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

Percentage Accurate: 96.0% → 99.3%

Time: 10.8s

Alternatives: 7

Speedup: 0.3×

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: 96.0% 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: 99.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} t_0 := \left(-z\right) \cdot x\\ \mathbf{if}\;z \cdot y \leq -1 \cdot 10^{+286}:\\ \;\;\;\;t\_0 \cdot y\\ \mathbf{elif}\;z \cdot y \leq 5 \cdot 10^{+83}:\\ \;\;\;\;\left(1 - z \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\mathsf{fma}\left(t\_0, y, x\right)}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- z) x)))
   (if (<= (* z y) -1e+286)
     (* t_0 y)
     (if (<= (* z y) 5e+83)
       (* (- 1.0 (* z y)) x)
       (/ 1.0 (/ 1.0 (fma t_0 y x)))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double t_0 = -z * x;
	double tmp;
	if ((z * y) <= -1e+286) {
		tmp = t_0 * y;
	} else if ((z * y) <= 5e+83) {
		tmp = (1.0 - (z * y)) * x;
	} else {
		tmp = 1.0 / (1.0 / fma(t_0, y, x));
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	t_0 = Float64(Float64(-z) * x)
	tmp = 0.0
	if (Float64(z * y) <= -1e+286)
		tmp = Float64(t_0 * y);
	elseif (Float64(z * y) <= 5e+83)
		tmp = Float64(Float64(1.0 - Float64(z * y)) * x);
	else
		tmp = Float64(1.0 / Float64(1.0 / fma(t_0, y, x)));
	end
	return tmp
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := Block[{t$95$0 = N[((-z) * x), $MachinePrecision]}, If[LessEqual[N[(z * y), $MachinePrecision], -1e+286], N[(t$95$0 * y), $MachinePrecision], If[LessEqual[N[(z * y), $MachinePrecision], 5e+83], N[(N[(1.0 - N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], N[(1.0 / N[(1.0 / N[(t$95$0 * y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 y z) < -1.00000000000000003e286

   1. Initial program 63.9%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. associate-*r* N/A

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

      9. *-rgt-identity N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 99.6

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

   4. Applied rewrites 99.6%

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

   6. Applied rewrites 0.4%

      \[\leadsto \color{blue}{\frac{\frac{{\left(\left(\left(-z\right) \cdot x\right) \cdot y\right)}^{2}}{x \cdot \mathsf{fma}\left(-z, y, -1\right)} \cdot \frac{{\left(\left(\left(-z\right) \cdot x\right) \cdot y\right)}^{2}}{x \cdot \mathsf{fma}\left(-z, y, -1\right)} - \frac{\left(-x\right) \cdot x}{\mathsf{fma}\left(y \cdot x, z, x\right)} \cdot \frac{\left(-x\right) \cdot x}{\mathsf{fma}\left(y \cdot x, z, x\right)}}{\mathsf{fma}\left({\left(\left(\left(-z\right) \cdot x\right) \cdot y\right)}^{2}, \frac{-1}{\mathsf{fma}\left(y \cdot x, z, x\right)}, \frac{\left(-x\right) \cdot x}{\mathsf{fma}\left(y \cdot x, z, x\right)}\right)}} \]

   7. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 99.6

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

   9. Applied rewrites 99.6%

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

   11. Applied rewrites 99.9%

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

   if -1.00000000000000003e286 < (*.f64 y z) < 5.00000000000000029e83

   1. Initial program 99.8%

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

   if 5.00000000000000029e83 < (*.f64 y z)

   1. Initial program 81.9%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. associate-*r* N/A

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

      9. *-rgt-identity N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 99.0

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

   4. Applied rewrites 99.0%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-neg.f64 N/A

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

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

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

      6. *-commutative N/A

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

      7. distribute-lft-neg-out N/A

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

      8. lift-neg.f64 N/A

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

      9. associate-*l* N/A

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

      10. lift-*.f64 N/A

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

      11. flip3-+ N/A

          \[\leadsto \color{blue}{\frac{{\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3} + {x}^{3}}{\left(\left(x \cdot \left(-z\right)\right) \cdot y\right) \cdot \left(\left(x \cdot \left(-z\right)\right) \cdot y\right) + \left(x \cdot x - \left(\left(x \cdot \left(-z\right)\right) \cdot y\right) \cdot x\right)}} \]

      12. clear-num N/A

          \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(x \cdot \left(-z\right)\right) \cdot y\right) \cdot \left(\left(x \cdot \left(-z\right)\right) \cdot y\right) + \left(x \cdot x - \left(\left(x \cdot \left(-z\right)\right) \cdot y\right) \cdot x\right)}{{\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3} + {x}^{3}}}} \]

      13. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(x \cdot \left(-z\right)\right) \cdot y\right) \cdot \left(\left(x \cdot \left(-z\right)\right) \cdot y\right) + \left(x \cdot x - \left(\left(x \cdot \left(-z\right)\right) \cdot y\right) \cdot x\right)}{{\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3} + {x}^{3}}}} \]

   6. Applied rewrites 96.8%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot y \leq -1 \cdot 10^{+286}:\\ \;\;\;\;\left(\left(-z\right) \cdot x\right) \cdot y\\ \mathbf{elif}\;z \cdot y \leq 5 \cdot 10^{+83}:\\ \;\;\;\;\left(1 - z \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\mathsf{fma}\left(\left(-z\right) \cdot x, y, x\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} t_0 := \left(1 - z \cdot y\right) \cdot x\\ t_1 := \mathsf{fma}\left(\left(-z\right) \cdot x, y, x\right)\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-23}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{+307}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- 1.0 (* z y)) x)) (t_1 (fma (* (- z) x) y x)))
   (if (<= t_0 -5e-23) t_1 (if (<= t_0 1e+307) t_0 t_1))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double t_0 = (1.0 - (z * y)) * x;
	double t_1 = fma((-z * x), y, x);
	double tmp;
	if (t_0 <= -5e-23) {
		tmp = t_1;
	} else if (t_0 <= 1e+307) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	t_0 = Float64(Float64(1.0 - Float64(z * y)) * x)
	t_1 = fma(Float64(Float64(-z) * x), y, x)
	tmp = 0.0
	if (t_0 <= -5e-23)
		tmp = t_1;
	elseif (t_0 <= 1e+307)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := Block[{t$95$0 = N[(N[(1.0 - N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$1 = N[(N[((-z) * x), $MachinePrecision] * y + x), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-23], t$95$1, If[LessEqual[t$95$0, 1e+307], t$95$0, t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x (-.f64 #s(literal 1 binary64) (*.f64 y z))) < -5.0000000000000002e-23 or 9.99999999999999986e306 < (*.f64 x (-.f64 #s(literal 1 binary64) (*.f64 y z)))

   1. Initial program 87.5%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. associate-*r* N/A

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

      10. *-rgt-identity N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 93.9

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

   4. Applied rewrites 93.9%

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

   if -5.0000000000000002e-23 < (*.f64 x (-.f64 #s(literal 1 binary64) (*.f64 y z))) < 9.99999999999999986e306

   1. Initial program 99.8%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - z \cdot y\right) \cdot x \leq -5 \cdot 10^{-23}:\\ \;\;\;\;\mathsf{fma}\left(\left(-z\right) \cdot x, y, x\right)\\ \mathbf{elif}\;\left(1 - z \cdot y\right) \cdot x \leq 10^{+307}:\\ \;\;\;\;\left(1 - z \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(-z\right) \cdot x, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} t_0 := \left(1 - z \cdot y\right) \cdot x\\ t_1 := \left(\left(-z\right) \cdot x\right) \cdot y\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+100}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+307}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- 1.0 (* z y)) x)) (t_1 (* (* (- z) x) y)))
   (if (<= t_0 -5e+100) t_1 (if (<= t_0 2e+307) t_0 t_1))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double t_0 = (1.0 - (z * y)) * x;
	double t_1 = (-z * x) * y;
	double tmp;
	if (t_0 <= -5e+100) {
		tmp = t_1;
	} else if (t_0 <= 2e+307) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
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 = (1.0d0 - (z * y)) * x
    t_1 = (-z * x) * y
    if (t_0 <= (-5d+100)) then
        tmp = t_1
    else if (t_0 <= 2d+307) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double t_0 = (1.0 - (z * y)) * x;
	double t_1 = (-z * x) * y;
	double tmp;
	if (t_0 <= -5e+100) {
		tmp = t_1;
	} else if (t_0 <= 2e+307) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	t_0 = (1.0 - (z * y)) * x
	t_1 = (-z * x) * y
	tmp = 0
	if t_0 <= -5e+100:
		tmp = t_1
	elif t_0 <= 2e+307:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	t_0 = Float64(Float64(1.0 - Float64(z * y)) * x)
	t_1 = Float64(Float64(Float64(-z) * x) * y)
	tmp = 0.0
	if (t_0 <= -5e+100)
		tmp = t_1;
	elseif (t_0 <= 2e+307)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	t_0 = (1.0 - (z * y)) * x;
	t_1 = (-z * x) * y;
	tmp = 0.0;
	if (t_0 <= -5e+100)
		tmp = t_1;
	elseif (t_0 <= 2e+307)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := Block[{t$95$0 = N[(N[(1.0 - N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$1 = N[(N[((-z) * x), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+100], t$95$1, If[LessEqual[t$95$0, 2e+307], t$95$0, t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x (-.f64 #s(literal 1 binary64) (*.f64 y z))) < -4.9999999999999999e100 or 1.99999999999999997e307 < (*.f64 x (-.f64 #s(literal 1 binary64) (*.f64 y z)))

   1. Initial program 84.0%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. associate-*r* N/A

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

      9. *-rgt-identity N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 91.9

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

   4. Applied rewrites 91.9%

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

   6. Applied rewrites 10.8%

      \[\leadsto \color{blue}{\frac{\frac{{\left(\left(\left(-z\right) \cdot x\right) \cdot y\right)}^{2}}{x \cdot \mathsf{fma}\left(-z, y, -1\right)} \cdot \frac{{\left(\left(\left(-z\right) \cdot x\right) \cdot y\right)}^{2}}{x \cdot \mathsf{fma}\left(-z, y, -1\right)} - \frac{\left(-x\right) \cdot x}{\mathsf{fma}\left(y \cdot x, z, x\right)} \cdot \frac{\left(-x\right) \cdot x}{\mathsf{fma}\left(y \cdot x, z, x\right)}}{\mathsf{fma}\left({\left(\left(\left(-z\right) \cdot x\right) \cdot y\right)}^{2}, \frac{-1}{\mathsf{fma}\left(y \cdot x, z, x\right)}, \frac{\left(-x\right) \cdot x}{\mathsf{fma}\left(y \cdot x, z, x\right)}\right)}} \]

   7. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 74.2

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

   9. Applied rewrites 74.2%

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

   11. Applied rewrites 72.2%

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

   if -4.9999999999999999e100 < (*.f64 x (-.f64 #s(literal 1 binary64) (*.f64 y z))) < 1.99999999999999997e307

   1. Initial program 99.8%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - z \cdot y\right) \cdot x \leq -5 \cdot 10^{+100}:\\ \;\;\;\;\left(\left(-z\right) \cdot x\right) \cdot y\\ \mathbf{elif}\;\left(1 - z \cdot y\right) \cdot x \leq 2 \cdot 10^{+307}:\\ \;\;\;\;\left(1 - z \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-z\right) \cdot x\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} t_0 := \left(\left(-z\right) \cdot x\right) \cdot y\\ \mathbf{if}\;z \cdot y \leq -1 \cdot 10^{+286}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \cdot y \leq -500:\\ \;\;\;\;\left(\left(-y\right) \cdot z\right) \cdot x\\ \mathbf{elif}\;z \cdot y \leq 5 \cdot 10^{-5}:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (* (- z) x) y)))
   (if (<= (* z y) -1e+286)
     t_0
     (if (<= (* z y) -500.0)
       (* (* (- y) z) x)
       (if (<= (* z y) 5e-5) (* 1.0 x) t_0)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double t_0 = (-z * x) * y;
	double tmp;
	if ((z * y) <= -1e+286) {
		tmp = t_0;
	} else if ((z * y) <= -500.0) {
		tmp = (-y * z) * x;
	} else if ((z * y) <= 5e-5) {
		tmp = 1.0 * x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
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) :: tmp
    t_0 = (-z * x) * y
    if ((z * y) <= (-1d+286)) then
        tmp = t_0
    else if ((z * y) <= (-500.0d0)) then
        tmp = (-y * z) * x
    else if ((z * y) <= 5d-5) then
        tmp = 1.0d0 * x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double t_0 = (-z * x) * y;
	double tmp;
	if ((z * y) <= -1e+286) {
		tmp = t_0;
	} else if ((z * y) <= -500.0) {
		tmp = (-y * z) * x;
	} else if ((z * y) <= 5e-5) {
		tmp = 1.0 * x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	t_0 = (-z * x) * y
	tmp = 0
	if (z * y) <= -1e+286:
		tmp = t_0
	elif (z * y) <= -500.0:
		tmp = (-y * z) * x
	elif (z * y) <= 5e-5:
		tmp = 1.0 * x
	else:
		tmp = t_0
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	t_0 = Float64(Float64(Float64(-z) * x) * y)
	tmp = 0.0
	if (Float64(z * y) <= -1e+286)
		tmp = t_0;
	elseif (Float64(z * y) <= -500.0)
		tmp = Float64(Float64(Float64(-y) * z) * x);
	elseif (Float64(z * y) <= 5e-5)
		tmp = Float64(1.0 * x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	t_0 = (-z * x) * y;
	tmp = 0.0;
	if ((z * y) <= -1e+286)
		tmp = t_0;
	elseif ((z * y) <= -500.0)
		tmp = (-y * z) * x;
	elseif ((z * y) <= 5e-5)
		tmp = 1.0 * x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := Block[{t$95$0 = N[(N[((-z) * x), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[N[(z * y), $MachinePrecision], -1e+286], t$95$0, If[LessEqual[N[(z * y), $MachinePrecision], -500.0], N[(N[((-y) * z), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[N[(z * y), $MachinePrecision], 5e-5], N[(1.0 * x), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 y z) < -1.00000000000000003e286 or 5.00000000000000024e-5 < (*.f64 y z)

   1. Initial program 80.0%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. associate-*r* N/A

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

      9. *-rgt-identity N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 97.9

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

   4. Applied rewrites 97.9%

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

   6. Applied rewrites 20.5%

      \[\leadsto \color{blue}{\frac{\frac{{\left(\left(\left(-z\right) \cdot x\right) \cdot y\right)}^{2}}{x \cdot \mathsf{fma}\left(-z, y, -1\right)} \cdot \frac{{\left(\left(\left(-z\right) \cdot x\right) \cdot y\right)}^{2}}{x \cdot \mathsf{fma}\left(-z, y, -1\right)} - \frac{\left(-x\right) \cdot x}{\mathsf{fma}\left(y \cdot x, z, x\right)} \cdot \frac{\left(-x\right) \cdot x}{\mathsf{fma}\left(y \cdot x, z, x\right)}}{\mathsf{fma}\left({\left(\left(\left(-z\right) \cdot x\right) \cdot y\right)}^{2}, \frac{-1}{\mathsf{fma}\left(y \cdot x, z, x\right)}, \frac{\left(-x\right) \cdot x}{\mathsf{fma}\left(y \cdot x, z, x\right)}\right)}} \]

   7. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 95.8

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

   9. Applied rewrites 95.8%

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

   11. Applied rewrites 90.7%

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

   if -1.00000000000000003e286 < (*.f64 y z) < -500

   1. Initial program 99.5%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 95.2

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

   5. Applied rewrites 95.2%

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

   if -500 < (*.f64 y z) < 5.00000000000000024e-5

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 97.5%

      \[\leadsto x \cdot \color{blue}{1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot y \leq -1 \cdot 10^{+286}:\\ \;\;\;\;\left(\left(-z\right) \cdot x\right) \cdot y\\ \mathbf{elif}\;z \cdot y \leq -500:\\ \;\;\;\;\left(\left(-y\right) \cdot z\right) \cdot x\\ \mathbf{elif}\;z \cdot y \leq 5 \cdot 10^{-5}:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-z\right) \cdot x\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} t_0 := 1 - z \cdot y\\ \mathbf{if}\;t\_0 \leq -2:\\ \;\;\;\;\left(\left(-z\right) \cdot x\right) \cdot y\\ \mathbf{elif}\;t\_0 \leq 10000:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-y\right) \cdot x\right) \cdot z\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (* z y))))
   (if (<= t_0 -2.0)
     (* (* (- z) x) y)
     (if (<= t_0 10000.0) (* 1.0 x) (* (* (- y) x) z)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double t_0 = 1.0 - (z * y);
	double tmp;
	if (t_0 <= -2.0) {
		tmp = (-z * x) * y;
	} else if (t_0 <= 10000.0) {
		tmp = 1.0 * x;
	} else {
		tmp = (-y * x) * z;
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
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) :: tmp
    t_0 = 1.0d0 - (z * y)
    if (t_0 <= (-2.0d0)) then
        tmp = (-z * x) * y
    else if (t_0 <= 10000.0d0) then
        tmp = 1.0d0 * x
    else
        tmp = (-y * x) * z
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double t_0 = 1.0 - (z * y);
	double tmp;
	if (t_0 <= -2.0) {
		tmp = (-z * x) * y;
	} else if (t_0 <= 10000.0) {
		tmp = 1.0 * x;
	} else {
		tmp = (-y * x) * z;
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	t_0 = 1.0 - (z * y)
	tmp = 0
	if t_0 <= -2.0:
		tmp = (-z * x) * y
	elif t_0 <= 10000.0:
		tmp = 1.0 * x
	else:
		tmp = (-y * x) * z
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	t_0 = Float64(1.0 - Float64(z * y))
	tmp = 0.0
	if (t_0 <= -2.0)
		tmp = Float64(Float64(Float64(-z) * x) * y);
	elseif (t_0 <= 10000.0)
		tmp = Float64(1.0 * x);
	else
		tmp = Float64(Float64(Float64(-y) * x) * z);
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	t_0 = 1.0 - (z * y);
	tmp = 0.0;
	if (t_0 <= -2.0)
		tmp = (-z * x) * y;
	elseif (t_0 <= 10000.0)
		tmp = 1.0 * x;
	else
		tmp = (-y * x) * z;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := Block[{t$95$0 = N[(1.0 - N[(z * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2.0], N[(N[((-z) * x), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$0, 10000.0], N[(1.0 * x), $MachinePrecision], N[(N[((-y) * x), $MachinePrecision] * z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 87.3%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. associate-*r* N/A

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

      9. *-rgt-identity N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 97.1

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

   4. Applied rewrites 97.1%

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

   6. Applied rewrites 29.7%

      \[\leadsto \color{blue}{\frac{\frac{{\left(\left(\left(-z\right) \cdot x\right) \cdot y\right)}^{2}}{x \cdot \mathsf{fma}\left(-z, y, -1\right)} \cdot \frac{{\left(\left(\left(-z\right) \cdot x\right) \cdot y\right)}^{2}}{x \cdot \mathsf{fma}\left(-z, y, -1\right)} - \frac{\left(-x\right) \cdot x}{\mathsf{fma}\left(y \cdot x, z, x\right)} \cdot \frac{\left(-x\right) \cdot x}{\mathsf{fma}\left(y \cdot x, z, x\right)}}{\mathsf{fma}\left({\left(\left(\left(-z\right) \cdot x\right) \cdot y\right)}^{2}, \frac{-1}{\mathsf{fma}\left(y \cdot x, z, x\right)}, \frac{\left(-x\right) \cdot x}{\mathsf{fma}\left(y \cdot x, z, x\right)}\right)}} \]

   7. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 94.1

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

   9. Applied rewrites 94.1%

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

   11. Applied rewrites 86.5%

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

   if -2 < (-.f64 #s(literal 1 binary64) (*.f64 y z)) < 1e4

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 96.3%

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

   if 1e4 < (-.f64 #s(literal 1 binary64) (*.f64 y z))

   1. Initial program 88.0%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. associate-*r* N/A

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

      9. *-rgt-identity N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 89.0

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

   4. Applied rewrites 89.0%

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

   6. Applied rewrites 19.9%

      \[\leadsto \color{blue}{\frac{\frac{{\left(\left(\left(-z\right) \cdot x\right) \cdot y\right)}^{2}}{x \cdot \mathsf{fma}\left(-z, y, -1\right)} \cdot \frac{{\left(\left(\left(-z\right) \cdot x\right) \cdot y\right)}^{2}}{x \cdot \mathsf{fma}\left(-z, y, -1\right)} - \frac{\left(-x\right) \cdot x}{\mathsf{fma}\left(y \cdot x, z, x\right)} \cdot \frac{\left(-x\right) \cdot x}{\mathsf{fma}\left(y \cdot x, z, x\right)}}{\mathsf{fma}\left({\left(\left(\left(-z\right) \cdot x\right) \cdot y\right)}^{2}, \frac{-1}{\mathsf{fma}\left(y \cdot x, z, x\right)}, \frac{\left(-x\right) \cdot x}{\mathsf{fma}\left(y \cdot x, z, x\right)}\right)}} \]

   7. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 87.9

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

   9. Applied rewrites 87.9%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - z \cdot y \leq -2:\\ \;\;\;\;\left(\left(-z\right) \cdot x\right) \cdot y\\ \mathbf{elif}\;1 - z \cdot y \leq 10000:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-y\right) \cdot x\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.7% accurate, 0.3× speedup

Math

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

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (* z y))) (t_1 (* (* (- z) x) y)))
   (if (<= t_0 -2.0) t_1 (if (<= t_0 2.0) (* 1.0 x) t_1))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double t_0 = 1.0 - (z * y);
	double t_1 = (-z * x) * y;
	double tmp;
	if (t_0 <= -2.0) {
		tmp = t_1;
	} else if (t_0 <= 2.0) {
		tmp = 1.0 * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

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

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double t_0 = 1.0 - (z * y);
	double t_1 = (-z * x) * y;
	double tmp;
	if (t_0 <= -2.0) {
		tmp = t_1;
	} else if (t_0 <= 2.0) {
		tmp = 1.0 * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	t_0 = 1.0 - (z * y)
	t_1 = (-z * x) * y
	tmp = 0
	if t_0 <= -2.0:
		tmp = t_1
	elif t_0 <= 2.0:
		tmp = 1.0 * x
	else:
		tmp = t_1
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	t_0 = Float64(1.0 - Float64(z * y))
	t_1 = Float64(Float64(Float64(-z) * x) * y)
	tmp = 0.0
	if (t_0 <= -2.0)
		tmp = t_1;
	elseif (t_0 <= 2.0)
		tmp = Float64(1.0 * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	t_0 = 1.0 - (z * y);
	t_1 = (-z * x) * y;
	tmp = 0.0;
	if (t_0 <= -2.0)
		tmp = t_1;
	elseif (t_0 <= 2.0)
		tmp = 1.0 * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := Block[{t$95$0 = N[(1.0 - N[(z * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[((-z) * x), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$0, -2.0], t$95$1, If[LessEqual[t$95$0, 2.0], N[(1.0 * x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 87.9%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. associate-*r* N/A

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

      9. *-rgt-identity N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 91.6

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

   4. Applied rewrites 91.6%

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

   6. Applied rewrites 23.6%

      \[\leadsto \color{blue}{\frac{\frac{{\left(\left(\left(-z\right) \cdot x\right) \cdot y\right)}^{2}}{x \cdot \mathsf{fma}\left(-z, y, -1\right)} \cdot \frac{{\left(\left(\left(-z\right) \cdot x\right) \cdot y\right)}^{2}}{x \cdot \mathsf{fma}\left(-z, y, -1\right)} - \frac{\left(-x\right) \cdot x}{\mathsf{fma}\left(y \cdot x, z, x\right)} \cdot \frac{\left(-x\right) \cdot x}{\mathsf{fma}\left(y \cdot x, z, x\right)}}{\mathsf{fma}\left({\left(\left(\left(-z\right) \cdot x\right) \cdot y\right)}^{2}, \frac{-1}{\mathsf{fma}\left(y \cdot x, z, x\right)}, \frac{\left(-x\right) \cdot x}{\mathsf{fma}\left(y \cdot x, z, x\right)}\right)}} \]

   7. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 89.2

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

   9. Applied rewrites 89.2%

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

   11. Applied rewrites 84.3%

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

   if -2 < (-.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 z around 0

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

   5. Applied rewrites 97.5%

      \[\leadsto x \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.7%

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

Alternative 7: 51.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ 1 \cdot x \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z) :precision binary64 (* 1.0 x))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	return 1.0 * x;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 1.0d0 * x
end function

Java

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

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	return 1.0 * x

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	return Float64(1.0 * x)
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp = code(x, y, z)
	tmp = 1.0 * x;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := N[(1.0 * x), $MachinePrecision]

Derivation

1. Initial program 94.6%

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

3. Taylor expanded in z around 0

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

5. Applied rewrites 56.1%

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

6. Final simplification 56.1%

   \[\leadsto 1 \cdot x \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024259 
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSV:hsv from colour-2.3.3, I"
  :precision binary64
  (* x (- 1.0 (* y z))))