Falkner and Boettcher, Equation (22+)

Percentage Accurate: 98.5% → 99.6%

Time: 1.9s

Alternatives: 5

Speedup: 2.5×

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: 99.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1 + v \cdot v}{\pi} \cdot 1.3333333333333333}{\sqrt{\mathsf{fma}\left(-6 \cdot v, v, 2\right)}} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (/
  (* (/ (+ 1.0 (* v v)) PI) 1.3333333333333333)
  (sqrt (fma (* -6.0 v) v 2.0))))

C

double code(double v) {
	return (((1.0 + (v * v)) / ((double) M_PI)) * 1.3333333333333333) / sqrt(fma((-6.0 * v), v, 2.0));
}

Julia

function code(v)
	return Float64(Float64(Float64(Float64(1.0 + Float64(v * v)) / pi) * 1.3333333333333333) / sqrt(fma(Float64(-6.0 * v), v, 2.0)))
end

Wolfram

code[v_] := N[(N[(N[(N[(1.0 + N[(v * v), $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision] * 1.3333333333333333), $MachinePrecision] / N[Sqrt[N[(N[(-6.0 * v), $MachinePrecision] * v + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.5%

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

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

3. Applied rewrites 100.0%

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

4. Taylor expanded in v around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. associate-*r/ N/A

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

   8. metadata-eval N/A

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

   9. lower-/.f64 N/A

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

   10. lift-PI.f64 99.6

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

6. Applied rewrites 99.6%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{v \cdot v}{\pi}, 1.3333333333333333, \frac{1.3333333333333333}{\pi}\right)}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}} \]
7. Step-by-step derivation

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. pow2 N/A

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

   6. *-commutative N/A

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

   7. lift-PI.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. metadata-eval N/A

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

   10. associate-*r/ N/A

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

   11. distribute-lft-out N/A

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

   12. lower-*.f64 N/A

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

   13. pow2 N/A

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

   14. associate-/l* N/A

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

   15. lower-fma.f64 N/A

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

   16. lower-/.f64 N/A

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

   17. lift-PI.f64 N/A

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

   18. inv-pow N/A

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

   19. lower-pow.f64 N/A

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

   20. lift-PI.f64 99.6

       \[\leadsto \frac{1.3333333333333333 \cdot \mathsf{fma}\left(v, \frac{v}{\pi}, {\pi}^{-1}\right)}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}} \]

8. Applied rewrites 99.6%

   \[\leadsto \frac{1.3333333333333333 \cdot \color{blue}{\mathsf{fma}\left(v, \frac{v}{\pi}, {\pi}^{-1}\right)}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}} \]
9. Step-by-step derivation

10. Applied rewrites 99.6%

    \[\leadsto \color{blue}{\frac{\frac{1 + v \cdot v}{\pi} \cdot 1.3333333333333333}{\sqrt{\mathsf{fma}\left(-6 \cdot v, v, 2\right)}}} \]
11. Add Preprocessing

Alternative 2: 99.1% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.5%

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

2. Taylor expanded in v around 0

   \[\leadsto \frac{4}{\color{blue}{\left(3 \cdot \mathsf{PI}\left(\right)\right)} \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]
3. 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 97.6

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

4. Applied rewrites 97.6%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-/r* N/A

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

   8. pow2 N/A

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

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

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

   10. metadata-eval N/A

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

   11. pow2 N/A

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

   12. lift-*.f64 N/A

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

6. Applied rewrites 99.1%

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

Alternative 3: 99.1% accurate, 1.8× 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 98.5%

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

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

3. Applied rewrites 100.0%

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

4. Taylor expanded in v around 0

   \[\leadsto \frac{\color{blue}{\frac{\frac{4}{3}}{\mathsf{PI}\left(\right)}}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}} \]
5. 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 99.1

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

6. Applied rewrites 99.1%

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

Alternative 4: 99.0% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.5%

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

2. Taylor expanded in v around 0

   \[\leadsto \frac{4}{\color{blue}{\left(3 \cdot \mathsf{PI}\left(\right)\right)} \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]
3. 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 97.6

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

4. Applied rewrites 97.6%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-/r* N/A

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

   8. pow2 N/A

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

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

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

   10. metadata-eval N/A

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

   11. pow2 N/A

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

   12. lift-*.f64 N/A

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

6. Applied rewrites 99.1%

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

7. Taylor expanded in v around 0

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

   1. associate-*r* 99.0

      \[\leadsto \frac{\frac{4}{\pi \cdot 3}}{\sqrt{2}} \]

9. Applied rewrites 99.0%

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

Alternative 5: 97.5% accurate, 3.3× 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 98.5%

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

2. Taylor expanded in v around 0

   \[\leadsto \color{blue}{\frac{4}{3} \cdot \frac{\sqrt{\frac{1}{2}}}{\mathsf{PI}\left(\right)}} \]
3. 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 97.5

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

4. Applied rewrites 97.5%

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

Reproduce

herbie shell --seed 2025101 
(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)))))))