Falkner and Boettcher, Equation (22+)

Percentage Accurate: 98.5% → 99.7%

Time: 1.9s

Alternatives: 6

Speedup: 1.5×

Specification

Math

\[\begin{array}{l} \\ \frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (/ 4.0 (* (* (* 3.0 PI) (- 1.0 (* v v))) (sqrt (- 2.0 (* 6.0 (* v v)))))))

C

double code(double v) {
	return 4.0 / (((3.0 * ((double) M_PI)) * (1.0 - (v * v))) * sqrt((2.0 - (6.0 * (v * v)))));
}

Java

public static double code(double v) {
	return 4.0 / (((3.0 * Math.PI) * (1.0 - (v * v))) * Math.sqrt((2.0 - (6.0 * (v * v)))));
}

Python

def code(v):
	return 4.0 / (((3.0 * math.pi) * (1.0 - (v * v))) * math.sqrt((2.0 - (6.0 * (v * v)))))

Julia

function code(v)
	return Float64(4.0 / Float64(Float64(Float64(3.0 * pi) * Float64(1.0 - Float64(v * v))) * sqrt(Float64(2.0 - Float64(6.0 * Float64(v * v))))))
end

MATLAB

function tmp = code(v)
	tmp = 4.0 / (((3.0 * pi) * (1.0 - (v * v))) * sqrt((2.0 - (6.0 * (v * v)))));
end

Wolfram

code[v_] := N[(4.0 / N[(N[(N[(3.0 * Pi), $MachinePrecision] * N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(2.0 - N[(6.0 * N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (/ 4.0 (* (* (* 3.0 PI) (- 1.0 (* v v))) (sqrt (- 2.0 (* 6.0 (* v v)))))))

C

double code(double v) {
	return 4.0 / (((3.0 * ((double) M_PI)) * (1.0 - (v * v))) * sqrt((2.0 - (6.0 * (v * v)))));
}

Java

public static double code(double v) {
	return 4.0 / (((3.0 * Math.PI) * (1.0 - (v * v))) * Math.sqrt((2.0 - (6.0 * (v * v)))));
}

Python

def code(v):
	return 4.0 / (((3.0 * math.pi) * (1.0 - (v * v))) * math.sqrt((2.0 - (6.0 * (v * v)))))

Julia

function code(v)
	return Float64(4.0 / Float64(Float64(Float64(3.0 * pi) * Float64(1.0 - Float64(v * v))) * sqrt(Float64(2.0 - Float64(6.0 * Float64(v * v))))))
end

MATLAB

function tmp = code(v)
	tmp = 4.0 / (((3.0 * pi) * (1.0 - (v * v))) * sqrt((2.0 - (6.0 * (v * v)))));
end

Wolfram

code[v_] := N[(4.0 / N[(N[(N[(3.0 * Pi), $MachinePrecision] * N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(2.0 - N[(6.0 * N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{v \cdot v}{\pi}, 1.3333333333333333, \frac{1.3333333333333333}{\pi}\right), v \cdot v, \frac{1.3333333333333333}{\pi}\right)}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (/
  (fma
   (fma (/ (* v v) PI) 1.3333333333333333 (/ 1.3333333333333333 PI))
   (* v v)
   (/ 1.3333333333333333 PI))
  (sqrt (fma -6.0 (* v v) 2.0))))

C

double code(double v) {
	return fma(fma(((v * v) / ((double) M_PI)), 1.3333333333333333, (1.3333333333333333 / ((double) M_PI))), (v * v), (1.3333333333333333 / ((double) M_PI))) / sqrt(fma(-6.0, (v * v), 2.0));
}

Julia

function code(v)
	return Float64(fma(fma(Float64(Float64(v * v) / pi), 1.3333333333333333, Float64(1.3333333333333333 / pi)), Float64(v * v), Float64(1.3333333333333333 / pi)) / sqrt(fma(-6.0, Float64(v * v), 2.0)))
end

Wolfram

code[v_] := N[(N[(N[(N[(N[(v * v), $MachinePrecision] / Pi), $MachinePrecision] * 1.3333333333333333 + N[(1.3333333333333333 / Pi), $MachinePrecision]), $MachinePrecision] * N[(v * v), $MachinePrecision] + N[(1.3333333333333333 / Pi), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(-6.0 * N[(v * v), $MachinePrecision] + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.4%

   \[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{4}{\color{blue}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{4}{\color{blue}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right)} \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]

   4. lift-PI.f64 N/A

      \[\leadsto \frac{4}{\left(\left(3 \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{4}{\left(\color{blue}{\left(3 \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]

   6. lift--.f64 N/A

      \[\leadsto \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left(1 - v \cdot v\right)}\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - \color{blue}{v \cdot v}\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]

   8. lift-sqrt.f64 N/A

      \[\leadsto \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \color{blue}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}} \]

   9. lift--.f64 N/A

      \[\leadsto \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{\color{blue}{2 - 6 \cdot \left(v \cdot v\right)}}} \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - \color{blue}{6 \cdot \left(v \cdot v\right)}}} \]

   11. lift-*.f64 N/A

       \[\leadsto \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \color{blue}{\left(v \cdot v\right)}}} \]

   12. associate-/r* N/A

       \[\leadsto \color{blue}{\frac{\frac{4}{\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}} \]

   13. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{\frac{4}{\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}} \]

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{\frac{4}{\left(1 - v \cdot v\right) \cdot \left(\pi \cdot 3\right)}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}}} \]

5. Taylor expanded in v around 0

   \[\leadsto \frac{\color{blue}{{v}^{2} \cdot \left(\frac{4}{3} \cdot \frac{{v}^{2}}{\mathsf{PI}\left(\right)} + \frac{4}{3} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right) + \frac{4}{3} \cdot \frac{1}{\mathsf{PI}\left(\right)}}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\left(\frac{4}{3} \cdot \frac{{v}^{2}}{\mathsf{PI}\left(\right)} + \frac{4}{3} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right) \cdot {v}^{2} + \color{blue}{\frac{4}{3}} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}} \]

   2. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{4}{3} \cdot \frac{{v}^{2}}{\mathsf{PI}\left(\right)} + \frac{4}{3} \cdot \frac{1}{\mathsf{PI}\left(\right)}, \color{blue}{{v}^{2}}, \frac{4}{3} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}} \]

   3. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{{v}^{2}}{\mathsf{PI}\left(\right)} \cdot \frac{4}{3} + \frac{4}{3} \cdot \frac{1}{\mathsf{PI}\left(\right)}, {v}^{2}, \frac{4}{3} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}} \]

   4. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{{v}^{2}}{\mathsf{PI}\left(\right)}, \frac{4}{3}, \frac{4}{3} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right), {\color{blue}{v}}^{2}, \frac{4}{3} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}} \]

   5. lower-/.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{{v}^{2}}{\mathsf{PI}\left(\right)}, \frac{4}{3}, \frac{4}{3} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right), {v}^{2}, \frac{4}{3} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}} \]

   6. pow2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{v \cdot v}{\mathsf{PI}\left(\right)}, \frac{4}{3}, \frac{4}{3} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right), {v}^{2}, \frac{4}{3} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{v \cdot v}{\mathsf{PI}\left(\right)}, \frac{4}{3}, \frac{4}{3} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right), {v}^{2}, \frac{4}{3} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}} \]

   8. lift-PI.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{v \cdot v}{\pi}, \frac{4}{3}, \frac{4}{3} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right), {v}^{2}, \frac{4}{3} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}} \]

   9. associate-*r/ N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{v \cdot v}{\pi}, \frac{4}{3}, \frac{\frac{4}{3} \cdot 1}{\mathsf{PI}\left(\right)}\right), {v}^{2}, \frac{4}{3} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}} \]

   10. metadata-eval N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{v \cdot v}{\pi}, \frac{4}{3}, \frac{\frac{4}{3}}{\mathsf{PI}\left(\right)}\right), {v}^{2}, \frac{4}{3} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}} \]

   11. lower-/.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{v \cdot v}{\pi}, \frac{4}{3}, \frac{\frac{4}{3}}{\mathsf{PI}\left(\right)}\right), {v}^{2}, \frac{4}{3} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}} \]

   12. lift-PI.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{v \cdot v}{\pi}, \frac{4}{3}, \frac{\frac{4}{3}}{\pi}\right), {v}^{2}, \frac{4}{3} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}} \]

   13. pow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{v \cdot v}{\pi}, \frac{4}{3}, \frac{\frac{4}{3}}{\pi}\right), v \cdot \color{blue}{v}, \frac{4}{3} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}} \]

   14. lift-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{v \cdot v}{\pi}, \frac{4}{3}, \frac{\frac{4}{3}}{\pi}\right), v \cdot \color{blue}{v}, \frac{4}{3} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}} \]

   15. associate-*r/ N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{v \cdot v}{\pi}, \frac{4}{3}, \frac{\frac{4}{3}}{\pi}\right), v \cdot v, \frac{\frac{4}{3} \cdot 1}{\mathsf{PI}\left(\right)}\right)}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}} \]

7. Applied rewrites 100.0%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{v \cdot v}{\pi}, 1.3333333333333333, \frac{1.3333333333333333}{\pi}\right), v \cdot v, \frac{1.3333333333333333}{\pi}\right)}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}} \]
8. Add Preprocessing

Alternative 2: 100.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{4}{\left(1 - v \cdot v\right) \cdot \left(\pi \cdot 3\right)}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (/ (/ 4.0 (* (- 1.0 (* v v)) (* PI 3.0))) (sqrt (fma -6.0 (* v v) 2.0))))

C

double code(double v) {
	return (4.0 / ((1.0 - (v * v)) * (((double) M_PI) * 3.0))) / sqrt(fma(-6.0, (v * v), 2.0));
}

Julia

function code(v)
	return Float64(Float64(4.0 / Float64(Float64(1.0 - Float64(v * v)) * Float64(pi * 3.0))) / sqrt(fma(-6.0, Float64(v * v), 2.0)))
end

Wolfram

code[v_] := N[(N[(4.0 / N[(N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision] * N[(Pi * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(-6.0 * N[(v * v), $MachinePrecision] + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.4%

   \[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{4}{\color{blue}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{4}{\color{blue}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right)} \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]

   4. lift-PI.f64 N/A

      \[\leadsto \frac{4}{\left(\left(3 \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{4}{\left(\color{blue}{\left(3 \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]

   6. lift--.f64 N/A

      \[\leadsto \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left(1 - v \cdot v\right)}\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - \color{blue}{v \cdot v}\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]

   8. lift-sqrt.f64 N/A

      \[\leadsto \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \color{blue}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}} \]

   9. lift--.f64 N/A

      \[\leadsto \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{\color{blue}{2 - 6 \cdot \left(v \cdot v\right)}}} \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - \color{blue}{6 \cdot \left(v \cdot v\right)}}} \]

   11. lift-*.f64 N/A

       \[\leadsto \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \color{blue}{\left(v \cdot v\right)}}} \]

   12. associate-/r* N/A

       \[\leadsto \color{blue}{\frac{\frac{4}{\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}} \]

   13. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{\frac{4}{\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}} \]

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{\frac{4}{\left(1 - v \cdot v\right) \cdot \left(\pi \cdot 3\right)}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}}} \]
5. Add Preprocessing

Alternative 3: 99.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{4}{\sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}}}{\pi \cdot 3} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (/ (/ 4.0 (sqrt (fma (* v v) -6.0 2.0))) (* PI 3.0)))

C

double code(double v) {
	return (4.0 / sqrt(fma((v * v), -6.0, 2.0))) / (((double) M_PI) * 3.0);
}

Julia

function code(v)
	return Float64(Float64(4.0 / sqrt(fma(Float64(v * v), -6.0, 2.0))) / Float64(pi * 3.0))
end

Wolfram

code[v_] := N[(N[(4.0 / N[Sqrt[N[(N[(v * v), $MachinePrecision] * -6.0 + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(Pi * 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.4%

   \[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]
2. Add Preprocessing

3. Taylor expanded in v around 0

   \[\leadsto \frac{4}{\color{blue}{\left(3 \cdot \mathsf{PI}\left(\right)\right)} \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{4}{\left(\mathsf{PI}\left(\right) \cdot \color{blue}{3}\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{4}{\left(\mathsf{PI}\left(\right) \cdot \color{blue}{3}\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]

   3. lift-PI.f64 97.9

      \[\leadsto \frac{4}{\left(\pi \cdot 3\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]

5. Applied rewrites 97.9%

   \[\leadsto \frac{4}{\color{blue}{\left(\pi \cdot 3\right)} \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{4}{\color{blue}{\left(\pi \cdot 3\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}} \]

   2. lift-sqrt.f64 N/A

      \[\leadsto \frac{4}{\left(\pi \cdot 3\right) \cdot \color{blue}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}} \]

   3. lift--.f64 N/A

      \[\leadsto \frac{4}{\left(\pi \cdot 3\right) \cdot \sqrt{\color{blue}{2 - 6 \cdot \left(v \cdot v\right)}}} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{4}{\left(\pi \cdot 3\right) \cdot \sqrt{2 - \color{blue}{6 \cdot \left(v \cdot v\right)}}} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{4}{\left(\pi \cdot 3\right) \cdot \sqrt{2 - 6 \cdot \color{blue}{\left(v \cdot v\right)}}} \]

   6. *-commutative N/A

      \[\leadsto \frac{4}{\color{blue}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)} \cdot \left(\pi \cdot 3\right)}} \]

   7. pow2 N/A

      \[\leadsto \frac{4}{\sqrt{2 - 6 \cdot \color{blue}{{v}^{2}}} \cdot \left(\pi \cdot 3\right)} \]

   8. fp-cancel-sub-sign-inv N/A

      \[\leadsto \frac{4}{\sqrt{\color{blue}{2 + \left(\mathsf{neg}\left(6\right)\right) \cdot {v}^{2}}} \cdot \left(\pi \cdot 3\right)} \]

   9. metadata-eval N/A

      \[\leadsto \frac{4}{\sqrt{2 + \color{blue}{-6} \cdot {v}^{2}} \cdot \left(\pi \cdot 3\right)} \]

   10. +-commutative N/A

       \[\leadsto \frac{4}{\sqrt{\color{blue}{-6 \cdot {v}^{2} + 2}} \cdot \left(\pi \cdot 3\right)} \]

   11. pow2 N/A

       \[\leadsto \frac{4}{\sqrt{-6 \cdot \color{blue}{\left(v \cdot v\right)} + 2} \cdot \left(\pi \cdot 3\right)} \]

7. Applied rewrites 97.9%

   \[\leadsto \color{blue}{\frac{4}{\sqrt{\mathsf{fma}\left(-6 \cdot v, v, 2\right)} \cdot \left(\pi \cdot 3\right)}} \]
8. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{4}{\sqrt{\mathsf{fma}\left(-6 \cdot v, v, 2\right)} \cdot \left(\pi \cdot 3\right)}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{4}{\color{blue}{\sqrt{\mathsf{fma}\left(-6 \cdot v, v, 2\right)} \cdot \left(\pi \cdot 3\right)}} \]

   3. lift-sqrt.f64 N/A

      \[\leadsto \frac{4}{\color{blue}{\sqrt{\mathsf{fma}\left(-6 \cdot v, v, 2\right)}} \cdot \left(\pi \cdot 3\right)} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{4}{\sqrt{\mathsf{fma}\left(\color{blue}{-6 \cdot v}, v, 2\right)} \cdot \left(\pi \cdot 3\right)} \]

   5. lift-fma.f64 N/A

      \[\leadsto \frac{4}{\sqrt{\color{blue}{\left(-6 \cdot v\right) \cdot v + 2}} \cdot \left(\pi \cdot 3\right)} \]

   6. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{4}{\sqrt{\left(-6 \cdot v\right) \cdot v + 2}}}{\pi \cdot 3}} \]

   7. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{4}{\sqrt{\left(-6 \cdot v\right) \cdot v + 2}}}{\pi \cdot 3}} \]

9. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\frac{\frac{4}{\sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}}}{\pi \cdot 3}} \]
10. Add Preprocessing

Alternative 4: 99.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1.3333333333333333}{\pi}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (/ (/ 1.3333333333333333 PI) (sqrt (fma -6.0 (* v v) 2.0))))

C

double code(double v) {
	return (1.3333333333333333 / ((double) M_PI)) / sqrt(fma(-6.0, (v * v), 2.0));
}

Julia

function code(v)
	return Float64(Float64(1.3333333333333333 / pi) / sqrt(fma(-6.0, Float64(v * v), 2.0)))
end

Wolfram

code[v_] := N[(N[(1.3333333333333333 / Pi), $MachinePrecision] / N[Sqrt[N[(-6.0 * N[(v * v), $MachinePrecision] + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.4%

   \[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{4}{\color{blue}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{4}{\color{blue}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right)} \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]

   4. lift-PI.f64 N/A

      \[\leadsto \frac{4}{\left(\left(3 \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{4}{\left(\color{blue}{\left(3 \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]

   6. lift--.f64 N/A

      \[\leadsto \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left(1 - v \cdot v\right)}\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - \color{blue}{v \cdot v}\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]

   8. lift-sqrt.f64 N/A

      \[\leadsto \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \color{blue}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}} \]

   9. lift--.f64 N/A

      \[\leadsto \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{\color{blue}{2 - 6 \cdot \left(v \cdot v\right)}}} \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - \color{blue}{6 \cdot \left(v \cdot v\right)}}} \]

   11. lift-*.f64 N/A

       \[\leadsto \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \color{blue}{\left(v \cdot v\right)}}} \]

   12. associate-/r* N/A

       \[\leadsto \color{blue}{\frac{\frac{4}{\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}} \]

   13. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{\frac{4}{\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}} \]

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{\frac{4}{\left(1 - v \cdot v\right) \cdot \left(\pi \cdot 3\right)}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}}} \]

5. Taylor expanded in v around 0

   \[\leadsto \frac{\color{blue}{\frac{\frac{4}{3}}{\mathsf{PI}\left(\right)}}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}} \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\frac{\frac{4}{3}}{\color{blue}{\mathsf{PI}\left(\right)}}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}} \]

   2. lift-PI.f64 99.4

      \[\leadsto \frac{\frac{1.3333333333333333}{\pi}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}} \]

7. Applied rewrites 99.4%

   \[\leadsto \frac{\color{blue}{\frac{1.3333333333333333}{\pi}}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}} \]
8. Add Preprocessing

Alternative 5: 99.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{4}{\sqrt{2}}}{\pi \cdot 3} \end{array} \]

FPCore

(FPCore (v) :precision binary64 (/ (/ 4.0 (sqrt 2.0)) (* PI 3.0)))

C

double code(double v) {
	return (4.0 / sqrt(2.0)) / (((double) M_PI) * 3.0);
}

Java

public static double code(double v) {
	return (4.0 / Math.sqrt(2.0)) / (Math.PI * 3.0);
}

Python

def code(v):
	return (4.0 / math.sqrt(2.0)) / (math.pi * 3.0)

Julia

function code(v)
	return Float64(Float64(4.0 / sqrt(2.0)) / Float64(pi * 3.0))
end

MATLAB

function tmp = code(v)
	tmp = (4.0 / sqrt(2.0)) / (pi * 3.0);
end

Wolfram

code[v_] := N[(N[(4.0 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[(Pi * 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.4%

   \[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]
2. Add Preprocessing

3. Taylor expanded in v around 0

   \[\leadsto \frac{4}{\color{blue}{\left(3 \cdot \mathsf{PI}\left(\right)\right)} \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{4}{\left(\mathsf{PI}\left(\right) \cdot \color{blue}{3}\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{4}{\left(\mathsf{PI}\left(\right) \cdot \color{blue}{3}\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]

   3. lift-PI.f64 97.9

      \[\leadsto \frac{4}{\left(\pi \cdot 3\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]

5. Applied rewrites 97.9%

   \[\leadsto \frac{4}{\color{blue}{\left(\pi \cdot 3\right)} \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{4}{\color{blue}{\left(\pi \cdot 3\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}} \]

   2. lift-sqrt.f64 N/A

      \[\leadsto \frac{4}{\left(\pi \cdot 3\right) \cdot \color{blue}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}} \]

   3. lift--.f64 N/A

      \[\leadsto \frac{4}{\left(\pi \cdot 3\right) \cdot \sqrt{\color{blue}{2 - 6 \cdot \left(v \cdot v\right)}}} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{4}{\left(\pi \cdot 3\right) \cdot \sqrt{2 - \color{blue}{6 \cdot \left(v \cdot v\right)}}} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{4}{\left(\pi \cdot 3\right) \cdot \sqrt{2 - 6 \cdot \color{blue}{\left(v \cdot v\right)}}} \]

   6. *-commutative N/A

      \[\leadsto \frac{4}{\color{blue}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)} \cdot \left(\pi \cdot 3\right)}} \]

   7. pow2 N/A

      \[\leadsto \frac{4}{\sqrt{2 - 6 \cdot \color{blue}{{v}^{2}}} \cdot \left(\pi \cdot 3\right)} \]

   8. fp-cancel-sub-sign-inv N/A

      \[\leadsto \frac{4}{\sqrt{\color{blue}{2 + \left(\mathsf{neg}\left(6\right)\right) \cdot {v}^{2}}} \cdot \left(\pi \cdot 3\right)} \]

   9. metadata-eval N/A

      \[\leadsto \frac{4}{\sqrt{2 + \color{blue}{-6} \cdot {v}^{2}} \cdot \left(\pi \cdot 3\right)} \]

   10. +-commutative N/A

       \[\leadsto \frac{4}{\sqrt{\color{blue}{-6 \cdot {v}^{2} + 2}} \cdot \left(\pi \cdot 3\right)} \]

   11. pow2 N/A

       \[\leadsto \frac{4}{\sqrt{-6 \cdot \color{blue}{\left(v \cdot v\right)} + 2} \cdot \left(\pi \cdot 3\right)} \]

7. Applied rewrites 97.9%

   \[\leadsto \color{blue}{\frac{4}{\sqrt{\mathsf{fma}\left(-6 \cdot v, v, 2\right)} \cdot \left(\pi \cdot 3\right)}} \]
8. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{4}{\sqrt{\mathsf{fma}\left(-6 \cdot v, v, 2\right)} \cdot \left(\pi \cdot 3\right)}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{4}{\color{blue}{\sqrt{\mathsf{fma}\left(-6 \cdot v, v, 2\right)} \cdot \left(\pi \cdot 3\right)}} \]

   3. lift-sqrt.f64 N/A

      \[\leadsto \frac{4}{\color{blue}{\sqrt{\mathsf{fma}\left(-6 \cdot v, v, 2\right)}} \cdot \left(\pi \cdot 3\right)} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{4}{\sqrt{\mathsf{fma}\left(\color{blue}{-6 \cdot v}, v, 2\right)} \cdot \left(\pi \cdot 3\right)} \]

   5. lift-fma.f64 N/A

      \[\leadsto \frac{4}{\sqrt{\color{blue}{\left(-6 \cdot v\right) \cdot v + 2}} \cdot \left(\pi \cdot 3\right)} \]

   6. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{4}{\sqrt{\left(-6 \cdot v\right) \cdot v + 2}}}{\pi \cdot 3}} \]

   7. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{4}{\sqrt{\left(-6 \cdot v\right) \cdot v + 2}}}{\pi \cdot 3}} \]

9. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\frac{\frac{4}{\sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}}}{\pi \cdot 3}} \]

10. Taylor expanded in v around 0

    \[\leadsto \frac{\frac{4}{\sqrt{\color{blue}{2}}}}{\pi \cdot 3} \]
11. Step-by-step derivation

12. Applied rewrites 99.4%

    \[\leadsto \frac{\frac{4}{\sqrt{\color{blue}{2}}}}{\pi \cdot 3} \]
13. Add Preprocessing

Alternative 6: 97.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \frac{\sqrt{0.5}}{\pi} \cdot 1.3333333333333333 \end{array} \]

FPCore

(FPCore (v) :precision binary64 (* (/ (sqrt 0.5) PI) 1.3333333333333333))

C

double code(double v) {
	return (sqrt(0.5) / ((double) M_PI)) * 1.3333333333333333;
}

Java

public static double code(double v) {
	return (Math.sqrt(0.5) / Math.PI) * 1.3333333333333333;
}

Python

def code(v):
	return (math.sqrt(0.5) / math.pi) * 1.3333333333333333

Julia

function code(v)
	return Float64(Float64(sqrt(0.5) / pi) * 1.3333333333333333)
end

MATLAB

function tmp = code(v)
	tmp = (sqrt(0.5) / pi) * 1.3333333333333333;
end

Wolfram

code[v_] := N[(N[(N[Sqrt[0.5], $MachinePrecision] / Pi), $MachinePrecision] * 1.3333333333333333), $MachinePrecision]

Derivation

1. Initial program 98.4%

   \[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]
2. Add Preprocessing

3. Taylor expanded in v around 0

   \[\leadsto \color{blue}{\frac{4}{3} \cdot \frac{\sqrt{\frac{1}{2}}}{\mathsf{PI}\left(\right)}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\sqrt{\frac{1}{2}}}{\mathsf{PI}\left(\right)} \cdot \color{blue}{\frac{4}{3}} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\sqrt{\frac{1}{2}}}{\mathsf{PI}\left(\right)} \cdot \color{blue}{\frac{4}{3}} \]

   3. lower-/.f64 N/A

      \[\leadsto \frac{\sqrt{\frac{1}{2}}}{\mathsf{PI}\left(\right)} \cdot \frac{4}{3} \]

   4. lower-sqrt.f64 N/A

      \[\leadsto \frac{\sqrt{\frac{1}{2}}}{\mathsf{PI}\left(\right)} \cdot \frac{4}{3} \]

   5. lift-PI.f64 97.9

      \[\leadsto \frac{\sqrt{0.5}}{\pi} \cdot 1.3333333333333333 \]

5. Applied rewrites 97.9%

   \[\leadsto \color{blue}{\frac{\sqrt{0.5}}{\pi} \cdot 1.3333333333333333} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2025082 
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  :precision binary64
  (/ 4.0 (* (* (* 3.0 PI) (- 1.0 (* v v))) (sqrt (- 2.0 (* 6.0 (* v v)))))))