Falkner and Boettcher, Equation (22+)

Percentage Accurate: 98.5% → 100.0%

Time: 1.9s

Alternatives: 4

Speedup: 3.2×

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: 100.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   4. lower-/.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*l* N/A

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

   8. associate-/r* N/A

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

   9. lower-/.f64 N/A

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

   10. metadata-eval N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 100.0

       \[\leadsto \frac{\frac{1.3333333333333333}{\color{blue}{\left(1 - v \cdot v\right) \cdot \pi}}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]

   13. lift--.f64 N/A

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

   14. sub-flip N/A

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

   15. +-commutative N/A

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

   16. lift-*.f64 N/A

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

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

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

3. Applied rewrites 100.0%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-*.f64 N/A

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

   5. lift-fma.f64 100.0

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

5. Applied rewrites 100.0%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. frac-2neg N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-fma.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*l* N/A

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

   10. lift-*.f64 N/A

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

   11. lift-fma.f64 N/A

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

   12. associate-*l* N/A

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

   13. *-commutative N/A

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

   14. lift-*.f64 N/A

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

   15. lift-*.f64 N/A

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

7. Applied rewrites 100.0%

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

Alternative 2: 99.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[v_] := N[(N[(4.0 / N[Sqrt[N[(-6.0 * N[(v * v), $MachinePrecision] + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * Pi), $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. lower-*.f64 N/A

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

   2. lower-PI.f64 97.5

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

4. Applied rewrites 97.5%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f64 N/A

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

   6. lower-/.f64 99.0

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

   7. lift--.f64 N/A

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

   8. lift-*.f64 N/A

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

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

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

   10. metadata-eval N/A

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

   11. +-commutative N/A

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

   12. lift-fma.f64 99.0

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

6. Applied rewrites 99.0%

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

Alternative 3: 99.0% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   4. lower-/.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*l* N/A

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

   8. associate-/r* N/A

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

   9. lower-/.f64 N/A

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

   10. metadata-eval N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 100.0

       \[\leadsto \frac{\frac{1.3333333333333333}{\color{blue}{\left(1 - v \cdot v\right) \cdot \pi}}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]

   13. lift--.f64 N/A

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

   14. sub-flip N/A

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

   15. +-commutative N/A

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

   16. lift-*.f64 N/A

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

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

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

3. Applied rewrites 100.0%

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

4. Taylor expanded in v around 0

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

   1. lower-PI.f64 99.0

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

6. Applied rewrites 99.0%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 99.0

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

   7. lift-fma.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*l* N/A

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

   10. lift-*.f64 N/A

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

   11. lift-fma.f64 99.0

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

8. Applied rewrites 99.0%

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

Alternative 4: 98.9% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \frac{1.3333333333333333}{\pi \cdot \sqrt{2}} \end{array} \]

FPCore

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

C

double code(double v) {
	return 1.3333333333333333 / (((double) M_PI) * sqrt(2.0));
}

Java

public static double code(double v) {
	return 1.3333333333333333 / (Math.PI * Math.sqrt(2.0));
}

Python

def code(v):
	return 1.3333333333333333 / (math.pi * math.sqrt(2.0))

Julia

function code(v)
	return Float64(1.3333333333333333 / Float64(pi * sqrt(2.0)))
end

MATLAB

function tmp = code(v)
	tmp = 1.3333333333333333 / (pi * sqrt(2.0));
end

Wolfram

code[v_] := N[(1.3333333333333333 / N[(Pi * N[Sqrt[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

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-PI.f64 N/A

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

   4. lower-sqrt.f64 98.9

      \[\leadsto \frac{1.3333333333333333}{\pi \cdot \sqrt{2}} \]

4. Applied rewrites 98.9%

   \[\leadsto \color{blue}{\frac{1.3333333333333333}{\pi \cdot \sqrt{2}}} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025150 
(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)))))))