Falkner and Boettcher, Equation (22+)

Percentage Accurate: 98.5% → 100.0%

Time: 3.4s

Alternatives: 7

Speedup: 30.0×

Specification

Math

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

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

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

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

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

FPCore

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

C

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

Julia

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

Wolfram

code[v_] := N[(-1.3333333333333333 / N[(N[(N[(v * v + -1.0), $MachinePrecision] * Pi), $MachinePrecision] * N[Sqrt[N[(-6.0 * N[(v * v), $MachinePrecision] + 2.0), $MachinePrecision]], $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-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. associate-*l* N/A

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

   7. associate-/r* N/A

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

   8. lower-/.f64 N/A

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

   9. metadata-eval N/A

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

   10. lower-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 100.0%

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

   13. lift--.f64 N/A

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

   14. lift-*.f64 N/A

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

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

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

   16. +-commutative N/A

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

3. Applied rewrites 100.0%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. lower-/.f64 N/A

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

   4. metadata-eval N/A

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

   5. lift-*.f64 N/A

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

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

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

   7. lower-*.f64 N/A

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

   8. lift-*.f64 N/A

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

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

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

   10. lift--.f64 N/A

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

   11. sub-negate-rev N/A

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

   12. lower-*.f64 N/A

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

   13. sub-flip N/A

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

   14. lift-*.f64 N/A

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

   15. lower-fma.f64 N/A

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

   16. metadata-eval 100.0%

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

5. Applied rewrites 100.0%

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

Alternative 2: 100.0% accurate, 1.3× speedup

Math

\[\frac{\frac{-0.4244131815783876}{\mathsf{fma}\left(v, v, -1\right)}}{\sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}} \]

FPCore

(FPCore (v)
  :precision binary64
  (/
 (/ -0.4244131815783876 (fma v v -1.0))
 (sqrt (fma (* v v) -6.0 2.0))))

C

double code(double v) {
	return (-0.4244131815783876 / fma(v, v, -1.0)) / sqrt(fma((v * v), -6.0, 2.0));
}

Julia

function code(v)
	return Float64(Float64(-0.4244131815783876 / fma(v, v, -1.0)) / sqrt(fma(Float64(v * v), -6.0, 2.0)))
end

Wolfram

code[v_] := N[(N[(-0.4244131815783876 / N[(v * v + -1.0), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(N[(v * v), $MachinePrecision] * -6.0 + 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. associate-*l* N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-/r* N/A

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

   9. lower-/.f64 N/A

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

   10. metadata-eval N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 100.0%

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

3. Applied rewrites 100.0%

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

4. Evaluated real constant 100.0%

   \[\leadsto \frac{\color{blue}{0.4244131815783876}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)} \cdot \left(1 - v \cdot v\right)} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{1911387046407553}{4503599627370496}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)} \cdot \left(1 - v \cdot v\right)}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\frac{1911387046407553}{4503599627370496}}{\color{blue}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)} \cdot \left(1 - v \cdot v\right)}} \]

   3. lift--.f64 N/A

      \[\leadsto \frac{\frac{1911387046407553}{4503599627370496}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)} \cdot \color{blue}{\left(1 - v \cdot v\right)}} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\frac{1911387046407553}{4503599627370496}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)} \cdot \left(1 - \color{blue}{v \cdot v}\right)} \]

   5. *-commutative N/A

      \[\leadsto \frac{\frac{1911387046407553}{4503599627370496}}{\color{blue}{\left(1 - v \cdot v\right) \cdot \sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}}} \]

   6. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{\frac{1911387046407553}{4503599627370496}}{1 - v \cdot v}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}}} \]

   7. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{\frac{1911387046407553}{4503599627370496}}{1 - v \cdot v}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}}} \]

   8. frac-2neg N/A

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{neg}\left(\frac{1911387046407553}{4503599627370496}\right)}{\mathsf{neg}\left(\left(1 - v \cdot v\right)\right)}}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{\frac{\mathsf{neg}\left(\frac{1911387046407553}{4503599627370496}\right)}{\mathsf{neg}\left(\left(1 - \color{blue}{v \cdot v}\right)\right)}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}} \]

   10. sub-negate-rev N/A

       \[\leadsto \frac{\frac{\mathsf{neg}\left(\frac{1911387046407553}{4503599627370496}\right)}{\color{blue}{v \cdot v - 1}}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}} \]

   11. metadata-eval N/A

       \[\leadsto \frac{\frac{\mathsf{neg}\left(\frac{1911387046407553}{4503599627370496}\right)}{v \cdot v - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)}}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}} \]

   12. add-flip N/A

       \[\leadsto \frac{\frac{\mathsf{neg}\left(\frac{1911387046407553}{4503599627370496}\right)}{\color{blue}{v \cdot v + -1}}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}} \]

   13. lift-*.f64 N/A

       \[\leadsto \frac{\frac{\mathsf{neg}\left(\frac{1911387046407553}{4503599627370496}\right)}{\color{blue}{v \cdot v} + -1}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}} \]

   14. lift-fma.f64 N/A

       \[\leadsto \frac{\frac{\mathsf{neg}\left(\frac{1911387046407553}{4503599627370496}\right)}{\color{blue}{\mathsf{fma}\left(v, v, -1\right)}}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}} \]

   15. lower-/.f64 N/A

       \[\leadsto \frac{\color{blue}{\frac{\mathsf{neg}\left(\frac{1911387046407553}{4503599627370496}\right)}{\mathsf{fma}\left(v, v, -1\right)}}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}} \]

   16. metadata-eval 100.0%

       \[\leadsto \frac{\frac{\color{blue}{-0.4244131815783876}}{\mathsf{fma}\left(v, v, -1\right)}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}} \]

   17. lift-fma.f64 N/A

       \[\leadsto \frac{\frac{\frac{-1911387046407553}{4503599627370496}}{\mathsf{fma}\left(v, v, -1\right)}}{\sqrt{\color{blue}{-6 \cdot \left(v \cdot v\right) + 2}}} \]

   18. *-commutative N/A

       \[\leadsto \frac{\frac{\frac{-1911387046407553}{4503599627370496}}{\mathsf{fma}\left(v, v, -1\right)}}{\sqrt{\color{blue}{\left(v \cdot v\right) \cdot -6} + 2}} \]

   19. lower-fma.f64 100.0%

       \[\leadsto \frac{\frac{-0.4244131815783876}{\mathsf{fma}\left(v, v, -1\right)}}{\sqrt{\color{blue}{\mathsf{fma}\left(v \cdot v, -6, 2\right)}}} \]

6. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{\frac{-0.4244131815783876}{\mathsf{fma}\left(v, v, -1\right)}}{\sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}}} \]
7. Add Preprocessing

Alternative 3: 100.0% accurate, 1.3× speedup

Math

\[\frac{-0.4244131815783876}{\sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)} \cdot \mathsf{fma}\left(v, v, -1\right)} \]

FPCore

(FPCore (v)
  :precision binary64
  (/
 -0.4244131815783876
 (* (sqrt (fma (* v v) -6.0 2.0)) (fma v v -1.0))))

C

double code(double v) {
	return -0.4244131815783876 / (sqrt(fma((v * v), -6.0, 2.0)) * fma(v, v, -1.0));
}

Julia

function code(v)
	return Float64(-0.4244131815783876 / Float64(sqrt(fma(Float64(v * v), -6.0, 2.0)) * fma(v, v, -1.0)))
end

Wolfram

code[v_] := N[(-0.4244131815783876 / N[(N[Sqrt[N[(N[(v * v), $MachinePrecision] * -6.0 + 2.0), $MachinePrecision]], $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. 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. associate-*l* N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-/r* N/A

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

   9. lower-/.f64 N/A

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

   10. metadata-eval N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 100.0%

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

3. Applied rewrites 100.0%

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

4. Evaluated real constant 100.0%

   \[\leadsto \frac{\color{blue}{0.4244131815783876}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)} \cdot \left(1 - v \cdot v\right)} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{1911387046407553}{4503599627370496}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)} \cdot \left(1 - v \cdot v\right)}} \]

   2. frac-2neg N/A

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\frac{1911387046407553}{4503599627370496}\right)}{\mathsf{neg}\left(\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)} \cdot \left(1 - v \cdot v\right)\right)}} \]

   3. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\frac{1911387046407553}{4503599627370496}\right)}{\mathsf{neg}\left(\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)} \cdot \left(1 - v \cdot v\right)\right)}} \]

   4. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{\frac{-1911387046407553}{4503599627370496}}}{\mathsf{neg}\left(\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)} \cdot \left(1 - v \cdot v\right)\right)} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{\frac{-1911387046407553}{4503599627370496}}{\mathsf{neg}\left(\color{blue}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)} \cdot \left(1 - v \cdot v\right)}\right)} \]

   6. lift--.f64 N/A

      \[\leadsto \frac{\frac{-1911387046407553}{4503599627370496}}{\mathsf{neg}\left(\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)} \cdot \color{blue}{\left(1 - v \cdot v\right)}\right)} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\frac{-1911387046407553}{4503599627370496}}{\mathsf{neg}\left(\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)} \cdot \left(1 - \color{blue}{v \cdot v}\right)\right)} \]

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

      \[\leadsto \frac{\frac{-1911387046407553}{4503599627370496}}{\color{blue}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)} \cdot \left(\mathsf{neg}\left(\left(1 - v \cdot v\right)\right)\right)}} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{\frac{-1911387046407553}{4503599627370496}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)} \cdot \left(\mathsf{neg}\left(\left(1 - \color{blue}{v \cdot v}\right)\right)\right)} \]

   10. sub-negate-rev N/A

       \[\leadsto \frac{\frac{-1911387046407553}{4503599627370496}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)} \cdot \color{blue}{\left(v \cdot v - 1\right)}} \]

   11. metadata-eval N/A

       \[\leadsto \frac{\frac{-1911387046407553}{4503599627370496}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)} \cdot \left(v \cdot v - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)}\right)} \]

   12. add-flip N/A

       \[\leadsto \frac{\frac{-1911387046407553}{4503599627370496}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)} \cdot \color{blue}{\left(v \cdot v + -1\right)}} \]

   13. lift-*.f64 N/A

       \[\leadsto \frac{\frac{-1911387046407553}{4503599627370496}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)} \cdot \left(\color{blue}{v \cdot v} + -1\right)} \]

   14. lift-fma.f64 N/A

       \[\leadsto \frac{\frac{-1911387046407553}{4503599627370496}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)} \cdot \color{blue}{\mathsf{fma}\left(v, v, -1\right)}} \]

   15. lower-*.f64 100.0%

       \[\leadsto \frac{-0.4244131815783876}{\color{blue}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)} \cdot \mathsf{fma}\left(v, v, -1\right)}} \]

   16. lift-fma.f64 N/A

       \[\leadsto \frac{\frac{-1911387046407553}{4503599627370496}}{\sqrt{\color{blue}{-6 \cdot \left(v \cdot v\right) + 2}} \cdot \mathsf{fma}\left(v, v, -1\right)} \]

   17. *-commutative N/A

       \[\leadsto \frac{\frac{-1911387046407553}{4503599627370496}}{\sqrt{\color{blue}{\left(v \cdot v\right) \cdot -6} + 2} \cdot \mathsf{fma}\left(v, v, -1\right)} \]

   18. lower-fma.f64 100.0%

       \[\leadsto \frac{-0.4244131815783876}{\sqrt{\color{blue}{\mathsf{fma}\left(v \cdot v, -6, 2\right)}} \cdot \mathsf{fma}\left(v, v, -1\right)} \]

6. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{-0.4244131815783876}{\sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)} \cdot \mathsf{fma}\left(v, v, -1\right)}} \]
7. Add Preprocessing

Alternative 4: 98.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   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. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*r* N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-*.f64 98.5%

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

   12. lift--.f64 N/A

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

   13. lift-*.f64 N/A

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

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

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

   15. +-commutative N/A

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

   16. lower-fma.f64 N/A

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

   17. metadata-eval 98.5%

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

3. Applied rewrites 98.5%

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

4. Taylor expanded in v around 0

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

6. Applied rewrites 97.4%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

8. Applied rewrites 98.9%

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

Alternative 5: 98.9% accurate, 1.7× speedup

Math

\[\frac{0.4244131815783876}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)} \cdot 1} \]

FPCore

(FPCore (v)
  :precision binary64
  (/ 0.4244131815783876 (* (sqrt (fma -6.0 (* v v) 2.0)) 1.0)))

C

double code(double v) {
	return 0.4244131815783876 / (sqrt(fma(-6.0, (v * v), 2.0)) * 1.0);
}

Julia

function code(v)
	return Float64(0.4244131815783876 / Float64(sqrt(fma(-6.0, Float64(v * v), 2.0)) * 1.0))
end

Wolfram

code[v_] := N[(0.4244131815783876 / N[(N[Sqrt[N[(-6.0 * N[(v * v), $MachinePrecision] + 2.0), $MachinePrecision]], $MachinePrecision] * 1.0), $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. 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. associate-*l* N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-/r* N/A

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

   9. lower-/.f64 N/A

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

   10. metadata-eval N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 100.0%

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

3. Applied rewrites 100.0%

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

4. Evaluated real constant 100.0%

   \[\leadsto \frac{\color{blue}{0.4244131815783876}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)} \cdot \left(1 - v \cdot v\right)} \]

5. Taylor expanded in v around 0

   \[\leadsto \frac{0.4244131815783876}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)} \cdot \color{blue}{1}} \]
6. Step-by-step derivation

7. Applied rewrites 98.9%

   \[\leadsto \frac{0.4244131815783876}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)} \cdot \color{blue}{1}} \]
8. Add Preprocessing

Alternative 6: 98.9% accurate, 2.0× speedup

Math

\[\frac{-0.4244131815783876}{\sqrt{2} \cdot \mathsf{fma}\left(v, v, -1\right)} \]

FPCore

(FPCore (v)
  :precision binary64
  (/ -0.4244131815783876 (* (sqrt 2.0) (fma v v -1.0))))

C

double code(double v) {
	return -0.4244131815783876 / (sqrt(2.0) * fma(v, v, -1.0));
}

Julia

function code(v)
	return Float64(-0.4244131815783876 / Float64(sqrt(2.0) * fma(v, v, -1.0)))
end

Wolfram

code[v_] := N[(-0.4244131815783876 / N[(N[Sqrt[2.0], $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. 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. associate-*l* N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-/r* N/A

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

   9. lower-/.f64 N/A

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

   10. metadata-eval N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 100.0%

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

3. Applied rewrites 100.0%

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

4. Evaluated real constant 100.0%

   \[\leadsto \frac{\color{blue}{0.4244131815783876}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)} \cdot \left(1 - v \cdot v\right)} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{1911387046407553}{4503599627370496}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)} \cdot \left(1 - v \cdot v\right)}} \]

   2. frac-2neg N/A

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\frac{1911387046407553}{4503599627370496}\right)}{\mathsf{neg}\left(\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)} \cdot \left(1 - v \cdot v\right)\right)}} \]

   3. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\frac{1911387046407553}{4503599627370496}\right)}{\mathsf{neg}\left(\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)} \cdot \left(1 - v \cdot v\right)\right)}} \]

   4. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{\frac{-1911387046407553}{4503599627370496}}}{\mathsf{neg}\left(\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)} \cdot \left(1 - v \cdot v\right)\right)} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{\frac{-1911387046407553}{4503599627370496}}{\mathsf{neg}\left(\color{blue}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)} \cdot \left(1 - v \cdot v\right)}\right)} \]

   6. lift--.f64 N/A

      \[\leadsto \frac{\frac{-1911387046407553}{4503599627370496}}{\mathsf{neg}\left(\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)} \cdot \color{blue}{\left(1 - v \cdot v\right)}\right)} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\frac{-1911387046407553}{4503599627370496}}{\mathsf{neg}\left(\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)} \cdot \left(1 - \color{blue}{v \cdot v}\right)\right)} \]

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

      \[\leadsto \frac{\frac{-1911387046407553}{4503599627370496}}{\color{blue}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)} \cdot \left(\mathsf{neg}\left(\left(1 - v \cdot v\right)\right)\right)}} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{\frac{-1911387046407553}{4503599627370496}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)} \cdot \left(\mathsf{neg}\left(\left(1 - \color{blue}{v \cdot v}\right)\right)\right)} \]

   10. sub-negate-rev N/A

       \[\leadsto \frac{\frac{-1911387046407553}{4503599627370496}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)} \cdot \color{blue}{\left(v \cdot v - 1\right)}} \]

   11. metadata-eval N/A

       \[\leadsto \frac{\frac{-1911387046407553}{4503599627370496}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)} \cdot \left(v \cdot v - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)}\right)} \]

   12. add-flip N/A

       \[\leadsto \frac{\frac{-1911387046407553}{4503599627370496}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)} \cdot \color{blue}{\left(v \cdot v + -1\right)}} \]

   13. lift-*.f64 N/A

       \[\leadsto \frac{\frac{-1911387046407553}{4503599627370496}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)} \cdot \left(\color{blue}{v \cdot v} + -1\right)} \]

   14. lift-fma.f64 N/A

       \[\leadsto \frac{\frac{-1911387046407553}{4503599627370496}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)} \cdot \color{blue}{\mathsf{fma}\left(v, v, -1\right)}} \]

   15. lower-*.f64 100.0%

       \[\leadsto \frac{-0.4244131815783876}{\color{blue}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)} \cdot \mathsf{fma}\left(v, v, -1\right)}} \]

   16. lift-fma.f64 N/A

       \[\leadsto \frac{\frac{-1911387046407553}{4503599627370496}}{\sqrt{\color{blue}{-6 \cdot \left(v \cdot v\right) + 2}} \cdot \mathsf{fma}\left(v, v, -1\right)} \]

   17. *-commutative N/A

       \[\leadsto \frac{\frac{-1911387046407553}{4503599627370496}}{\sqrt{\color{blue}{\left(v \cdot v\right) \cdot -6} + 2} \cdot \mathsf{fma}\left(v, v, -1\right)} \]

   18. lower-fma.f64 100.0%

       \[\leadsto \frac{-0.4244131815783876}{\sqrt{\color{blue}{\mathsf{fma}\left(v \cdot v, -6, 2\right)}} \cdot \mathsf{fma}\left(v, v, -1\right)} \]

6. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{-0.4244131815783876}{\sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)} \cdot \mathsf{fma}\left(v, v, -1\right)}} \]

7. Taylor expanded in v around 0

   \[\leadsto \frac{-0.4244131815783876}{\sqrt{\color{blue}{2}} \cdot \mathsf{fma}\left(v, v, -1\right)} \]
8. Step-by-step derivation

9. Applied rewrites 98.9%

   \[\leadsto \frac{-0.4244131815783876}{\sqrt{\color{blue}{2}} \cdot \mathsf{fma}\left(v, v, -1\right)} \]
10. Add Preprocessing

Alternative 7: 98.8% accurate, 30.0× speedup

Math

\[0.30010543871903533 \]

FPCore

(FPCore (v)
  :precision binary64
  0.30010543871903533)

C

double code(double v) {
	return 0.30010543871903533;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(v)
use fmin_fmax_functions
    real(8), intent (in) :: v
    code = 0.30010543871903533d0
end function

Java

public static double code(double v) {
	return 0.30010543871903533;
}

Python

def code(v):
	return 0.30010543871903533

Julia

function code(v)
	return 0.30010543871903533
end

MATLAB

function tmp = code(v)
	tmp = 0.30010543871903533;
end

Wolfram

code[v_] := 0.30010543871903533

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

   \[\leadsto \color{blue}{\frac{\frac{4}{3}}{\pi \cdot \sqrt{2}}} \]
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 98.8%

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

4. Applied rewrites 98.8%

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

5. Evaluated real constant 98.8%

   \[\leadsto 0.30010543871903533 \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2025210 
(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)))))))