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

Percentage Accurate: 96.2% → 98.9%

Time: 10.9s

Alternatives: 7

Speedup: 0.7×

Specification

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return x * (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 - (y * z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return x * (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 - (y * z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z] = \mathsf{sort}([x_m, y, z])\\ \\ \begin{array}{l} t_0 := x\_m \cdot \left(1 - y \cdot z\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+244}:\\ \;\;\;\;\left(-z\right) \cdot \left(x\_m \cdot y\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+303}:\\ \;\;\;\;\mathsf{fma}\left(-y \cdot z, x\_m, x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;-y \cdot \left(x\_m \cdot z\right)\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (* x_m (- 1.0 (* y z)))))
   (*
    x_s
    (if (<= t_0 -2e+244)
      (* (- z) (* x_m y))
      (if (<= t_0 2e+303) (fma (- (* y z)) x_m x_m) (- (* y (* x_m z))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z);
double code(double x_s, double x_m, double y, double z) {
	double t_0 = x_m * (1.0 - (y * z));
	double tmp;
	if (t_0 <= -2e+244) {
		tmp = -z * (x_m * y);
	} else if (t_0 <= 2e+303) {
		tmp = fma(-(y * z), x_m, x_m);
	} else {
		tmp = -(y * (x_m * z));
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z = sort([x_m, y, z])
function code(x_s, x_m, y, z)
	t_0 = Float64(x_m * Float64(1.0 - Float64(y * z)))
	tmp = 0.0
	if (t_0 <= -2e+244)
		tmp = Float64(Float64(-z) * Float64(x_m * y));
	elseif (t_0 <= 2e+303)
		tmp = fma(Float64(-Float64(y * z)), x_m, x_m);
	else
		tmp = Float64(-Float64(y * 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]
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_] := Block[{t$95$0 = N[(x$95$m * N[(1.0 - N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$0, -2e+244], N[((-z) * N[(x$95$m * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+303], N[((-N[(y * z), $MachinePrecision]) * x$95$m + x$95$m), $MachinePrecision], (-N[(y * N[(x$95$m * z), $MachinePrecision]), $MachinePrecision])]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x (-.f64 #s(literal 1 binary64) (*.f64 y z))) < -2.00000000000000015e244

   1. Initial program 78.0%

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

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

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

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 69.1

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

   5. Applied rewrites 69.1%

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

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

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

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

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

      3. lift-neg.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 91.0

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

   7. Applied rewrites 91.0%

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

   if -2.00000000000000015e244 < (*.f64 x (-.f64 #s(literal 1 binary64) (*.f64 y z))) < 2e303

   1. Initial program 99.9%

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

      1. lift-*.f64 N/A

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

      2. sub-neg N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 99.9

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

   4. Applied rewrites 99.9%

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

   if 2e303 < (*.f64 x (-.f64 #s(literal 1 binary64) (*.f64 y z)))

   1. Initial program 83.2%

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

   3. Taylor expanded in y around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 99.9

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

   5. Applied rewrites 99.9%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \left(1 - y \cdot z\right) \leq -2 \cdot 10^{+244}:\\ \;\;\;\;\left(-z\right) \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;x \cdot \left(1 - y \cdot z\right) \leq 2 \cdot 10^{+303}:\\ \;\;\;\;\mathsf{fma}\left(-y \cdot z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;-y \cdot \left(x \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z] = \mathsf{sort}([x_m, y, z])\\ \\ \begin{array}{l} t_0 := -x\_m \cdot \left(y \cdot z\right)\\ t_1 := 1 - y \cdot z\\ t_2 := -y \cdot \left(x\_m \cdot z\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -500000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;x\_m\\ \mathbf{elif}\;t\_1 \leq 10^{+292}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (- (* x_m (* y z))))
        (t_1 (- 1.0 (* y z)))
        (t_2 (- (* y (* x_m z)))))
   (*
    x_s
    (if (<= t_1 (- INFINITY))
      t_2
      (if (<= t_1 -500000.0)
        t_0
        (if (<= t_1 2.0) x_m (if (<= t_1 1e+292) t_0 t_2)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z);
double code(double x_s, double x_m, double y, double z) {
	double t_0 = -(x_m * (y * z));
	double t_1 = 1.0 - (y * z);
	double t_2 = -(y * (x_m * z));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_1 <= -500000.0) {
		tmp = t_0;
	} else if (t_1 <= 2.0) {
		tmp = x_m;
	} else if (t_1 <= 1e+292) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return x_s * tmp;
}

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z;
public static double code(double x_s, double x_m, double y, double z) {
	double t_0 = -(x_m * (y * z));
	double t_1 = 1.0 - (y * z);
	double t_2 = -(y * (x_m * z));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else if (t_1 <= -500000.0) {
		tmp = t_0;
	} else if (t_1 <= 2.0) {
		tmp = x_m;
	} else if (t_1 <= 1e+292) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z] = sort([x_m, y, z])
def code(x_s, x_m, y, z):
	t_0 = -(x_m * (y * z))
	t_1 = 1.0 - (y * z)
	t_2 = -(y * (x_m * z))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = t_2
	elif t_1 <= -500000.0:
		tmp = t_0
	elif t_1 <= 2.0:
		tmp = x_m
	elif t_1 <= 1e+292:
		tmp = t_0
	else:
		tmp = t_2
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z = sort([x_m, y, z])
function code(x_s, x_m, y, z)
	t_0 = Float64(-Float64(x_m * Float64(y * z)))
	t_1 = Float64(1.0 - Float64(y * z))
	t_2 = Float64(-Float64(y * Float64(x_m * z)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_1 <= -500000.0)
		tmp = t_0;
	elseif (t_1 <= 2.0)
		tmp = x_m;
	elseif (t_1 <= 1e+292)
		tmp = t_0;
	else
		tmp = t_2;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z = num2cell(sort([x_m, y, z])){:}
function tmp_2 = code(x_s, x_m, y, z)
	t_0 = -(x_m * (y * z));
	t_1 = 1.0 - (y * z);
	t_2 = -(y * (x_m * z));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = t_2;
	elseif (t_1 <= -500000.0)
		tmp = t_0;
	elseif (t_1 <= 2.0)
		tmp = x_m;
	elseif (t_1 <= 1e+292)
		tmp = t_0;
	else
		tmp = t_2;
	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]
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_] := Block[{t$95$0 = (-N[(x$95$m * N[(y * z), $MachinePrecision]), $MachinePrecision])}, Block[{t$95$1 = N[(1.0 - N[(y * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = (-N[(y * N[(x$95$m * z), $MachinePrecision]), $MachinePrecision])}, N[(x$95$s * If[LessEqual[t$95$1, (-Infinity)], t$95$2, If[LessEqual[t$95$1, -500000.0], t$95$0, If[LessEqual[t$95$1, 2.0], x$95$m, If[LessEqual[t$95$1, 1e+292], t$95$0, t$95$2]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 #s(literal 1 binary64) (*.f64 y z)) < -inf.0 or 1e292 < (-.f64 #s(literal 1 binary64) (*.f64 y z))

   1. Initial program 68.5%

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

   3. Taylor expanded in y around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 99.9

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

   5. Applied rewrites 99.9%

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

   if -inf.0 < (-.f64 #s(literal 1 binary64) (*.f64 y z)) < -5e5 or 2 < (-.f64 #s(literal 1 binary64) (*.f64 y z)) < 1e292

   1. Initial program 99.8%

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

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

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

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 96.1

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

   5. Applied rewrites 96.1%

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

   if -5e5 < (-.f64 #s(literal 1 binary64) (*.f64 y z)) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 98.0%

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

      1. *-rgt-identity 98.0

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

   7. Applied rewrites 98.0%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - y \cdot z \leq -\infty:\\ \;\;\;\;-y \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;1 - y \cdot z \leq -500000:\\ \;\;\;\;-x \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;1 - y \cdot z \leq 2:\\ \;\;\;\;x\\ \mathbf{elif}\;1 - y \cdot z \leq 10^{+292}:\\ \;\;\;\;-x \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;-y \cdot \left(x \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z] = \mathsf{sort}([x_m, y, z])\\ \\ \begin{array}{l} t_0 := 1 - y \cdot z\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -500000:\\ \;\;\;\;\left(-z\right) \cdot \left(x\_m \cdot y\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;x\_m\\ \mathbf{elif}\;t\_0 \leq 10^{+292}:\\ \;\;\;\;-x\_m \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;-y \cdot \left(x\_m \cdot z\right)\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (* y z))))
   (*
    x_s
    (if (<= t_0 -500000.0)
      (* (- z) (* x_m y))
      (if (<= t_0 2.0)
        x_m
        (if (<= t_0 1e+292) (- (* x_m (* y z))) (- (* y (* x_m z)))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z);
double code(double x_s, double x_m, double y, double z) {
	double t_0 = 1.0 - (y * z);
	double tmp;
	if (t_0 <= -500000.0) {
		tmp = -z * (x_m * y);
	} else if (t_0 <= 2.0) {
		tmp = x_m;
	} else if (t_0 <= 1e+292) {
		tmp = -(x_m * (y * z));
	} else {
		tmp = -(y * (x_m * z));
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
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 = 1.0d0 - (y * z)
    if (t_0 <= (-500000.0d0)) then
        tmp = -z * (x_m * y)
    else if (t_0 <= 2.0d0) then
        tmp = x_m
    else if (t_0 <= 1d+292) then
        tmp = -(x_m * (y * 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);
assert x_m < y && y < z;
public static double code(double x_s, double x_m, double y, double z) {
	double t_0 = 1.0 - (y * z);
	double tmp;
	if (t_0 <= -500000.0) {
		tmp = -z * (x_m * y);
	} else if (t_0 <= 2.0) {
		tmp = x_m;
	} else if (t_0 <= 1e+292) {
		tmp = -(x_m * (y * z));
	} else {
		tmp = -(y * (x_m * z));
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z] = sort([x_m, y, z])
def code(x_s, x_m, y, z):
	t_0 = 1.0 - (y * z)
	tmp = 0
	if t_0 <= -500000.0:
		tmp = -z * (x_m * y)
	elif t_0 <= 2.0:
		tmp = x_m
	elif t_0 <= 1e+292:
		tmp = -(x_m * (y * z))
	else:
		tmp = -(y * (x_m * z))
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z = sort([x_m, y, z])
function code(x_s, x_m, y, z)
	t_0 = Float64(1.0 - Float64(y * z))
	tmp = 0.0
	if (t_0 <= -500000.0)
		tmp = Float64(Float64(-z) * Float64(x_m * y));
	elseif (t_0 <= 2.0)
		tmp = x_m;
	elseif (t_0 <= 1e+292)
		tmp = Float64(-Float64(x_m * Float64(y * z)));
	else
		tmp = Float64(-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);
x_m, y, z = num2cell(sort([x_m, y, z])){:}
function tmp_2 = code(x_s, x_m, y, z)
	t_0 = 1.0 - (y * z);
	tmp = 0.0;
	if (t_0 <= -500000.0)
		tmp = -z * (x_m * y);
	elseif (t_0 <= 2.0)
		tmp = x_m;
	elseif (t_0 <= 1e+292)
		tmp = -(x_m * (y * 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]
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_] := Block[{t$95$0 = N[(1.0 - N[(y * z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$0, -500000.0], N[((-z) * N[(x$95$m * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2.0], x$95$m, If[LessEqual[t$95$0, 1e+292], (-N[(x$95$m * N[(y * z), $MachinePrecision]), $MachinePrecision]), (-N[(y * N[(x$95$m * z), $MachinePrecision]), $MachinePrecision])]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if (-.f64 #s(literal 1 binary64) (*.f64 y z)) < -5e5

   1. Initial program 89.5%

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

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

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

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 87.2

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

   5. Applied rewrites 87.2%

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

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

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

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

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

      3. lift-neg.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 93.1

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

   7. Applied rewrites 93.1%

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

   if -5e5 < (-.f64 #s(literal 1 binary64) (*.f64 y z)) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 98.0%

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

      1. *-rgt-identity 98.0

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

   7. Applied rewrites 98.0%

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

   if 2 < (-.f64 #s(literal 1 binary64) (*.f64 y z)) < 1e292

   1. Initial program 99.8%

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

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

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

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 95.3

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

   5. Applied rewrites 95.3%

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

   if 1e292 < (-.f64 #s(literal 1 binary64) (*.f64 y z))

   1. Initial program 74.5%

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

   3. Taylor expanded in y around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 99.9

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

   5. Applied rewrites 99.9%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - y \cdot z \leq -500000:\\ \;\;\;\;\left(-z\right) \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;1 - y \cdot z \leq 2:\\ \;\;\;\;x\\ \mathbf{elif}\;1 - y \cdot z \leq 10^{+292}:\\ \;\;\;\;-x \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;-y \cdot \left(x \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z] = \mathsf{sort}([x_m, y, z])\\ \\ \begin{array}{l} t_0 := x\_m \cdot \left(1 - y \cdot z\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+244}:\\ \;\;\;\;\left(-z\right) \cdot \left(x\_m \cdot y\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+303}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;-y \cdot \left(x\_m \cdot z\right)\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (* x_m (- 1.0 (* y z)))))
   (*
    x_s
    (if (<= t_0 -2e+244)
      (* (- z) (* x_m y))
      (if (<= t_0 2e+303) t_0 (- (* y (* x_m z))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z);
double code(double x_s, double x_m, double y, double z) {
	double t_0 = x_m * (1.0 - (y * z));
	double tmp;
	if (t_0 <= -2e+244) {
		tmp = -z * (x_m * y);
	} else if (t_0 <= 2e+303) {
		tmp = t_0;
	} else {
		tmp = -(y * (x_m * z));
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
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 * (1.0d0 - (y * z))
    if (t_0 <= (-2d+244)) then
        tmp = -z * (x_m * y)
    else if (t_0 <= 2d+303) then
        tmp = t_0
    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);
assert x_m < y && y < z;
public static double code(double x_s, double x_m, double y, double z) {
	double t_0 = x_m * (1.0 - (y * z));
	double tmp;
	if (t_0 <= -2e+244) {
		tmp = -z * (x_m * y);
	} else if (t_0 <= 2e+303) {
		tmp = t_0;
	} else {
		tmp = -(y * (x_m * z));
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z] = sort([x_m, y, z])
def code(x_s, x_m, y, z):
	t_0 = x_m * (1.0 - (y * z))
	tmp = 0
	if t_0 <= -2e+244:
		tmp = -z * (x_m * y)
	elif t_0 <= 2e+303:
		tmp = t_0
	else:
		tmp = -(y * (x_m * z))
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z = sort([x_m, y, z])
function code(x_s, x_m, y, z)
	t_0 = Float64(x_m * Float64(1.0 - Float64(y * z)))
	tmp = 0.0
	if (t_0 <= -2e+244)
		tmp = Float64(Float64(-z) * Float64(x_m * y));
	elseif (t_0 <= 2e+303)
		tmp = t_0;
	else
		tmp = Float64(-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);
x_m, y, z = num2cell(sort([x_m, y, z])){:}
function tmp_2 = code(x_s, x_m, y, z)
	t_0 = x_m * (1.0 - (y * z));
	tmp = 0.0;
	if (t_0 <= -2e+244)
		tmp = -z * (x_m * y);
	elseif (t_0 <= 2e+303)
		tmp = t_0;
	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]
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_] := Block[{t$95$0 = N[(x$95$m * N[(1.0 - N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$0, -2e+244], N[((-z) * N[(x$95$m * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+303], t$95$0, (-N[(y * N[(x$95$m * z), $MachinePrecision]), $MachinePrecision])]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x (-.f64 #s(literal 1 binary64) (*.f64 y z))) < -2.00000000000000015e244

   1. Initial program 78.0%

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

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

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

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 69.1

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

   5. Applied rewrites 69.1%

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

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

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

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

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

      3. lift-neg.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 91.0

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

   7. Applied rewrites 91.0%

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

   if -2.00000000000000015e244 < (*.f64 x (-.f64 #s(literal 1 binary64) (*.f64 y z))) < 2e303

   1. Initial program 99.9%

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

   if 2e303 < (*.f64 x (-.f64 #s(literal 1 binary64) (*.f64 y z)))

   1. Initial program 83.2%

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

   3. Taylor expanded in y around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 99.9

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

   5. Applied rewrites 99.9%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \left(1 - y \cdot z\right) \leq -2 \cdot 10^{+244}:\\ \;\;\;\;\left(-z\right) \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;x \cdot \left(1 - y \cdot z\right) \leq 2 \cdot 10^{+303}:\\ \;\;\;\;x \cdot \left(1 - y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;-y \cdot \left(x \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z] = \mathsf{sort}([x_m, y, z])\\ \\ \begin{array}{l} t_0 := 1 - y \cdot z\\ t_1 := -x\_m \cdot \left(y \cdot z\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -500000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;x\_m\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (* y z))) (t_1 (- (* x_m (* y z)))))
   (* x_s (if (<= t_0 -500000.0) t_1 (if (<= t_0 2.0) x_m t_1)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z);
double code(double x_s, double x_m, double y, double z) {
	double t_0 = 1.0 - (y * z);
	double t_1 = -(x_m * (y * z));
	double tmp;
	if (t_0 <= -500000.0) {
		tmp = t_1;
	} else if (t_0 <= 2.0) {
		tmp = x_m;
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
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 = 1.0d0 - (y * z)
    t_1 = -(x_m * (y * z))
    if (t_0 <= (-500000.0d0)) then
        tmp = t_1
    else if (t_0 <= 2.0d0) then
        tmp = x_m
    else
        tmp = t_1
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z;
public static double code(double x_s, double x_m, double y, double z) {
	double t_0 = 1.0 - (y * z);
	double t_1 = -(x_m * (y * z));
	double tmp;
	if (t_0 <= -500000.0) {
		tmp = t_1;
	} else if (t_0 <= 2.0) {
		tmp = x_m;
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z] = sort([x_m, y, z])
def code(x_s, x_m, y, z):
	t_0 = 1.0 - (y * z)
	t_1 = -(x_m * (y * z))
	tmp = 0
	if t_0 <= -500000.0:
		tmp = t_1
	elif t_0 <= 2.0:
		tmp = x_m
	else:
		tmp = t_1
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z = sort([x_m, y, z])
function code(x_s, x_m, y, z)
	t_0 = Float64(1.0 - Float64(y * z))
	t_1 = Float64(-Float64(x_m * Float64(y * z)))
	tmp = 0.0
	if (t_0 <= -500000.0)
		tmp = t_1;
	elseif (t_0 <= 2.0)
		tmp = x_m;
	else
		tmp = t_1;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z = num2cell(sort([x_m, y, z])){:}
function tmp_2 = code(x_s, x_m, y, z)
	t_0 = 1.0 - (y * z);
	t_1 = -(x_m * (y * z));
	tmp = 0.0;
	if (t_0 <= -500000.0)
		tmp = t_1;
	elseif (t_0 <= 2.0)
		tmp = x_m;
	else
		tmp = t_1;
	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]
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_] := Block[{t$95$0 = N[(1.0 - N[(y * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-N[(x$95$m * N[(y * z), $MachinePrecision]), $MachinePrecision])}, N[(x$95$s * If[LessEqual[t$95$0, -500000.0], t$95$1, If[LessEqual[t$95$0, 2.0], x$95$m, t$95$1]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (*.f64 y z)) < -5e5 or 2 < (-.f64 #s(literal 1 binary64) (*.f64 y z))

   1. Initial program 89.7%

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

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

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

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 87.2

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

   5. Applied rewrites 87.2%

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

   if -5e5 < (-.f64 #s(literal 1 binary64) (*.f64 y z)) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 98.0%

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

      1. *-rgt-identity 98.0

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

   7. Applied rewrites 98.0%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - y \cdot z \leq -500000:\\ \;\;\;\;-x \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;1 - y \cdot z \leq 2:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-x \cdot \left(y \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z] = \mathsf{sort}([x_m, y, z])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.2 \cdot 10^{-100}:\\ \;\;\;\;\mathsf{fma}\left(-x\_m \cdot y, z, x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-y \cdot z, x\_m, x\_m\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= x_m 1.2e-100) (fma (- (* x_m y)) z x_m) (fma (- (* y z)) x_m x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (x_m <= 1.2e-100) {
		tmp = fma(-(x_m * y), z, x_m);
	} else {
		tmp = fma(-(y * z), x_m, x_m);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z = sort([x_m, y, z])
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (x_m <= 1.2e-100)
		tmp = fma(Float64(-Float64(x_m * y)), z, x_m);
	else
		tmp = fma(Float64(-Float64(y * z)), x_m, 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]
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[x$95$m, 1.2e-100], N[((-N[(x$95$m * y), $MachinePrecision]) * z + x$95$m), $MachinePrecision], N[((-N[(y * z), $MachinePrecision]) * x$95$m + x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.2000000000000001e-100

   1. Initial program 91.1%

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

      1. lift-*.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. associate-*r* N/A

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

      8. *-rgt-identity N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-neg.f64 96.4

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

   4. Applied rewrites 96.4%

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

   if 1.2000000000000001e-100 < x

   1. Initial program 99.9%

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

      1. lift-*.f64 N/A

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

      2. sub-neg N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 99.9

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

   4. Applied rewrites 99.9%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.2 \cdot 10^{-100}:\\ \;\;\;\;\mathsf{fma}\left(-x \cdot y, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-y \cdot z, x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 49.8% accurate, 14.0× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z) :precision binary64 (* x_s x_m))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z);
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)
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
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);
assert x_m < y && y < z;
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)
[x_m, y, z] = sort([x_m, y, z])
def code(x_s, x_m, y, z):
	return x_s * x_m

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z = sort([x_m, y, z])
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);
x_m, y, z = num2cell(sort([x_m, y, z])){:}
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]
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * x$95$m), $MachinePrecision]

Derivation

1. Initial program 94.4%

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

3. Taylor expanded in y around 0

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

5. Applied rewrites 46.2%

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

   1. *-rgt-identity 46.2

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

7. Applied rewrites 46.2%

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

Reproduce

herbie shell --seed 2024214 
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSV:hsv from colour-2.3.3, I"
  :precision binary64
  (* x (- 1.0 (* y z))))