.4e+66) {
		tmp = x_m * (1.0 - z);
	} else {
		tmp = 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 y <= -3.1e+52:
		tmp = z * (x_m * y)
	elif y <= 2.4e+66:
		tmp = x_m * (1.0 - z)
	else:
		tmp = 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 (y <= -3.1e+52)
		tmp = Float64(z * Float64(x_m * y));
	elseif (y <= 2.4e+66)
		tmp = Float64(x_m * Float64(1.0 - z));
	else
		tmp = 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 (y <= -3.1e+52)
		tmp = z * (x_m * y);
	elseif (y <= 2.4e+66)
		tmp = x_m * (1.0 - z);
	else
		tmp = 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[y, -3.1e+52], N[(z * N[(x$95$m * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.4e+66], N[(x$95$m * N[(1.0 - z), $MachinePrecision]), $MachinePrecision], N[(y * N[(x$95$m * z), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -3.1e52

   1. Initial program 91.5%

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

   3. Taylor expanded in y around inf 88.0%

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

      1. distribute-rgt-in 88.0%

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

      2. associate-*r* 86.2%

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

      3. *-commutative 86.2%

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

      4. associate-*r* 89.6%

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

      5. *-commutative 89.6%

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

      6. associate-/l* 89.5%

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

      7. associate-*r* 91.2%

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

      8. *-commutative 91.2%

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

      9. distribute-lft-out 94.8%

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

   5. Simplified 94.8%

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

   6. Taylor expanded in y around inf 70.3%

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

   if -3.1e52 < y < 2.4000000000000002e66

   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 93.1%

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

   if 2.4000000000000002e66 < y

   1. Initial program 88.8%

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

   3. Taylor expanded in y around inf 72.3%

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

      1. *-commutative 72.3%

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

   5. Simplified 72.3%

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

      1. associate-*r* 81.4%

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

      2. *-commutative 81.4%

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

   7. Applied egg-rr 81.4%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.1 \cdot 10^{+52}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{+66}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 63.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -60000000000000 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;x\_m \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (* x_s (if (or (<= z -60000000000000.0) (not (<= z 1.0))) (* 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 <= -60000000000000.0) || !(z <= 1.0)) {
		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 <= (-60000000000000.0d0)) .or. (.not. (z <= 1.0d0))) 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 <= -60000000000000.0) || !(z <= 1.0)) {
		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 <= -60000000000000.0) or not (z <= 1.0):
		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 <= -60000000000000.0) || !(z <= 1.0))
		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 <= -60000000000000.0) || ~((z <= 1.0)))
		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, -60000000000000.0], N[Not[LessEqual[z, 1.0]], $MachinePrecision]], N[(x$95$m * (-z)), $MachinePrecision], x$95$m]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -6e13 or 1 < z

   1. Initial program 92.4%

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

   3. Taylor expanded in z around inf 91.4%

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

   4. Taylor expanded in y around 0 57.3%

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

      1. mul-1-neg 57.3%

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

      2. distribute-rgt-neg-in 57.3%

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

   6. Simplified 57.3%

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

   if -6e13 < z < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0 74.8%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -60000000000000 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;x \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 38.0% 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 95.9%

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

3. Taylor expanded in z around 0 36.4%

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

Developer target: 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 2024096 
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSV:hsv from colour-2.3.3, J"
  :precision binary64

  :alt
  (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))))