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

Percentage Accurate: 96.3% → 99.8%

Time: 9.7s

Alternatives: 12

Speedup: 0.8×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.3% 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: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2 \cdot 10^{+31}:\\ \;\;\;\;x\_m + z \cdot \left(x\_m \cdot \left(y + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y + -1, x\_m \cdot z, x\_m\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= x_m 2e+31)
    (+ x_m (* z (* x_m (+ y -1.0))))
    (fma (+ y -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 (x_m <= 2e+31) {
		tmp = x_m + (z * (x_m * (y + -1.0)));
	} else {
		tmp = fma((y + -1.0), (x_m * z), 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 (x_m <= 2e+31)
		tmp = Float64(x_m + Float64(z * Float64(x_m * Float64(y + -1.0))));
	else
		tmp = fma(Float64(y + -1.0), Float64(x_m * z), x_m);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[x$95$m, 2e+31], N[(x$95$m + N[(z * N[(x$95$m * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y + -1.0), $MachinePrecision] * N[(x$95$m * z), $MachinePrecision] + x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.9999999999999999e31

   1. Initial program 93.8%

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

   4. Applied egg-rr 98.9%

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

   if 1.9999999999999999e31 < x

   1. Initial program 100.0%

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

   4. Applied egg-rr 100.0%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+31}:\\ \;\;\;\;x + z \cdot \left(x \cdot \left(y + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y + -1, x \cdot z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x\_m \cdot z, y, x\_m\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;1 - y \leq -2:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;1 - y \leq 1.01:\\ \;\;\;\;x\_m - x\_m \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 (fma (* x_m z) y x_m)))
   (*
    x_s
    (if (<= (- 1.0 y) -2.0)
      t_0
      (if (<= (- 1.0 y) 1.01) (- x_m (* 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 = fma((x_m * z), y, x_m);
	double tmp;
	if ((1.0 - y) <= -2.0) {
		tmp = t_0;
	} else if ((1.0 - y) <= 1.01) {
		tmp = x_m - (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 = fma(Float64(x_m * z), y, x_m)
	tmp = 0.0
	if (Float64(1.0 - y) <= -2.0)
		tmp = t_0;
	elseif (Float64(1.0 - y) <= 1.01)
		tmp = Float64(x_m - Float64(x_m * z));
	else
		tmp = t_0;
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := Block[{t$95$0 = N[(N[(x$95$m * z), $MachinePrecision] * y + x$95$m), $MachinePrecision]}, N[(x$95$s * If[LessEqual[N[(1.0 - y), $MachinePrecision], -2.0], t$95$0, If[LessEqual[N[(1.0 - y), $MachinePrecision], 1.01], N[(x$95$m - N[(x$95$m * z), $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) y) < -2 or 1.01000000000000001 < (-.f64 #s(literal 1 binary64) y)

   1. Initial program 91.2%

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

   4. Applied egg-rr 96.5%

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

   5. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 90.5

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

   7. Simplified 90.5%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 94.9

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

   9. Applied egg-rr 94.9%

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

   if -2 < (-.f64 #s(literal 1 binary64) y) < 1.01000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. distribute-lft-out-- N/A

         \[\leadsto \color{blue}{x \cdot 1 - x \cdot z} \]

      2. *-rgt-identity N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 99.4

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

   5. Simplified 99.4%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - y \leq -2:\\ \;\;\;\;\mathsf{fma}\left(x \cdot z, y, x\right)\\ \mathbf{elif}\;1 - y \leq 1.01:\\ \;\;\;\;x - x \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot z, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \mathsf{fma}\left(z \cdot y, x\_m, x\_m\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;1 - y \leq -2:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;1 - y \leq 1.01:\\ \;\;\;\;x\_m - x\_m \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 (fma (* z y) x_m x_m)))
   (*
    x_s
    (if (<= (- 1.0 y) -2.0)
      t_0
      (if (<= (- 1.0 y) 1.01) (- x_m (* 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 = fma((z * y), x_m, x_m);
	double tmp;
	if ((1.0 - y) <= -2.0) {
		tmp = t_0;
	} else if ((1.0 - y) <= 1.01) {
		tmp = x_m - (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 = fma(Float64(z * y), x_m, x_m)
	tmp = 0.0
	if (Float64(1.0 - y) <= -2.0)
		tmp = t_0;
	elseif (Float64(1.0 - y) <= 1.01)
		tmp = Float64(x_m - Float64(x_m * z));
	else
		tmp = t_0;
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := Block[{t$95$0 = N[(N[(z * y), $MachinePrecision] * x$95$m + x$95$m), $MachinePrecision]}, N[(x$95$s * If[LessEqual[N[(1.0 - y), $MachinePrecision], -2.0], t$95$0, If[LessEqual[N[(1.0 - y), $MachinePrecision], 1.01], N[(x$95$m - N[(x$95$m * z), $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) y) < -2 or 1.01000000000000001 < (-.f64 #s(literal 1 binary64) y)

   1. Initial program 91.2%

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

   4. Applied egg-rr 96.5%

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

   5. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 90.5

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

   7. Simplified 90.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 90.5

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 90.5

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

   9. Applied egg-rr 90.5%

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

   if -2 < (-.f64 #s(literal 1 binary64) y) < 1.01000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. distribute-lft-out-- N/A

         \[\leadsto \color{blue}{x \cdot 1 - x \cdot z} \]

      2. *-rgt-identity N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 99.4

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

   5. Simplified 99.4%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - y \leq -2:\\ \;\;\;\;\mathsf{fma}\left(z \cdot y, x, x\right)\\ \mathbf{elif}\;1 - y \leq 1.01:\\ \;\;\;\;x - x \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot y, x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := z \cdot \left(x\_m \cdot y\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;1 - y \leq -4 \cdot 10^{+83}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;1 - y \leq 2 \cdot 10^{+62}:\\ \;\;\;\;x\_m - x\_m \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 (* z (* x_m y))))
   (*
    x_s
    (if (<= (- 1.0 y) -4e+83)
      t_0
      (if (<= (- 1.0 y) 2e+62) (- x_m (* 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 = z * (x_m * y);
	double tmp;
	if ((1.0 - y) <= -4e+83) {
		tmp = t_0;
	} else if ((1.0 - y) <= 2e+62) {
		tmp = x_m - (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 = z * (x_m * y)
    if ((1.0d0 - y) <= (-4d+83)) then
        tmp = t_0
    else if ((1.0d0 - y) <= 2d+62) then
        tmp = x_m - (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 = z * (x_m * y);
	double tmp;
	if ((1.0 - y) <= -4e+83) {
		tmp = t_0;
	} else if ((1.0 - y) <= 2e+62) {
		tmp = x_m - (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 = z * (x_m * y)
	tmp = 0
	if (1.0 - y) <= -4e+83:
		tmp = t_0
	elif (1.0 - y) <= 2e+62:
		tmp = x_m - (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(z * Float64(x_m * y))
	tmp = 0.0
	if (Float64(1.0 - y) <= -4e+83)
		tmp = t_0;
	elseif (Float64(1.0 - y) <= 2e+62)
		tmp = Float64(x_m - Float64(x_m * 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 = z * (x_m * y);
	tmp = 0.0;
	if ((1.0 - y) <= -4e+83)
		tmp = t_0;
	elseif ((1.0 - y) <= 2e+62)
		tmp = x_m - (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[(z * N[(x$95$m * y), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[N[(1.0 - y), $MachinePrecision], -4e+83], t$95$0, If[LessEqual[N[(1.0 - y), $MachinePrecision], 2e+62], N[(x$95$m - N[(x$95$m * z), $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) y) < -4.00000000000000012e83 or 2.00000000000000007e62 < (-.f64 #s(literal 1 binary64) y)

   1. Initial program 88.4%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 81.3

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

   5. Simplified 81.3%

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

   if -4.00000000000000012e83 < (-.f64 #s(literal 1 binary64) y) < 2.00000000000000007e62

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. distribute-lft-out-- N/A

         \[\leadsto \color{blue}{x \cdot 1 - x \cdot z} \]

      2. *-rgt-identity N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 93.1

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

   5. Simplified 93.1%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - y \leq -4 \cdot 10^{+83}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;1 - y \leq 2 \cdot 10^{+62}:\\ \;\;\;\;x - x \cdot z\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := x\_m \cdot \left(z \cdot y\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;1 - y \leq -4 \cdot 10^{+83}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;1 - y \leq 10^{+122}:\\ \;\;\;\;x\_m - x\_m \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (* x_m (* z y))))
   (*
    x_s
    (if (<= (- 1.0 y) -4e+83)
      t_0
      (if (<= (- 1.0 y) 1e+122) (- x_m (* 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 ((1.0 - y) <= -4e+83) {
		tmp = t_0;
	} else if ((1.0 - y) <= 1e+122) {
		tmp = x_m - (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 ((1.0d0 - y) <= (-4d+83)) then
        tmp = t_0
    else if ((1.0d0 - y) <= 1d+122) then
        tmp = x_m - (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 ((1.0 - y) <= -4e+83) {
		tmp = t_0;
	} else if ((1.0 - y) <= 1e+122) {
		tmp = x_m - (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 (1.0 - y) <= -4e+83:
		tmp = t_0
	elif (1.0 - y) <= 1e+122:
		tmp = x_m - (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 (Float64(1.0 - y) <= -4e+83)
		tmp = t_0;
	elseif (Float64(1.0 - y) <= 1e+122)
		tmp = Float64(x_m - Float64(x_m * 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 ((1.0 - y) <= -4e+83)
		tmp = t_0;
	elseif ((1.0 - y) <= 1e+122)
		tmp = x_m - (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[N[(1.0 - y), $MachinePrecision], -4e+83], t$95$0, If[LessEqual[N[(1.0 - y), $MachinePrecision], 1e+122], N[(x$95$m - N[(x$95$m * z), $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) y) < -4.00000000000000012e83 or 1.00000000000000001e122 < (-.f64 #s(literal 1 binary64) y)

   1. Initial program 89.3%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 76.5

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

   5. Simplified 76.5%

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

   if -4.00000000000000012e83 < (-.f64 #s(literal 1 binary64) y) < 1.00000000000000001e122

   1. Initial program 98.2%

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

   3. Taylor expanded in y around 0

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

      1. distribute-lft-out-- N/A

         \[\leadsto \color{blue}{x \cdot 1 - x \cdot z} \]

      2. *-rgt-identity N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 89.3

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

   5. Simplified 89.3%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - y \leq -4 \cdot 10^{+83}:\\ \;\;\;\;x \cdot \left(z \cdot y\right)\\ \mathbf{elif}\;1 - y \leq 10^{+122}:\\ \;\;\;\;x - x \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -17:\\ \;\;\;\;z \cdot \left(x\_m \cdot y - x\_m\right)\\ \mathbf{elif}\;z \leq 0.0015:\\ \;\;\;\;\mathsf{fma}\left(z \cdot y, x\_m, x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y + -1\right) \cdot \left(x\_m \cdot z\right)\\ \end{array} \end{array} \]

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 (<= z -17.0)
    (* z (- (* x_m y) x_m))
    (if (<= z 0.0015) (fma (* z y) x_m x_m) (* (+ y -1.0) (* 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 <= -17.0) {
		tmp = z * ((x_m * y) - x_m);
	} else if (z <= 0.0015) {
		tmp = fma((z * y), x_m, x_m);
	} else {
		tmp = (y + -1.0) * (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 (z <= -17.0)
		tmp = Float64(z * Float64(Float64(x_m * y) - x_m));
	elseif (z <= 0.0015)
		tmp = fma(Float64(z * y), x_m, x_m);
	else
		tmp = Float64(Float64(y + -1.0) * Float64(x_m * z));
	end
	return Float64(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[z, -17.0], N[(z * N[(N[(x$95$m * y), $MachinePrecision] - x$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 0.0015], N[(N[(z * y), $MachinePrecision] * x$95$m + x$95$m), $MachinePrecision], N[(N[(y + -1.0), $MachinePrecision] * N[(x$95$m * z), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -17

   1. Initial program 97.1%

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

   3. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-commutative N/A

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

      7. cancel-sign-sub N/A

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

      8. mul-1-neg N/A

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

      9. *-rgt-identity N/A

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

      10. distribute-lft-out-- N/A

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

      11. mul-1-neg N/A

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

      12. *-commutative N/A

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

      13. distribute-lft-neg-in N/A

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

      14. associate-*r* N/A

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

      15. distribute-lft-neg-in N/A

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

      16. distribute-rgt-neg-in N/A

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

      17. lower-*.f64 N/A

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

      18. distribute-rgt-out-- N/A

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

      19. *-lft-identity N/A

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

      20. unsub-neg N/A

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

      21. +-commutative N/A

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

      22. distribute-neg-in N/A

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

      23. remove-double-neg N/A

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

      24. unsub-neg N/A

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

      25. lower--.f64 N/A

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

      26. lower-*.f64 98.7

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

   5. Simplified 98.7%

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

   if -17 < z < 0.0015

   1. Initial program 99.9%

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

   4. Applied egg-rr 96.0%

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

   5. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 98.6

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

   7. Simplified 98.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 98.6

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 98.6

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

   9. Applied egg-rr 98.6%

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

   if 0.0015 < z

   1. Initial program 85.1%

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

   3. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-commutative N/A

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

      7. cancel-sign-sub N/A

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

      8. mul-1-neg N/A

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

      9. *-rgt-identity N/A

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

      10. distribute-lft-out-- N/A

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

      11. mul-1-neg N/A

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

      12. *-commutative N/A

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

      13. distribute-lft-neg-in N/A

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

      14. associate-*r* N/A

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

      15. distribute-lft-neg-in N/A

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

      16. distribute-rgt-neg-in N/A

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

      17. lower-*.f64 N/A

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

      18. distribute-rgt-out-- N/A

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

      19. *-lft-identity N/A

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

      20. unsub-neg N/A

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

      21. +-commutative N/A

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

      22. distribute-neg-in N/A

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

      23. remove-double-neg N/A

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

      24. unsub-neg N/A

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

      25. lower--.f64 N/A

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

      26. lower-*.f64 99.6

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

   5. Simplified 99.6%

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

      1. lift-*.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. neg-mul-1 N/A

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

      5. distribute-rgt-in N/A

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

      6. lift-+.f64 N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 99.7

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

   7. Applied egg-rr 99.7%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -17:\\ \;\;\;\;z \cdot \left(x \cdot y - x\right)\\ \mathbf{elif}\;z \leq 0.0015:\\ \;\;\;\;\mathsf{fma}\left(z \cdot y, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y + -1\right) \cdot \left(x \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := z \cdot \left(x\_m \cdot y - x\_m\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -17:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 0.00145:\\ \;\;\;\;\mathsf{fma}\left(z \cdot y, x\_m, x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (* z (- (* x_m y) x_m))))
   (*
    x_s
    (if (<= z -17.0) t_0 (if (<= z 0.00145) (fma (* z y) x_m x_m) t_0)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double t_0 = z * ((x_m * y) - x_m);
	double tmp;
	if (z <= -17.0) {
		tmp = t_0;
	} else if (z <= 0.00145) {
		tmp = fma((z * y), x_m, x_m);
	} else {
		tmp = t_0;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	t_0 = Float64(z * Float64(Float64(x_m * y) - x_m))
	tmp = 0.0
	if (z <= -17.0)
		tmp = t_0;
	elseif (z <= 0.00145)
		tmp = fma(Float64(z * y), x_m, x_m);
	else
		tmp = t_0;
	end
	return Float64(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[(z * N[(N[(x$95$m * y), $MachinePrecision] - x$95$m), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[z, -17.0], t$95$0, If[LessEqual[z, 0.00145], N[(N[(z * y), $MachinePrecision] * x$95$m + x$95$m), $MachinePrecision], t$95$0]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -17 or 0.00145 < z

   1. Initial program 91.0%

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

   3. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-commutative N/A

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

      7. cancel-sign-sub N/A

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

      8. mul-1-neg N/A

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

      9. *-rgt-identity N/A

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

      10. distribute-lft-out-- N/A

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

      11. mul-1-neg N/A

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

      12. *-commutative N/A

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

      13. distribute-lft-neg-in N/A

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

      14. associate-*r* N/A

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

      15. distribute-lft-neg-in N/A

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

      16. distribute-rgt-neg-in N/A

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

      17. lower-*.f64 N/A

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

      18. distribute-rgt-out-- N/A

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

      19. *-lft-identity N/A

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

      20. unsub-neg N/A

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

      21. +-commutative N/A

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

      22. distribute-neg-in N/A

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

      23. remove-double-neg N/A

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

      24. unsub-neg N/A

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

      25. lower--.f64 N/A

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

      26. lower-*.f64 99.2

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

   5. Simplified 99.2%

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

   if -17 < z < 0.00145

   1. Initial program 99.9%

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

   4. Applied egg-rr 95.9%

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

   5. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 98.6

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

   7. Simplified 98.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 98.6

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 98.6

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

   9. Applied egg-rr 98.6%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -17:\\ \;\;\;\;z \cdot \left(x \cdot y - x\right)\\ \mathbf{elif}\;z \leq 0.00145:\\ \;\;\;\;\mathsf{fma}\left(z \cdot y, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(x \cdot y - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 2000000000000:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \left(y + -1\right), x\_m, x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y + -1\right) \cdot \left(x\_m \cdot z\right)\\ \end{array} \end{array} \]

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 (<= z 2000000000000.0)
    (fma (* z (+ y -1.0)) x_m x_m)
    (* (+ y -1.0) (* 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 <= 2000000000000.0) {
		tmp = fma((z * (y + -1.0)), x_m, x_m);
	} else {
		tmp = (y + -1.0) * (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 (z <= 2000000000000.0)
		tmp = fma(Float64(z * Float64(y + -1.0)), x_m, x_m);
	else
		tmp = Float64(Float64(y + -1.0) * Float64(x_m * z));
	end
	return Float64(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[z, 2000000000000.0], N[(N[(z * N[(y + -1.0), $MachinePrecision]), $MachinePrecision] * x$95$m + x$95$m), $MachinePrecision], N[(N[(y + -1.0), $MachinePrecision] * N[(x$95$m * z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2e12

   1. Initial program 98.9%

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

   4. Applied egg-rr 98.9%

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

   if 2e12 < z

   1. Initial program 84.7%

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

   3. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-commutative N/A

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

      7. cancel-sign-sub N/A

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

      8. mul-1-neg N/A

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

      9. *-rgt-identity N/A

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

      10. distribute-lft-out-- N/A

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

      11. mul-1-neg N/A

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

      12. *-commutative N/A

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

      13. distribute-lft-neg-in N/A

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

      14. associate-*r* N/A

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

      15. distribute-lft-neg-in N/A

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

      16. distribute-rgt-neg-in N/A

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

      17. lower-*.f64 N/A

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

      18. distribute-rgt-out-- N/A

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

      19. *-lft-identity N/A

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

      20. unsub-neg N/A

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

      21. +-commutative N/A

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

      22. distribute-neg-in N/A

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

      23. remove-double-neg N/A

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

      24. unsub-neg N/A

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

      25. lower--.f64 N/A

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

      26. lower-*.f64 99.8

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

   5. Simplified 99.8%

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

      1. lift-*.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. neg-mul-1 N/A

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

      5. distribute-rgt-in N/A

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

      6. lift-+.f64 N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 99.9

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

   7. Applied egg-rr 99.9%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2000000000000:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \left(y + -1\right), x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y + -1\right) \cdot \left(x \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 64.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := z \cdot \left(-x\_m\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 0.00145:\\ \;\;\;\;x\_m\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (* z (- x_m))))
   (* x_s (if (<= z -1.0) t_0 (if (<= z 0.00145) x_m t_0)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double t_0 = z * -x_m;
	double tmp;
	if (z <= -1.0) {
		tmp = t_0;
	} else if (z <= 0.00145) {
		tmp = x_m;
	} else {
		tmp = t_0;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = z * -x_m
    if (z <= (-1.0d0)) then
        tmp = t_0
    else if (z <= 0.00145d0) then
        tmp = x_m
    else
        tmp = t_0
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double t_0 = z * -x_m;
	double tmp;
	if (z <= -1.0) {
		tmp = t_0;
	} else if (z <= 0.00145) {
		tmp = x_m;
	} else {
		tmp = t_0;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	t_0 = z * -x_m
	tmp = 0
	if z <= -1.0:
		tmp = t_0
	elif z <= 0.00145:
		tmp = x_m
	else:
		tmp = t_0
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	t_0 = Float64(z * Float64(-x_m))
	tmp = 0.0
	if (z <= -1.0)
		tmp = t_0;
	elseif (z <= 0.00145)
		tmp = x_m;
	else
		tmp = t_0;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	t_0 = z * -x_m;
	tmp = 0.0;
	if (z <= -1.0)
		tmp = t_0;
	elseif (z <= 0.00145)
		tmp = x_m;
	else
		tmp = t_0;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := Block[{t$95$0 = N[(z * (-x$95$m)), $MachinePrecision]}, N[(x$95$s * If[LessEqual[z, -1.0], t$95$0, If[LessEqual[z, 0.00145], x$95$m, t$95$0]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 0.00145 < z

   1. Initial program 91.1%

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

   3. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-commutative N/A

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

      7. cancel-sign-sub N/A

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

      8. mul-1-neg N/A

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

      9. *-rgt-identity N/A

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

      10. distribute-lft-out-- N/A

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

      11. mul-1-neg N/A

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

      12. *-commutative N/A

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

      13. distribute-lft-neg-in N/A

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

      14. associate-*r* N/A

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

      15. distribute-lft-neg-in N/A

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

      16. distribute-rgt-neg-in N/A

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

      17. lower-*.f64 N/A

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

      18. distribute-rgt-out-- N/A

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

      19. *-lft-identity N/A

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

      20. unsub-neg N/A

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

      21. +-commutative N/A

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

      22. distribute-neg-in N/A

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

      23. remove-double-neg N/A

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

      24. unsub-neg N/A

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

      25. lower--.f64 N/A

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

      26. lower-*.f64 99.2

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

   5. Simplified 99.2%

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

   6. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 55.0

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

   8. Simplified 55.0%

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

   if -1 < z < 0.00145

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

   5. Simplified 75.6%

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

      1. *-rgt-identity 75.6

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

   7. Applied egg-rr 75.6%

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

Alternative 10: 65.4% accurate, 1.9× speedup

Math

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

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 (* 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) {
	return x_s * (x_m - (x_m * z));
}

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 - (x_m * z))
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 - (x_m * z));
}

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 - (x_m * z))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	return Float64(x_s * Float64(x_m - Float64(x_m * z)))
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 - (x_m * z));
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 * N[(x$95$m - N[(x$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.1%

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

3. Taylor expanded in y around 0

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

   1. distribute-lft-out-- N/A

      \[\leadsto \color{blue}{x \cdot 1 - x \cdot z} \]

   2. *-rgt-identity N/A

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

   3. lower--.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 65.5

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

5. Simplified 65.5%

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

6. Final simplification 65.5%

   \[\leadsto x - x \cdot z \]
7. Add Preprocessing

Alternative 11: 65.4% accurate, 1.9× speedup

Math

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

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 (- 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) {
	return x_s * (x_m * (1.0 - z));
}

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 * (1.0d0 - z))
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 * (1.0 - z));
}

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 * (1.0 - z))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	return Float64(x_s * Float64(x_m * Float64(1.0 - z)))
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 * (1.0 - z));
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 * N[(x$95$m * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.1%

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

3. Taylor expanded in y around 0

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

   1. lower--.f64 65.4

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

5. Simplified 65.4%

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

Alternative 12: 39.0% accurate, 17.0× 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 95.1%

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

3. Taylor expanded in z around 0

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

5. Simplified 36.6%

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

   1. *-rgt-identity 36.6

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

7. Applied egg-rr 36.6%

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

Developer Target 1: 99.6% 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 2024212 
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSV:hsv from colour-2.3.3, J"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< (* x (- 1 (* (- 1 y) z))) -161819597360704900000000000000000000000000000000000) (+ x (* (- 1 y) (* (- z) x))) (if (< (* x (- 1 (* (- 1 y) z))) 389223764966390300000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- (* (* x y) z) (- (* x z) x)) (+ x (* (- 1 y) (* (- z) x))))))

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