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

Percentage Accurate: 96.2% → 98.9%

Time: 8.8s

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: 96.2% 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

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \left(1 - z \cdot y\right) \cdot x\_m\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{y}{\frac{-1}{z \cdot x\_m}}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+268}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(-y\right) \cdot x\_m, z, x\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (* (- 1.0 (* z y)) x_m)))
   (*
    x_s
    (if (<= t_0 (- INFINITY))
      (/ y (/ -1.0 (* z x_m)))
      (if (<= t_0 5e+268) t_0 (fma (* (- y) x_m) z x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double t_0 = (1.0 - (z * y)) * x_m;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = y / (-1.0 / (z * x_m));
	} else if (t_0 <= 5e+268) {
		tmp = t_0;
	} else {
		tmp = fma((-y * x_m), z, x_m);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	t_0 = Float64(Float64(1.0 - Float64(z * y)) * x_m)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(y / Float64(-1.0 / Float64(z * x_m)));
	elseif (t_0 <= 5e+268)
		tmp = t_0;
	else
		tmp = fma(Float64(Float64(-y) * x_m), z, x_m);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := Block[{t$95$0 = N[(N[(1.0 - N[(z * y), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$0, (-Infinity)], N[(y / N[(-1.0 / N[(z * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+268], t$95$0, N[(N[((-y) * x$95$m), $MachinePrecision] * z + x$95$m), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 75.5%

      \[x \cdot \left(1 - y \cdot z\right) \]
   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. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. flip3-- N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. lower--.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

      16. *-lft-identity N/A

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

      17. distribute-lft1-in N/A

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

      18. +-commutative N/A

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

   4. Applied rewrites 11.1%

      \[\leadsto \color{blue}{\frac{\left(1 - {\left(z \cdot y\right)}^{3}\right) \cdot x}{\mathsf{fma}\left(\mathsf{fma}\left(z, y, 1\right), z \cdot y, 1\right)}} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. cube-mult N/A

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

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

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

      7. lift-*.f64 N/A

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

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

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

      9. lift-neg.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-*r* N/A

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

      17. lift-*.f64 N/A

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

      18. associate-*r* N/A

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

      19. lower-fma.f64 N/A

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

   6. Applied rewrites 7.5%

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

   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. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-*.f64 99.9

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

   9. Applied rewrites 99.9%

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

   11. Applied rewrites 99.9%

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

   if -inf.0 < (*.f64 x (-.f64 #s(literal 1 binary64) (*.f64 y z))) < 5.0000000000000002e268

   1. Initial program 99.9%

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

   if 5.0000000000000002e268 < (*.f64 x (-.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 92.9

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

   4. Applied rewrites 92.9%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - z \cdot y\right) \cdot x \leq -\infty:\\ \;\;\;\;\frac{y}{\frac{-1}{z \cdot x}}\\ \mathbf{elif}\;\left(1 - z \cdot y\right) \cdot x \leq 5 \cdot 10^{+268}:\\ \;\;\;\;\left(1 - z \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(-y\right) \cdot x, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \left(1 - z \cdot y\right) \cdot x\_m\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\left(-z\right) \cdot x\_m\right) \cdot y\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+268}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(-y\right) \cdot x\_m, z, x\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (* (- 1.0 (* z y)) x_m)))
   (*
    x_s
    (if (<= t_0 (- INFINITY))
      (* (* (- z) x_m) y)
      (if (<= t_0 5e+268) t_0 (fma (* (- y) x_m) z x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double t_0 = (1.0 - (z * y)) * x_m;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (-z * x_m) * y;
	} else if (t_0 <= 5e+268) {
		tmp = t_0;
	} else {
		tmp = fma((-y * x_m), z, x_m);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	t_0 = Float64(Float64(1.0 - Float64(z * y)) * x_m)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(-z) * x_m) * y);
	elseif (t_0 <= 5e+268)
		tmp = t_0;
	else
		tmp = fma(Float64(Float64(-y) * x_m), z, x_m);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := Block[{t$95$0 = N[(N[(1.0 - N[(z * y), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$0, (-Infinity)], N[(N[((-z) * x$95$m), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$0, 5e+268], t$95$0, N[(N[((-y) * x$95$m), $MachinePrecision] * z + x$95$m), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 75.5%

      \[x \cdot \left(1 - y \cdot z\right) \]
   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. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. flip3-- N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. lower--.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

      16. *-lft-identity N/A

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

      17. distribute-lft1-in N/A

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

      18. +-commutative N/A

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

   4. Applied rewrites 11.1%

      \[\leadsto \color{blue}{\frac{\left(1 - {\left(z \cdot y\right)}^{3}\right) \cdot x}{\mathsf{fma}\left(\mathsf{fma}\left(z, y, 1\right), z \cdot y, 1\right)}} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. cube-mult N/A

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

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

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

      7. lift-*.f64 N/A

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

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

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

      9. lift-neg.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-*r* N/A

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

      17. lift-*.f64 N/A

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

      18. associate-*r* N/A

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

      19. lower-fma.f64 N/A

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

   6. Applied rewrites 7.5%

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

   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. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-*.f64 99.9

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

   9. Applied rewrites 99.9%

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

   if -inf.0 < (*.f64 x (-.f64 #s(literal 1 binary64) (*.f64 y z))) < 5.0000000000000002e268

   1. Initial program 99.9%

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

   if 5.0000000000000002e268 < (*.f64 x (-.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 92.9

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

   4. Applied rewrites 92.9%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - z \cdot y\right) \cdot x \leq -\infty:\\ \;\;\;\;\left(\left(-z\right) \cdot x\right) \cdot y\\ \mathbf{elif}\;\left(1 - z \cdot y\right) \cdot x \leq 5 \cdot 10^{+268}:\\ \;\;\;\;\left(1 - z \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(-y\right) \cdot x, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \left(\left(-z\right) \cdot x\_m\right) \cdot y\\ t_1 := \left(1 - z \cdot y\right) \cdot x\_m\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+298}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (* (* (- z) x_m) y)) (t_1 (* (- 1.0 (* z y)) x_m)))
   (* x_s (if (<= t_1 (- INFINITY)) t_0 (if (<= t_1 2e+298) t_1 t_0)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double t_0 = (-z * x_m) * y;
	double t_1 = (1.0 - (z * y)) * x_m;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_0;
	} else if (t_1 <= 2e+298) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return x_s * tmp;
}

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double t_0 = (-z * x_m) * y;
	double t_1 = (1.0 - (z * y)) * x_m;
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = t_0;
	} else if (t_1 <= 2e+298) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	t_0 = (-z * x_m) * y
	t_1 = (1.0 - (z * y)) * x_m
	tmp = 0
	if t_1 <= -math.inf:
		tmp = t_0
	elif t_1 <= 2e+298:
		tmp = t_1
	else:
		tmp = t_0
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	t_0 = Float64(Float64(Float64(-z) * x_m) * y)
	t_1 = Float64(Float64(1.0 - Float64(z * y)) * x_m)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_0;
	elseif (t_1 <= 2e+298)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	t_0 = (-z * x_m) * y;
	t_1 = (1.0 - (z * y)) * x_m;
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = t_0;
	elseif (t_1 <= 2e+298)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := Block[{t$95$0 = N[(N[((-z) * x$95$m), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 - N[(z * y), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$1, (-Infinity)], t$95$0, If[LessEqual[t$95$1, 2e+298], t$95$1, t$95$0]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x (-.f64 #s(literal 1 binary64) (*.f64 y z))) < -inf.0 or 1.9999999999999999e298 < (*.f64 x (-.f64 #s(literal 1 binary64) (*.f64 y z)))

   1. Initial program 80.8%

      \[x \cdot \left(1 - y \cdot z\right) \]
   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. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. flip3-- N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. lower--.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

      16. *-lft-identity N/A

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

      17. distribute-lft1-in N/A

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

      18. +-commutative N/A

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

   4. Applied rewrites 10.2%

      \[\leadsto \color{blue}{\frac{\left(1 - {\left(z \cdot y\right)}^{3}\right) \cdot x}{\mathsf{fma}\left(\mathsf{fma}\left(z, y, 1\right), z \cdot y, 1\right)}} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. cube-mult N/A

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

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

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

      7. lift-*.f64 N/A

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

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

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

      9. lift-neg.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-*r* N/A

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

      17. lift-*.f64 N/A

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

      18. associate-*r* N/A

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

      19. lower-fma.f64 N/A

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

   6. Applied rewrites 8.5%

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

   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. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-*.f64 99.9

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

   9. Applied rewrites 99.9%

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

   if -inf.0 < (*.f64 x (-.f64 #s(literal 1 binary64) (*.f64 y z))) < 1.9999999999999999e298

   1. Initial program 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - z \cdot y\right) \cdot x \leq -\infty:\\ \;\;\;\;\left(\left(-z\right) \cdot x\right) \cdot y\\ \mathbf{elif}\;\left(1 - z \cdot y\right) \cdot x \leq 2 \cdot 10^{+298}:\\ \;\;\;\;\left(1 - z \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-z\right) \cdot x\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot y \leq -500:\\ \;\;\;\;\left(\left(-y\right) \cdot x\_m\right) \cdot z\\ \mathbf{elif}\;z \cdot y \leq 0.02:\\ \;\;\;\;1 \cdot x\_m\\ \mathbf{elif}\;z \cdot y \leq 2 \cdot 10^{+289}:\\ \;\;\;\;\left(\left(-z\right) \cdot y\right) \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-z\right) \cdot x\_m\right) \cdot y\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= (* z y) -500.0)
    (* (* (- y) x_m) z)
    (if (<= (* z y) 0.02)
      (* 1.0 x_m)
      (if (<= (* z y) 2e+289) (* (* (- z) y) x_m) (* (* (- z) x_m) y))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((z * y) <= -500.0) {
		tmp = (-y * x_m) * z;
	} else if ((z * y) <= 0.02) {
		tmp = 1.0 * x_m;
	} else if ((z * y) <= 2e+289) {
		tmp = (-z * y) * x_m;
	} else {
		tmp = (-z * x_m) * y;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * y) <= (-500.0d0)) then
        tmp = (-y * x_m) * z
    else if ((z * y) <= 0.02d0) then
        tmp = 1.0d0 * x_m
    else if ((z * y) <= 2d+289) then
        tmp = (-z * y) * x_m
    else
        tmp = (-z * x_m) * y
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((z * y) <= -500.0) {
		tmp = (-y * x_m) * z;
	} else if ((z * y) <= 0.02) {
		tmp = 1.0 * x_m;
	} else if ((z * y) <= 2e+289) {
		tmp = (-z * y) * x_m;
	} else {
		tmp = (-z * x_m) * y;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if (z * y) <= -500.0:
		tmp = (-y * x_m) * z
	elif (z * y) <= 0.02:
		tmp = 1.0 * x_m
	elif (z * y) <= 2e+289:
		tmp = (-z * y) * x_m
	else:
		tmp = (-z * x_m) * y
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(z * y) <= -500.0)
		tmp = Float64(Float64(Float64(-y) * x_m) * z);
	elseif (Float64(z * y) <= 0.02)
		tmp = Float64(1.0 * x_m);
	elseif (Float64(z * y) <= 2e+289)
		tmp = Float64(Float64(Float64(-z) * y) * x_m);
	else
		tmp = Float64(Float64(Float64(-z) * x_m) * y);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if ((z * y) <= -500.0)
		tmp = (-y * x_m) * z;
	elseif ((z * y) <= 0.02)
		tmp = 1.0 * x_m;
	elseif ((z * y) <= 2e+289)
		tmp = (-z * y) * x_m;
	else
		tmp = (-z * x_m) * y;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(z * y), $MachinePrecision], -500.0], N[(N[((-y) * x$95$m), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[N[(z * y), $MachinePrecision], 0.02], N[(1.0 * x$95$m), $MachinePrecision], If[LessEqual[N[(z * y), $MachinePrecision], 2e+289], N[(N[((-z) * y), $MachinePrecision] * x$95$m), $MachinePrecision], N[(N[((-z) * x$95$m), $MachinePrecision] * y), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if (*.f64 y z) < -500

   1. Initial program 92.5%

      \[x \cdot \left(1 - y \cdot z\right) \]
   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. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. flip3-- N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. lower--.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

      16. *-lft-identity N/A

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

      17. distribute-lft1-in N/A

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

      18. +-commutative N/A

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

   4. Applied rewrites 33.4%

      \[\leadsto \color{blue}{\frac{\left(1 - {\left(z \cdot y\right)}^{3}\right) \cdot x}{\mathsf{fma}\left(\mathsf{fma}\left(z, y, 1\right), z \cdot y, 1\right)}} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. cube-mult N/A

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

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

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

      7. lift-*.f64 N/A

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

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

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

      9. lift-neg.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-*r* N/A

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

      17. lift-*.f64 N/A

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

      18. associate-*r* N/A

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

      19. lower-fma.f64 N/A

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

   6. Applied rewrites 18.6%

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

   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. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-*.f64 88.2

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

   9. Applied rewrites 88.2%

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

   11. Applied rewrites 94.0%

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

   if -500 < (*.f64 y z) < 0.0200000000000000004

   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 97.9%

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

   if 0.0200000000000000004 < (*.f64 y z) < 2.0000000000000001e289

   1. Initial program 99.8%

      \[x \cdot \left(1 - y \cdot z\right) \]
   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 96.1

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

   5. Applied rewrites 96.1%

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

   if 2.0000000000000001e289 < (*.f64 y z)

   1. Initial program 68.6%

      \[x \cdot \left(1 - y \cdot z\right) \]
   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. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. flip3-- N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. lower--.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

      16. *-lft-identity N/A

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

      17. distribute-lft1-in N/A

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

      18. +-commutative N/A

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

   4. Applied rewrites 0.0%

      \[\leadsto \color{blue}{\frac{\left(1 - {\left(z \cdot y\right)}^{3}\right) \cdot x}{\mathsf{fma}\left(\mathsf{fma}\left(z, y, 1\right), z \cdot y, 1\right)}} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. cube-mult N/A

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

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

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

      7. lift-*.f64 N/A

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

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

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

      9. lift-neg.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-*r* N/A

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

      17. lift-*.f64 N/A

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

      18. associate-*r* N/A

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

      19. lower-fma.f64 N/A

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

   6. Applied rewrites 0.0%

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

   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. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-*.f64 99.7

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

   9. Applied rewrites 99.7%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot y \leq -500:\\ \;\;\;\;\left(\left(-y\right) \cdot x\right) \cdot z\\ \mathbf{elif}\;z \cdot y \leq 0.02:\\ \;\;\;\;1 \cdot x\\ \mathbf{elif}\;z \cdot y \leq 2 \cdot 10^{+289}:\\ \;\;\;\;\left(\left(-z\right) \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-z\right) \cdot x\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := 1 - z \cdot y\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -100:\\ \;\;\;\;\left(\left(-z\right) \cdot x\_m\right) \cdot y\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1 \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-y\right) \cdot x\_m\right) \cdot z\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (* z y))))
   (*
    x_s
    (if (<= t_0 -100.0)
      (* (* (- z) x_m) y)
      (if (<= t_0 2.0) (* 1.0 x_m) (* (* (- y) x_m) z))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double t_0 = 1.0 - (z * y);
	double tmp;
	if (t_0 <= -100.0) {
		tmp = (-z * x_m) * y;
	} else if (t_0 <= 2.0) {
		tmp = 1.0 * x_m;
	} else {
		tmp = (-y * x_m) * z;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (z * y)
    if (t_0 <= (-100.0d0)) then
        tmp = (-z * x_m) * y
    else if (t_0 <= 2.0d0) then
        tmp = 1.0d0 * x_m
    else
        tmp = (-y * x_m) * z
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double t_0 = 1.0 - (z * y);
	double tmp;
	if (t_0 <= -100.0) {
		tmp = (-z * x_m) * y;
	} else if (t_0 <= 2.0) {
		tmp = 1.0 * x_m;
	} else {
		tmp = (-y * x_m) * z;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	t_0 = 1.0 - (z * y)
	tmp = 0
	if t_0 <= -100.0:
		tmp = (-z * x_m) * y
	elif t_0 <= 2.0:
		tmp = 1.0 * x_m
	else:
		tmp = (-y * x_m) * z
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	t_0 = Float64(1.0 - Float64(z * y))
	tmp = 0.0
	if (t_0 <= -100.0)
		tmp = Float64(Float64(Float64(-z) * x_m) * y);
	elseif (t_0 <= 2.0)
		tmp = Float64(1.0 * x_m);
	else
		tmp = Float64(Float64(Float64(-y) * x_m) * z);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	t_0 = 1.0 - (z * y);
	tmp = 0.0;
	if (t_0 <= -100.0)
		tmp = (-z * x_m) * y;
	elseif (t_0 <= 2.0)
		tmp = 1.0 * x_m;
	else
		tmp = (-y * x_m) * z;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := Block[{t$95$0 = N[(1.0 - N[(z * y), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$0, -100.0], N[(N[((-z) * x$95$m), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(1.0 * x$95$m), $MachinePrecision], N[(N[((-y) * x$95$m), $MachinePrecision] * z), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 #s(literal 1 binary64) (*.f64 y z)) < -100

   1. Initial program 90.8%

      \[x \cdot \left(1 - y \cdot z\right) \]
   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. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. flip3-- N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. lower--.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

      16. *-lft-identity N/A

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

      17. distribute-lft1-in N/A

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

      18. +-commutative N/A

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

   4. Applied rewrites 29.7%

      \[\leadsto \color{blue}{\frac{\left(1 - {\left(z \cdot y\right)}^{3}\right) \cdot x}{\mathsf{fma}\left(\mathsf{fma}\left(z, y, 1\right), z \cdot y, 1\right)}} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. cube-mult N/A

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

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

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

      7. lift-*.f64 N/A

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

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

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

      9. lift-neg.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-*r* N/A

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

      17. lift-*.f64 N/A

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

      18. associate-*r* N/A

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

      19. lower-fma.f64 N/A

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

   6. Applied rewrites 20.0%

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

   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. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-*.f64 94.8

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

   9. Applied rewrites 94.8%

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

   if -100 < (-.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 97.9%

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

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

   1. Initial program 92.5%

      \[x \cdot \left(1 - y \cdot z\right) \]
   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. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. flip3-- N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. lower--.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

      16. *-lft-identity N/A

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

      17. distribute-lft1-in N/A

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

      18. +-commutative N/A

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

   4. Applied rewrites 33.4%

      \[\leadsto \color{blue}{\frac{\left(1 - {\left(z \cdot y\right)}^{3}\right) \cdot x}{\mathsf{fma}\left(\mathsf{fma}\left(z, y, 1\right), z \cdot y, 1\right)}} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. cube-mult N/A

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

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

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

      7. lift-*.f64 N/A

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

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

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

      9. lift-neg.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-*r* N/A

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

      17. lift-*.f64 N/A

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

      18. associate-*r* N/A

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

      19. lower-fma.f64 N/A

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

   6. Applied rewrites 18.6%

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

   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. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-*.f64 88.2

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

   9. Applied rewrites 88.2%

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

   11. Applied rewrites 94.0%

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

4. Final simplification 96.1%

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

Alternative 6: 94.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := 1 - z \cdot y\\ t_1 := \left(\left(-z\right) \cdot x\_m\right) \cdot y\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -100:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 1000:\\ \;\;\;\;1 \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (* z y))) (t_1 (* (* (- z) x_m) y)))
   (* x_s (if (<= t_0 -100.0) t_1 (if (<= t_0 1000.0) (* 1.0 x_m) t_1)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double t_0 = 1.0 - (z * y);
	double t_1 = (-z * x_m) * y;
	double tmp;
	if (t_0 <= -100.0) {
		tmp = t_1;
	} else if (t_0 <= 1000.0) {
		tmp = 1.0 * x_m;
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 - (z * y)
    t_1 = (-z * x_m) * y
    if (t_0 <= (-100.0d0)) then
        tmp = t_1
    else if (t_0 <= 1000.0d0) then
        tmp = 1.0d0 * 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);
public static double code(double x_s, double x_m, double y, double z) {
	double t_0 = 1.0 - (z * y);
	double t_1 = (-z * x_m) * y;
	double tmp;
	if (t_0 <= -100.0) {
		tmp = t_1;
	} else if (t_0 <= 1000.0) {
		tmp = 1.0 * x_m;
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	t_0 = 1.0 - (z * y)
	t_1 = (-z * x_m) * y
	tmp = 0
	if t_0 <= -100.0:
		tmp = t_1
	elif t_0 <= 1000.0:
		tmp = 1.0 * x_m
	else:
		tmp = t_1
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	t_0 = Float64(1.0 - Float64(z * y))
	t_1 = Float64(Float64(Float64(-z) * x_m) * y)
	tmp = 0.0
	if (t_0 <= -100.0)
		tmp = t_1;
	elseif (t_0 <= 1000.0)
		tmp = Float64(1.0 * 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);
function tmp_2 = code(x_s, x_m, y, z)
	t_0 = 1.0 - (z * y);
	t_1 = (-z * x_m) * y;
	tmp = 0.0;
	if (t_0 <= -100.0)
		tmp = t_1;
	elseif (t_0 <= 1000.0)
		tmp = 1.0 * 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]
code[x$95$s_, x$95$m_, y_, z_] := Block[{t$95$0 = N[(1.0 - N[(z * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[((-z) * x$95$m), $MachinePrecision] * y), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$0, -100.0], t$95$1, If[LessEqual[t$95$0, 1000.0], N[(1.0 * x$95$m), $MachinePrecision], t$95$1]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (*.f64 y z)) < -100 or 1e3 < (-.f64 #s(literal 1 binary64) (*.f64 y z))

   1. Initial program 91.5%

      \[x \cdot \left(1 - y \cdot z\right) \]
   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. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. flip3-- N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. lower--.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

      16. *-lft-identity N/A

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

      17. distribute-lft1-in N/A

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

      18. +-commutative N/A

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

   4. Applied rewrites 30.9%

      \[\leadsto \color{blue}{\frac{\left(1 - {\left(z \cdot y\right)}^{3}\right) \cdot x}{\mathsf{fma}\left(\mathsf{fma}\left(z, y, 1\right), z \cdot y, 1\right)}} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. cube-mult N/A

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

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

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

      7. lift-*.f64 N/A

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

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

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

      9. lift-neg.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-*r* N/A

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

      17. lift-*.f64 N/A

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

      18. associate-*r* N/A

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

      19. lower-fma.f64 N/A

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

   6. Applied rewrites 19.4%

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

   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. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-*.f64 92.4

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

   9. Applied rewrites 92.4%

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

   if -100 < (-.f64 #s(literal 1 binary64) (*.f64 y z)) < 1e3

   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 97.2%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - z \cdot y \leq -100:\\ \;\;\;\;\left(\left(-z\right) \cdot x\right) \cdot y\\ \mathbf{elif}\;1 - z \cdot y \leq 1000:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-z\right) \cdot x\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 50.3% accurate, 2.3× speedup

Math

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

FPCore

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

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	return x_s * (1.0 * x_m);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x_s * (1.0d0 * x_m)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	return x_s * (1.0 * x_m);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	return x_s * (1.0 * x_m)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	return Float64(x_s * Float64(1.0 * x_m))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m, y, z)
	tmp = x_s * (1.0 * x_m);
end

Wolfram

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

Derivation

1. Initial program 95.5%

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

3. Taylor expanded in z around 0

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

5. Applied rewrites 47.4%

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

6. Final simplification 47.4%

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

Reproduce

herbie shell --seed 2024243 
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSV:hsv from colour-2.3.3, I"
  :precision binary64
  (* x (- 1.0 (* y z))))