Diagrams.TwoD.Arc:bezierFromSweepQ1 from diagrams-lib-1.3.0.3

Percentage Accurate: 94.0% → 99.6%

Time: 7.5s

Alternatives: 8

Speedup: 0.7×

• 25.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (* (- 1.0 x) (- 3.0 x)) (* y 3.0)))

C

double code(double x, double y) {
	return ((1.0 - x) * (3.0 - x)) / (y * 3.0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((1.0d0 - x) * (3.0d0 - x)) / (y * 3.0d0)
end function

Java

public static double code(double x, double y) {
	return ((1.0 - x) * (3.0 - x)) / (y * 3.0);
}

Python

def code(x, y):
	return ((1.0 - x) * (3.0 - x)) / (y * 3.0)

Julia

function code(x, y)
	return Float64(Float64(Float64(1.0 - x) * Float64(3.0 - x)) / Float64(y * 3.0))
end

MATLAB

function tmp = code(x, y)
	tmp = ((1.0 - x) * (3.0 - x)) / (y * 3.0);
end

Wolfram

code[x_, y_] := N[(N[(N[(1.0 - x), $MachinePrecision] * N[(3.0 - x), $MachinePrecision]), $MachinePrecision] / N[(y * 3.0), $MachinePrecision]), $MachinePrecision]

Initial Program: 94.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (* (- 1.0 x) (- 3.0 x)) (* y 3.0)))

C

double code(double x, double y) {
	return ((1.0 - x) * (3.0 - x)) / (y * 3.0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((1.0d0 - x) * (3.0d0 - x)) / (y * 3.0d0)
end function

Java

public static double code(double x, double y) {
	return ((1.0 - x) * (3.0 - x)) / (y * 3.0);
}

Python

def code(x, y):
	return ((1.0 - x) * (3.0 - x)) / (y * 3.0)

Julia

function code(x, y)
	return Float64(Float64(Float64(1.0 - x) * Float64(3.0 - x)) / Float64(y * 3.0))
end

MATLAB

function tmp = code(x, y)
	tmp = ((1.0 - x) * (3.0 - x)) / (y * 3.0);
end

Wolfram

code[x_, y_] := N[(N[(N[(1.0 - x), $MachinePrecision] * N[(3.0 - x), $MachinePrecision]), $MachinePrecision] / N[(y * 3.0), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{3 \cdot y\_m} \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\frac{\left(1 - x\right) \cdot \mathsf{fma}\left(x, -0.3333333333333333, 1\right)}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.3333333333333333}{y\_m}\right)\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m)
 :precision binary64
 (*
  y_s
  (if (<= (/ (* (- 1.0 x) (- 3.0 x)) (* 3.0 y_m)) 2e+306)
    (/ (* (- 1.0 x) (fma x -0.3333333333333333 1.0)) y_m)
    (* x (* x (/ 0.3333333333333333 y_m))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m) {
	double tmp;
	if ((((1.0 - x) * (3.0 - x)) / (3.0 * y_m)) <= 2e+306) {
		tmp = ((1.0 - x) * fma(x, -0.3333333333333333, 1.0)) / y_m;
	} else {
		tmp = x * (x * (0.3333333333333333 / y_m));
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m)
	tmp = 0.0
	if (Float64(Float64(Float64(1.0 - x) * Float64(3.0 - x)) / Float64(3.0 * y_m)) <= 2e+306)
		tmp = Float64(Float64(Float64(1.0 - x) * fma(x, -0.3333333333333333, 1.0)) / y_m);
	else
		tmp = Float64(x * Float64(x * Float64(0.3333333333333333 / y_m)));
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_] := N[(y$95$s * If[LessEqual[N[(N[(N[(1.0 - x), $MachinePrecision] * N[(3.0 - x), $MachinePrecision]), $MachinePrecision] / N[(3.0 * y$95$m), $MachinePrecision]), $MachinePrecision], 2e+306], N[(N[(N[(1.0 - x), $MachinePrecision] * N[(x * -0.3333333333333333 + 1.0), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision], N[(x * N[(x * N[(0.3333333333333333 / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) (-.f64 #s(literal 3 binary64) x)) (*.f64 y #s(literal 3 binary64))) < 2.00000000000000003e306

   1. Initial program 95.8%

      \[\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. distribute-rgt-in N/A

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

      11. metadata-eval N/A

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

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

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

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

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

      14. lower-fma.f64 N/A

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

      15. metadata-eval 96.0

          \[\leadsto \frac{\left(1 - x\right) \cdot \mathsf{fma}\left(x, \color{blue}{-0.3333333333333333}, 1\right)}{y} \]

   5. Simplified 96.0%

      \[\leadsto \color{blue}{\frac{\left(1 - x\right) \cdot \mathsf{fma}\left(x, -0.3333333333333333, 1\right)}{y}} \]

   if 2.00000000000000003e306 < (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) (-.f64 #s(literal 3 binary64) x)) (*.f64 y #s(literal 3 binary64)))

   1. Initial program 86.5%

      \[\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. distribute-rgt-in N/A

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

      11. metadata-eval N/A

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

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

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

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

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

      14. lower-fma.f64 N/A

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

      15. metadata-eval 86.5

          \[\leadsto \frac{\left(1 - x\right) \cdot \mathsf{fma}\left(x, \color{blue}{-0.3333333333333333}, 1\right)}{y} \]

   5. Simplified 86.5%

      \[\leadsto \color{blue}{\frac{\left(1 - x\right) \cdot \mathsf{fma}\left(x, -0.3333333333333333, 1\right)}{y}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{{x}^{2}}{y}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{{x}^{2}}{y} \cdot \frac{1}{3}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{x \cdot x}}{y} \cdot \frac{1}{3} \]

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-lft-identity N/A

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

      8. associate-*l/ N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 100.0

          \[\leadsto x \cdot \left(x \cdot \color{blue}{\frac{0.3333333333333333}{y}}\right) \]

   8. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \frac{0.3333333333333333}{y}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{3 \cdot y} \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\frac{\left(1 - x\right) \cdot \mathsf{fma}\left(x, -0.3333333333333333, 1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.3333333333333333}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{3 \cdot y\_m} \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.3333333333333333, -1.3333333333333333\right), 1\right)}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.3333333333333333}{y\_m}\right)\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m)
 :precision binary64
 (*
  y_s
  (if (<= (/ (* (- 1.0 x) (- 3.0 x)) (* 3.0 y_m)) 2e+306)
    (/ (fma x (fma x 0.3333333333333333 -1.3333333333333333) 1.0) y_m)
    (* x (* x (/ 0.3333333333333333 y_m))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m) {
	double tmp;
	if ((((1.0 - x) * (3.0 - x)) / (3.0 * y_m)) <= 2e+306) {
		tmp = fma(x, fma(x, 0.3333333333333333, -1.3333333333333333), 1.0) / y_m;
	} else {
		tmp = x * (x * (0.3333333333333333 / y_m));
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m)
	tmp = 0.0
	if (Float64(Float64(Float64(1.0 - x) * Float64(3.0 - x)) / Float64(3.0 * y_m)) <= 2e+306)
		tmp = Float64(fma(x, fma(x, 0.3333333333333333, -1.3333333333333333), 1.0) / y_m);
	else
		tmp = Float64(x * Float64(x * Float64(0.3333333333333333 / y_m)));
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_] := N[(y$95$s * If[LessEqual[N[(N[(N[(1.0 - x), $MachinePrecision] * N[(3.0 - x), $MachinePrecision]), $MachinePrecision] / N[(3.0 * y$95$m), $MachinePrecision]), $MachinePrecision], 2e+306], N[(N[(x * N[(x * 0.3333333333333333 + -1.3333333333333333), $MachinePrecision] + 1.0), $MachinePrecision] / y$95$m), $MachinePrecision], N[(x * N[(x * N[(0.3333333333333333 / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) (-.f64 #s(literal 3 binary64) x)) (*.f64 y #s(literal 3 binary64))) < 2.00000000000000003e306

   1. Initial program 95.8%

      \[\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. distribute-rgt-in N/A

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

      11. metadata-eval N/A

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

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

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

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

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

      14. lower-fma.f64 N/A

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

      15. metadata-eval 96.0

          \[\leadsto \frac{\left(1 - x\right) \cdot \mathsf{fma}\left(x, \color{blue}{-0.3333333333333333}, 1\right)}{y} \]

   5. Simplified 96.0%

      \[\leadsto \color{blue}{\frac{\left(1 - x\right) \cdot \mathsf{fma}\left(x, -0.3333333333333333, 1\right)}{y}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{1 + x \cdot \left(\frac{1}{3} \cdot x - \frac{4}{3}\right)}}{y} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{x \cdot \left(\frac{1}{3} \cdot x - \frac{4}{3}\right) + 1}}{y} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, \frac{1}{3} \cdot x - \frac{4}{3}, 1\right)}}{y} \]

      3. sub-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{\frac{1}{3} \cdot x + \left(\mathsf{neg}\left(\frac{4}{3}\right)\right)}, 1\right)}{y} \]

      4. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{3}} + \left(\mathsf{neg}\left(\frac{4}{3}\right)\right), 1\right)}{y} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot \frac{1}{3} + \color{blue}{\frac{-4}{3}}, 1\right)}{y} \]

      6. lower-fma.f64 95.9

         \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.3333333333333333, -1.3333333333333333\right)}, 1\right)}{y} \]

   8. Simplified 95.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.3333333333333333, -1.3333333333333333\right), 1\right)}}{y} \]

   if 2.00000000000000003e306 < (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) (-.f64 #s(literal 3 binary64) x)) (*.f64 y #s(literal 3 binary64)))

   1. Initial program 86.5%

      \[\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. distribute-rgt-in N/A

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

      11. metadata-eval N/A

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

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

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

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

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

      14. lower-fma.f64 N/A

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

      15. metadata-eval 86.5

          \[\leadsto \frac{\left(1 - x\right) \cdot \mathsf{fma}\left(x, \color{blue}{-0.3333333333333333}, 1\right)}{y} \]

   5. Simplified 86.5%

      \[\leadsto \color{blue}{\frac{\left(1 - x\right) \cdot \mathsf{fma}\left(x, -0.3333333333333333, 1\right)}{y}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{{x}^{2}}{y}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{{x}^{2}}{y} \cdot \frac{1}{3}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{x \cdot x}}{y} \cdot \frac{1}{3} \]

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-lft-identity N/A

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

      8. associate-*l/ N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 100.0

          \[\leadsto x \cdot \left(x \cdot \color{blue}{\frac{0.3333333333333333}{y}}\right) \]

   8. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \frac{0.3333333333333333}{y}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{3 \cdot y} \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.3333333333333333, -1.3333333333333333\right), 1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.3333333333333333}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\left(1 - x\right) \cdot \left(3 - x\right) \leq 5:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1.3333333333333333, x, 1\right)}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\mathsf{fma}\left(x, 0.3333333333333333, -1.3333333333333333\right)}{y\_m}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m)
 :precision binary64
 (*
  y_s
  (if (<= (* (- 1.0 x) (- 3.0 x)) 5.0)
    (/ (fma -1.3333333333333333 x 1.0) y_m)
    (* x (/ (fma x 0.3333333333333333 -1.3333333333333333) y_m)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m) {
	double tmp;
	if (((1.0 - x) * (3.0 - x)) <= 5.0) {
		tmp = fma(-1.3333333333333333, x, 1.0) / y_m;
	} else {
		tmp = x * (fma(x, 0.3333333333333333, -1.3333333333333333) / y_m);
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m)
	tmp = 0.0
	if (Float64(Float64(1.0 - x) * Float64(3.0 - x)) <= 5.0)
		tmp = Float64(fma(-1.3333333333333333, x, 1.0) / y_m);
	else
		tmp = Float64(x * Float64(fma(x, 0.3333333333333333, -1.3333333333333333) / y_m));
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_] := N[(y$95$s * If[LessEqual[N[(N[(1.0 - x), $MachinePrecision] * N[(3.0 - x), $MachinePrecision]), $MachinePrecision], 5.0], N[(N[(-1.3333333333333333 * x + 1.0), $MachinePrecision] / y$95$m), $MachinePrecision], N[(x * N[(N[(x * 0.3333333333333333 + -1.3333333333333333), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 #s(literal 1 binary64) x) (-.f64 #s(literal 3 binary64) x)) < 5

   1. Initial program 99.6%

      \[\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. distribute-rgt-in N/A

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

      11. metadata-eval N/A

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

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

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

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

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

      14. lower-fma.f64 N/A

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

      15. metadata-eval 100.0

          \[\leadsto \frac{\left(1 - x\right) \cdot \mathsf{fma}\left(x, \color{blue}{-0.3333333333333333}, 1\right)}{y} \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\left(1 - x\right) \cdot \mathsf{fma}\left(x, -0.3333333333333333, 1\right)}{y}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-4}{3} \cdot \frac{x}{y} + \frac{1}{y}} \]
   7. Step-by-step derivation

      1. *-lft-identity N/A

         \[\leadsto \frac{-4}{3} \cdot \frac{\color{blue}{1 \cdot x}}{y} + \frac{1}{y} \]

      2. associate-*l/ N/A

         \[\leadsto \frac{-4}{3} \cdot \color{blue}{\left(\frac{1}{y} \cdot x\right)} + \frac{1}{y} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\frac{-4}{3} \cdot \frac{1}{y}\right) \cdot x} + \frac{1}{y} \]

      4. metadata-eval N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(\frac{4}{3}\right)\right)} \cdot \frac{1}{y}\right) \cdot x + \frac{1}{y} \]

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

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{y}\right)\right)} \cdot x + \frac{1}{y} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{y}\right)\right)} + \frac{1}{y} \]

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

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x \cdot \left(\frac{4}{3} \cdot \frac{1}{y}\right)\right)\right)} + \frac{1}{y} \]

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

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{4}{3} \cdot \frac{1}{y}\right)} + \frac{1}{y} \]

      9. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot x\right)} \cdot \left(\frac{4}{3} \cdot \frac{1}{y}\right) + \frac{1}{y} \]

      10. rgt-mult-inverse N/A

          \[\leadsto \left(-1 \cdot x\right) \cdot \left(\frac{4}{3} \cdot \frac{1}{y}\right) + \frac{\color{blue}{\left(-1 \cdot x\right) \cdot \frac{1}{-1 \cdot x}}}{y} \]

      11. mul-1-neg N/A

          \[\leadsto \left(-1 \cdot x\right) \cdot \left(\frac{4}{3} \cdot \frac{1}{y}\right) + \frac{\left(-1 \cdot x\right) \cdot \frac{1}{\color{blue}{\mathsf{neg}\left(x\right)}}}{y} \]

      12. distribute-neg-frac2 N/A

          \[\leadsto \left(-1 \cdot x\right) \cdot \left(\frac{4}{3} \cdot \frac{1}{y}\right) + \frac{\left(-1 \cdot x\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{x}\right)\right)}}{y} \]

      13. associate-*r/ N/A

          \[\leadsto \left(-1 \cdot x\right) \cdot \left(\frac{4}{3} \cdot \frac{1}{y}\right) + \color{blue}{\left(-1 \cdot x\right) \cdot \frac{\mathsf{neg}\left(\frac{1}{x}\right)}{y}} \]

      14. distribute-neg-frac N/A

          \[\leadsto \left(-1 \cdot x\right) \cdot \left(\frac{4}{3} \cdot \frac{1}{y}\right) + \left(-1 \cdot x\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{\frac{1}{x}}{y}\right)\right)} \]

      15. associate-/r* N/A

          \[\leadsto \left(-1 \cdot x\right) \cdot \left(\frac{4}{3} \cdot \frac{1}{y}\right) + \left(-1 \cdot x\right) \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{1}{x \cdot y}}\right)\right) \]

      16. distribute-lft-in N/A

          \[\leadsto \color{blue}{\left(-1 \cdot x\right) \cdot \left(\frac{4}{3} \cdot \frac{1}{y} + \left(\mathsf{neg}\left(\frac{1}{x \cdot y}\right)\right)\right)} \]

      17. sub-neg N/A

          \[\leadsto \left(-1 \cdot x\right) \cdot \color{blue}{\left(\frac{4}{3} \cdot \frac{1}{y} - \frac{1}{x \cdot y}\right)} \]

   8. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1.3333333333333333, x, 1\right)}{y}} \]

   if 5 < (*.f64 (-.f64 #s(literal 1 binary64) x) (-.f64 #s(literal 3 binary64) x))

   1. Initial program 89.4%

      \[\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. distribute-rgt-in N/A

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

      11. metadata-eval N/A

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

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

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

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

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

      14. lower-fma.f64 N/A

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

      15. metadata-eval 89.3

          \[\leadsto \frac{\left(1 - x\right) \cdot \mathsf{fma}\left(x, \color{blue}{-0.3333333333333333}, 1\right)}{y} \]

   5. Simplified 89.3%

      \[\leadsto \color{blue}{\frac{\left(1 - x\right) \cdot \mathsf{fma}\left(x, -0.3333333333333333, 1\right)}{y}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{3} \cdot \frac{1}{y} - \frac{4}{3} \cdot \frac{1}{x \cdot y}\right)} \]

   8. Simplified 98.2%

      \[\leadsto \color{blue}{x \cdot \frac{\mathsf{fma}\left(x, 0.3333333333333333, -1.3333333333333333\right)}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 98.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\left(1 - x\right) \cdot \left(3 - x\right) \leq 5:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1.3333333333333333, x, 1\right)}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.3333333333333333}{y\_m}\right)\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m)
 :precision binary64
 (*
  y_s
  (if (<= (* (- 1.0 x) (- 3.0 x)) 5.0)
    (/ (fma -1.3333333333333333 x 1.0) y_m)
    (* x (* x (/ 0.3333333333333333 y_m))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m) {
	double tmp;
	if (((1.0 - x) * (3.0 - x)) <= 5.0) {
		tmp = fma(-1.3333333333333333, x, 1.0) / y_m;
	} else {
		tmp = x * (x * (0.3333333333333333 / y_m));
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m)
	tmp = 0.0
	if (Float64(Float64(1.0 - x) * Float64(3.0 - x)) <= 5.0)
		tmp = Float64(fma(-1.3333333333333333, x, 1.0) / y_m);
	else
		tmp = Float64(x * Float64(x * Float64(0.3333333333333333 / y_m)));
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_] := N[(y$95$s * If[LessEqual[N[(N[(1.0 - x), $MachinePrecision] * N[(3.0 - x), $MachinePrecision]), $MachinePrecision], 5.0], N[(N[(-1.3333333333333333 * x + 1.0), $MachinePrecision] / y$95$m), $MachinePrecision], N[(x * N[(x * N[(0.3333333333333333 / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 #s(literal 1 binary64) x) (-.f64 #s(literal 3 binary64) x)) < 5

   1. Initial program 99.6%

      \[\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. distribute-rgt-in N/A

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

      11. metadata-eval N/A

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

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

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

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

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

      14. lower-fma.f64 N/A

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

      15. metadata-eval 100.0

          \[\leadsto \frac{\left(1 - x\right) \cdot \mathsf{fma}\left(x, \color{blue}{-0.3333333333333333}, 1\right)}{y} \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\left(1 - x\right) \cdot \mathsf{fma}\left(x, -0.3333333333333333, 1\right)}{y}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-4}{3} \cdot \frac{x}{y} + \frac{1}{y}} \]
   7. Step-by-step derivation

      1. *-lft-identity N/A

         \[\leadsto \frac{-4}{3} \cdot \frac{\color{blue}{1 \cdot x}}{y} + \frac{1}{y} \]

      2. associate-*l/ N/A

         \[\leadsto \frac{-4}{3} \cdot \color{blue}{\left(\frac{1}{y} \cdot x\right)} + \frac{1}{y} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\frac{-4}{3} \cdot \frac{1}{y}\right) \cdot x} + \frac{1}{y} \]

      4. metadata-eval N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(\frac{4}{3}\right)\right)} \cdot \frac{1}{y}\right) \cdot x + \frac{1}{y} \]

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

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{y}\right)\right)} \cdot x + \frac{1}{y} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{y}\right)\right)} + \frac{1}{y} \]

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

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x \cdot \left(\frac{4}{3} \cdot \frac{1}{y}\right)\right)\right)} + \frac{1}{y} \]

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

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{4}{3} \cdot \frac{1}{y}\right)} + \frac{1}{y} \]

      9. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot x\right)} \cdot \left(\frac{4}{3} \cdot \frac{1}{y}\right) + \frac{1}{y} \]

      10. rgt-mult-inverse N/A

          \[\leadsto \left(-1 \cdot x\right) \cdot \left(\frac{4}{3} \cdot \frac{1}{y}\right) + \frac{\color{blue}{\left(-1 \cdot x\right) \cdot \frac{1}{-1 \cdot x}}}{y} \]

      11. mul-1-neg N/A

          \[\leadsto \left(-1 \cdot x\right) \cdot \left(\frac{4}{3} \cdot \frac{1}{y}\right) + \frac{\left(-1 \cdot x\right) \cdot \frac{1}{\color{blue}{\mathsf{neg}\left(x\right)}}}{y} \]

      12. distribute-neg-frac2 N/A

          \[\leadsto \left(-1 \cdot x\right) \cdot \left(\frac{4}{3} \cdot \frac{1}{y}\right) + \frac{\left(-1 \cdot x\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{x}\right)\right)}}{y} \]

      13. associate-*r/ N/A

          \[\leadsto \left(-1 \cdot x\right) \cdot \left(\frac{4}{3} \cdot \frac{1}{y}\right) + \color{blue}{\left(-1 \cdot x\right) \cdot \frac{\mathsf{neg}\left(\frac{1}{x}\right)}{y}} \]

      14. distribute-neg-frac N/A

          \[\leadsto \left(-1 \cdot x\right) \cdot \left(\frac{4}{3} \cdot \frac{1}{y}\right) + \left(-1 \cdot x\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{\frac{1}{x}}{y}\right)\right)} \]

      15. associate-/r* N/A

          \[\leadsto \left(-1 \cdot x\right) \cdot \left(\frac{4}{3} \cdot \frac{1}{y}\right) + \left(-1 \cdot x\right) \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{1}{x \cdot y}}\right)\right) \]

      16. distribute-lft-in N/A

          \[\leadsto \color{blue}{\left(-1 \cdot x\right) \cdot \left(\frac{4}{3} \cdot \frac{1}{y} + \left(\mathsf{neg}\left(\frac{1}{x \cdot y}\right)\right)\right)} \]

      17. sub-neg N/A

          \[\leadsto \left(-1 \cdot x\right) \cdot \color{blue}{\left(\frac{4}{3} \cdot \frac{1}{y} - \frac{1}{x \cdot y}\right)} \]

   8. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1.3333333333333333, x, 1\right)}{y}} \]

   if 5 < (*.f64 (-.f64 #s(literal 1 binary64) x) (-.f64 #s(literal 3 binary64) x))

   1. Initial program 89.4%

      \[\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. distribute-rgt-in N/A

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

      11. metadata-eval N/A

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

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

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

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

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

      14. lower-fma.f64 N/A

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

      15. metadata-eval 89.3

          \[\leadsto \frac{\left(1 - x\right) \cdot \mathsf{fma}\left(x, \color{blue}{-0.3333333333333333}, 1\right)}{y} \]

   5. Simplified 89.3%

      \[\leadsto \color{blue}{\frac{\left(1 - x\right) \cdot \mathsf{fma}\left(x, -0.3333333333333333, 1\right)}{y}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{{x}^{2}}{y}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{{x}^{2}}{y} \cdot \frac{1}{3}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{x \cdot x}}{y} \cdot \frac{1}{3} \]

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-lft-identity N/A

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

      8. associate-*l/ N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 96.7

          \[\leadsto x \cdot \left(x \cdot \color{blue}{\frac{0.3333333333333333}{y}}\right) \]

   8. Simplified 96.7%

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \frac{0.3333333333333333}{y}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 92.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\left(1 - x\right) \cdot \left(3 - x\right) \leq 5:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1.3333333333333333, x, 1\right)}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{x \cdot x}{y\_m}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m)
 :precision binary64
 (*
  y_s
  (if (<= (* (- 1.0 x) (- 3.0 x)) 5.0)
    (/ (fma -1.3333333333333333 x 1.0) y_m)
    (* 0.3333333333333333 (/ (* x x) y_m)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m) {
	double tmp;
	if (((1.0 - x) * (3.0 - x)) <= 5.0) {
		tmp = fma(-1.3333333333333333, x, 1.0) / y_m;
	} else {
		tmp = 0.3333333333333333 * ((x * x) / y_m);
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m)
	tmp = 0.0
	if (Float64(Float64(1.0 - x) * Float64(3.0 - x)) <= 5.0)
		tmp = Float64(fma(-1.3333333333333333, x, 1.0) / y_m);
	else
		tmp = Float64(0.3333333333333333 * Float64(Float64(x * x) / y_m));
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_] := N[(y$95$s * If[LessEqual[N[(N[(1.0 - x), $MachinePrecision] * N[(3.0 - x), $MachinePrecision]), $MachinePrecision], 5.0], N[(N[(-1.3333333333333333 * x + 1.0), $MachinePrecision] / y$95$m), $MachinePrecision], N[(0.3333333333333333 * N[(N[(x * x), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 #s(literal 1 binary64) x) (-.f64 #s(literal 3 binary64) x)) < 5

   1. Initial program 99.6%

      \[\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. distribute-rgt-in N/A

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

      11. metadata-eval N/A

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

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

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

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

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

      14. lower-fma.f64 N/A

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

      15. metadata-eval 100.0

          \[\leadsto \frac{\left(1 - x\right) \cdot \mathsf{fma}\left(x, \color{blue}{-0.3333333333333333}, 1\right)}{y} \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\left(1 - x\right) \cdot \mathsf{fma}\left(x, -0.3333333333333333, 1\right)}{y}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-4}{3} \cdot \frac{x}{y} + \frac{1}{y}} \]
   7. Step-by-step derivation

      1. *-lft-identity N/A

         \[\leadsto \frac{-4}{3} \cdot \frac{\color{blue}{1 \cdot x}}{y} + \frac{1}{y} \]

      2. associate-*l/ N/A

         \[\leadsto \frac{-4}{3} \cdot \color{blue}{\left(\frac{1}{y} \cdot x\right)} + \frac{1}{y} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\frac{-4}{3} \cdot \frac{1}{y}\right) \cdot x} + \frac{1}{y} \]

      4. metadata-eval N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(\frac{4}{3}\right)\right)} \cdot \frac{1}{y}\right) \cdot x + \frac{1}{y} \]

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

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{y}\right)\right)} \cdot x + \frac{1}{y} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{y}\right)\right)} + \frac{1}{y} \]

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

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x \cdot \left(\frac{4}{3} \cdot \frac{1}{y}\right)\right)\right)} + \frac{1}{y} \]

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

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{4}{3} \cdot \frac{1}{y}\right)} + \frac{1}{y} \]

      9. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot x\right)} \cdot \left(\frac{4}{3} \cdot \frac{1}{y}\right) + \frac{1}{y} \]

      10. rgt-mult-inverse N/A

          \[\leadsto \left(-1 \cdot x\right) \cdot \left(\frac{4}{3} \cdot \frac{1}{y}\right) + \frac{\color{blue}{\left(-1 \cdot x\right) \cdot \frac{1}{-1 \cdot x}}}{y} \]

      11. mul-1-neg N/A

          \[\leadsto \left(-1 \cdot x\right) \cdot \left(\frac{4}{3} \cdot \frac{1}{y}\right) + \frac{\left(-1 \cdot x\right) \cdot \frac{1}{\color{blue}{\mathsf{neg}\left(x\right)}}}{y} \]

      12. distribute-neg-frac2 N/A

          \[\leadsto \left(-1 \cdot x\right) \cdot \left(\frac{4}{3} \cdot \frac{1}{y}\right) + \frac{\left(-1 \cdot x\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{x}\right)\right)}}{y} \]

      13. associate-*r/ N/A

          \[\leadsto \left(-1 \cdot x\right) \cdot \left(\frac{4}{3} \cdot \frac{1}{y}\right) + \color{blue}{\left(-1 \cdot x\right) \cdot \frac{\mathsf{neg}\left(\frac{1}{x}\right)}{y}} \]

      14. distribute-neg-frac N/A

          \[\leadsto \left(-1 \cdot x\right) \cdot \left(\frac{4}{3} \cdot \frac{1}{y}\right) + \left(-1 \cdot x\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{\frac{1}{x}}{y}\right)\right)} \]

      15. associate-/r* N/A

          \[\leadsto \left(-1 \cdot x\right) \cdot \left(\frac{4}{3} \cdot \frac{1}{y}\right) + \left(-1 \cdot x\right) \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{1}{x \cdot y}}\right)\right) \]

      16. distribute-lft-in N/A

          \[\leadsto \color{blue}{\left(-1 \cdot x\right) \cdot \left(\frac{4}{3} \cdot \frac{1}{y} + \left(\mathsf{neg}\left(\frac{1}{x \cdot y}\right)\right)\right)} \]

      17. sub-neg N/A

          \[\leadsto \left(-1 \cdot x\right) \cdot \color{blue}{\left(\frac{4}{3} \cdot \frac{1}{y} - \frac{1}{x \cdot y}\right)} \]

   8. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1.3333333333333333, x, 1\right)}{y}} \]

   if 5 < (*.f64 (-.f64 #s(literal 1 binary64) x) (-.f64 #s(literal 3 binary64) x))

   1. Initial program 89.4%

      \[\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. distribute-rgt-in N/A

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

      11. metadata-eval N/A

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

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

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

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

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

      14. lower-fma.f64 N/A

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

      15. metadata-eval 89.3

          \[\leadsto \frac{\left(1 - x\right) \cdot \mathsf{fma}\left(x, \color{blue}{-0.3333333333333333}, 1\right)}{y} \]

   5. Simplified 89.3%

      \[\leadsto \color{blue}{\frac{\left(1 - x\right) \cdot \mathsf{fma}\left(x, -0.3333333333333333, 1\right)}{y}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{{x}^{2}}{y}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{{x}^{2}}{y} \cdot \frac{1}{3}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{x \cdot x}}{y} \cdot \frac{1}{3} \]

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-lft-identity N/A

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

      8. associate-*l/ N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 96.7

          \[\leadsto x \cdot \left(x \cdot \color{blue}{\frac{0.3333333333333333}{y}}\right) \]

   8. Simplified 96.7%

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \frac{0.3333333333333333}{y}\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{{x}^{2}}{y}} \]
   10. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot {x}^{2}}{y}} \]

       2. *-rgt-identity N/A

          \[\leadsto \frac{\color{blue}{\left(\frac{1}{3} \cdot {x}^{2}\right) \cdot 1}}{y} \]

       3. associate-*r/ N/A

          \[\leadsto \color{blue}{\left(\frac{1}{3} \cdot {x}^{2}\right) \cdot \frac{1}{y}} \]

       4. associate-*l* N/A

          \[\leadsto \color{blue}{\frac{1}{3} \cdot \left({x}^{2} \cdot \frac{1}{y}\right)} \]

       5. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{3} \cdot \left({x}^{2} \cdot \frac{1}{y}\right)} \]

       6. associate-*r/ N/A

          \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{{x}^{2} \cdot 1}{y}} \]

       7. *-rgt-identity N/A

          \[\leadsto \frac{1}{3} \cdot \frac{\color{blue}{{x}^{2}}}{y} \]

       8. lower-/.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{{x}^{2}}{y}} \]

       9. unpow2 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{\color{blue}{x \cdot x}}{y} \]

       10. lower-*.f64 86.4

           \[\leadsto 0.3333333333333333 \cdot \frac{\color{blue}{x \cdot x}}{y} \]

   11. Simplified 86.4%

       \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{x \cdot x}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 57.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq -0.75:\\ \;\;\;\;x \cdot \frac{-1.3333333333333333}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y\_m}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m)
 :precision binary64
 (* y_s (if (<= x -0.75) (* x (/ -1.3333333333333333 y_m)) (/ 1.0 y_m))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m) {
	double tmp;
	if (x <= -0.75) {
		tmp = x * (-1.3333333333333333 / y_m);
	} else {
		tmp = 1.0 / y_m;
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if (x <= (-0.75d0)) then
        tmp = x * ((-1.3333333333333333d0) / y_m)
    else
        tmp = 1.0d0 / y_m
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m) {
	double tmp;
	if (x <= -0.75) {
		tmp = x * (-1.3333333333333333 / y_m);
	} else {
		tmp = 1.0 / y_m;
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m):
	tmp = 0
	if x <= -0.75:
		tmp = x * (-1.3333333333333333 / y_m)
	else:
		tmp = 1.0 / y_m
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m)
	tmp = 0.0
	if (x <= -0.75)
		tmp = Float64(x * Float64(-1.3333333333333333 / y_m));
	else
		tmp = Float64(1.0 / y_m);
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m)
	tmp = 0.0;
	if (x <= -0.75)
		tmp = x * (-1.3333333333333333 / y_m);
	else
		tmp = 1.0 / y_m;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_] := N[(y$95$s * If[LessEqual[x, -0.75], N[(x * N[(-1.3333333333333333 / y$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 / y$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < -0.75

   1. Initial program 91.6%

      \[\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. distribute-rgt-in N/A

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

      11. metadata-eval N/A

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

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

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

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

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

      14. lower-fma.f64 N/A

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

      15. metadata-eval 91.5

          \[\leadsto \frac{\left(1 - x\right) \cdot \mathsf{fma}\left(x, \color{blue}{-0.3333333333333333}, 1\right)}{y} \]

   5. Simplified 91.5%

      \[\leadsto \color{blue}{\frac{\left(1 - x\right) \cdot \mathsf{fma}\left(x, -0.3333333333333333, 1\right)}{y}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{3} \cdot \frac{1}{y} - \frac{4}{3} \cdot \frac{1}{x \cdot y}\right)} \]

   8. Simplified 98.0%

      \[\leadsto \color{blue}{x \cdot \frac{\mathsf{fma}\left(x, 0.3333333333333333, -1.3333333333333333\right)}{y}} \]

   9. Taylor expanded in x around 0

      \[\leadsto x \cdot \color{blue}{\frac{\frac{-4}{3}}{y}} \]
   10. Step-by-step derivation

       1. lower-/.f64 26.3

          \[\leadsto x \cdot \color{blue}{\frac{-1.3333333333333333}{y}} \]

   11. Simplified 26.3%

       \[\leadsto x \cdot \color{blue}{\frac{-1.3333333333333333}{y}} \]

   if -0.75 < x

   1. Initial program 95.2%

      \[\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 66.5

         \[\leadsto \color{blue}{\frac{1}{y}} \]

   5. Simplified 66.5%

      \[\leadsto \color{blue}{\frac{1}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 56.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \frac{\mathsf{fma}\left(-1.3333333333333333, x, 1\right)}{y\_m} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m)
 :precision binary64
 (* y_s (/ (fma -1.3333333333333333 x 1.0) y_m)))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m) {
	return y_s * (fma(-1.3333333333333333, x, 1.0) / y_m);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m)
	return Float64(y_s * Float64(fma(-1.3333333333333333, x, 1.0) / y_m))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_] := N[(y$95$s * N[(N[(-1.3333333333333333 * x + 1.0), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.3%

   \[\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y}} \]
4. Step-by-step derivation

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower--.f64 N/A

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

   8. sub-neg N/A

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

   9. +-commutative N/A

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

   10. distribute-rgt-in N/A

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

   11. metadata-eval N/A

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

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

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

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

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

   14. lower-fma.f64 N/A

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

   15. metadata-eval 94.4

       \[\leadsto \frac{\left(1 - x\right) \cdot \mathsf{fma}\left(x, \color{blue}{-0.3333333333333333}, 1\right)}{y} \]

5. Simplified 94.4%

   \[\leadsto \color{blue}{\frac{\left(1 - x\right) \cdot \mathsf{fma}\left(x, -0.3333333333333333, 1\right)}{y}} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{-4}{3} \cdot \frac{x}{y} + \frac{1}{y}} \]
7. Step-by-step derivation

   1. *-lft-identity N/A

      \[\leadsto \frac{-4}{3} \cdot \frac{\color{blue}{1 \cdot x}}{y} + \frac{1}{y} \]

   2. associate-*l/ N/A

      \[\leadsto \frac{-4}{3} \cdot \color{blue}{\left(\frac{1}{y} \cdot x\right)} + \frac{1}{y} \]

   3. associate-*l* N/A

      \[\leadsto \color{blue}{\left(\frac{-4}{3} \cdot \frac{1}{y}\right) \cdot x} + \frac{1}{y} \]

   4. metadata-eval N/A

      \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(\frac{4}{3}\right)\right)} \cdot \frac{1}{y}\right) \cdot x + \frac{1}{y} \]

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

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{y}\right)\right)} \cdot x + \frac{1}{y} \]

   6. *-commutative N/A

      \[\leadsto \color{blue}{x \cdot \left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{y}\right)\right)} + \frac{1}{y} \]

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

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x \cdot \left(\frac{4}{3} \cdot \frac{1}{y}\right)\right)\right)} + \frac{1}{y} \]

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

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{4}{3} \cdot \frac{1}{y}\right)} + \frac{1}{y} \]

   9. mul-1-neg N/A

      \[\leadsto \color{blue}{\left(-1 \cdot x\right)} \cdot \left(\frac{4}{3} \cdot \frac{1}{y}\right) + \frac{1}{y} \]

   10. rgt-mult-inverse N/A

       \[\leadsto \left(-1 \cdot x\right) \cdot \left(\frac{4}{3} \cdot \frac{1}{y}\right) + \frac{\color{blue}{\left(-1 \cdot x\right) \cdot \frac{1}{-1 \cdot x}}}{y} \]

   11. mul-1-neg N/A

       \[\leadsto \left(-1 \cdot x\right) \cdot \left(\frac{4}{3} \cdot \frac{1}{y}\right) + \frac{\left(-1 \cdot x\right) \cdot \frac{1}{\color{blue}{\mathsf{neg}\left(x\right)}}}{y} \]

   12. distribute-neg-frac2 N/A

       \[\leadsto \left(-1 \cdot x\right) \cdot \left(\frac{4}{3} \cdot \frac{1}{y}\right) + \frac{\left(-1 \cdot x\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{x}\right)\right)}}{y} \]

   13. associate-*r/ N/A

       \[\leadsto \left(-1 \cdot x\right) \cdot \left(\frac{4}{3} \cdot \frac{1}{y}\right) + \color{blue}{\left(-1 \cdot x\right) \cdot \frac{\mathsf{neg}\left(\frac{1}{x}\right)}{y}} \]

   14. distribute-neg-frac N/A

       \[\leadsto \left(-1 \cdot x\right) \cdot \left(\frac{4}{3} \cdot \frac{1}{y}\right) + \left(-1 \cdot x\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{\frac{1}{x}}{y}\right)\right)} \]

   15. associate-/r* N/A

       \[\leadsto \left(-1 \cdot x\right) \cdot \left(\frac{4}{3} \cdot \frac{1}{y}\right) + \left(-1 \cdot x\right) \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{1}{x \cdot y}}\right)\right) \]

   16. distribute-lft-in N/A

       \[\leadsto \color{blue}{\left(-1 \cdot x\right) \cdot \left(\frac{4}{3} \cdot \frac{1}{y} + \left(\mathsf{neg}\left(\frac{1}{x \cdot y}\right)\right)\right)} \]

   17. sub-neg N/A

       \[\leadsto \left(-1 \cdot x\right) \cdot \color{blue}{\left(\frac{4}{3} \cdot \frac{1}{y} - \frac{1}{x \cdot y}\right)} \]

8. Simplified 54.8%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1.3333333333333333, x, 1\right)}{y}} \]
9. Add Preprocessing

Alternative 8: 51.2% accurate, 2.3× speedup

Math

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

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m) :precision binary64 (* y_s (/ 1.0 y_m)))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m) {
	return y_s * (1.0 / y_m);
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    code = y_s * (1.0d0 / y_m)
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m) {
	return y_s * (1.0 / y_m);
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m):
	return y_s * (1.0 / y_m)

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m)
	return Float64(y_s * Float64(1.0 / y_m))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp = code(y_s, x, y_m)
	tmp = y_s * (1.0 / y_m);
end

Wolfram

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

Derivation

1. Initial program 94.3%

   \[\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. lower-/.f64 50.2

      \[\leadsto \color{blue}{\frac{1}{y}} \]

5. Simplified 50.2%

   \[\leadsto \color{blue}{\frac{1}{y}} \]
6. Add Preprocessing

Developer Target 1: 99.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{1 - x}{y} \cdot \frac{3 - x}{3} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (/ (- 1.0 x) y) (/ (- 3.0 x) 3.0)))

C

double code(double x, double y) {
	return ((1.0 - x) / y) * ((3.0 - x) / 3.0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((1.0d0 - x) / y) * ((3.0d0 - x) / 3.0d0)
end function

Java

public static double code(double x, double y) {
	return ((1.0 - x) / y) * ((3.0 - x) / 3.0);
}

Python

def code(x, y):
	return ((1.0 - x) / y) * ((3.0 - x) / 3.0)

Julia

function code(x, y)
	return Float64(Float64(Float64(1.0 - x) / y) * Float64(Float64(3.0 - x) / 3.0))
end

MATLAB

function tmp = code(x, y)
	tmp = ((1.0 - x) / y) * ((3.0 - x) / 3.0);
end

Wolfram

code[x_, y_] := N[(N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision] * N[(N[(3.0 - x), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024215 
(FPCore (x y)
  :name "Diagrams.TwoD.Arc:bezierFromSweepQ1 from diagrams-lib-1.3.0.3"
  :precision binary64

  :alt
  (! :herbie-platform default (* (/ (- 1 x) y) (/ (- 3 x) 3)))

  (/ (* (- 1.0 x) (- 3.0 x)) (* y 3.0)))