Falkner and Boettcher, Equation (22+)

Percentage Accurate: 98.5% → 100.0%

Time: 3.8s

Alternatives: 7

Speedup: 29.1×

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, 1.1× speedup

Math

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

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

Wolfram

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

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

3. Applied rewrites 100.0%

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

Alternative 2: 100.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   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

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

3. Applied rewrites 100.0%

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

   1. lift-/.f64 N/A

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

   2. mult-flip N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/r* N/A

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

   6. associate-*l/ N/A

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

   7. lower-/.f64 N/A

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

   8. frac-times N/A

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

   9. metadata-eval N/A

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

   10. lower-/.f64 N/A

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

   11. lower-*.f64 100.0%

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

   12. lift-fma.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-fma.f64 100.0%

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

5. Applied rewrites 100.0%

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

Alternative 3: 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. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. associate-*l* N/A

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

   8. associate-/r* N/A

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

   9. lower-/.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

3. Applied rewrites 100.0%

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

4. Evaluated real constant 100.0%

   \[\leadsto \frac{\color{blue}{1.2732395447351628}}{\left(\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)} \cdot \left(1 - v \cdot v\right)\right) \cdot 3} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{5734161139222659}{4503599627370496}}{\left(\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)} \cdot \left(1 - v \cdot v\right)\right) \cdot 3}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\frac{5734161139222659}{4503599627370496}}{\color{blue}{\left(\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)} \cdot \left(1 - v \cdot v\right)\right) \cdot 3}} \]

   3. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{\frac{5734161139222659}{4503599627370496}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)} \cdot \left(1 - v \cdot v\right)}}{3}} \]

   4. mult-flip N/A

      \[\leadsto \color{blue}{\frac{\frac{5734161139222659}{4503599627370496}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)} \cdot \left(1 - v \cdot v\right)} \cdot \frac{1}{3}} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\frac{5734161139222659}{4503599627370496}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)} \cdot \left(1 - v \cdot v\right)} \cdot \color{blue}{\frac{1}{3}} \]

   6. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{5734161139222659}{4503599627370496}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)} \cdot \left(1 - v \cdot v\right)} \cdot \frac{1}{3}} \]

6. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{-1.2732395447351628}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)} \cdot \mathsf{fma}\left(v, v, -1\right)} \cdot 0.3333333333333333} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{-5734161139222659}{4503599627370496}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)} \cdot \mathsf{fma}\left(v, v, -1\right)} \cdot \frac{1}{3}} \]

   2. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{-5734161139222659}{4503599627370496}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)} \cdot \mathsf{fma}\left(v, v, -1\right)}} \cdot \frac{1}{3} \]

   3. associate-*l/ N/A

      \[\leadsto \color{blue}{\frac{\frac{-5734161139222659}{4503599627370496} \cdot \frac{1}{3}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)} \cdot \mathsf{fma}\left(v, v, -1\right)}} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\frac{-5734161139222659}{4503599627370496} \cdot \frac{1}{3}}{\color{blue}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)} \cdot \mathsf{fma}\left(v, v, -1\right)}} \]

   5. *-commutative N/A

      \[\leadsto \frac{\frac{-5734161139222659}{4503599627370496} \cdot \frac{1}{3}}{\color{blue}{\mathsf{fma}\left(v, v, -1\right) \cdot \sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}}} \]

   6. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{\frac{-5734161139222659}{4503599627370496} \cdot \frac{1}{3}}{\mathsf{fma}\left(v, v, -1\right)}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}}} \]

   7. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{\frac{-5734161139222659}{4503599627370496} \cdot \frac{1}{3}}{\mathsf{fma}\left(v, v, -1\right)}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}}} \]

   8. lower-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{\frac{-5734161139222659}{4503599627370496} \cdot \frac{1}{3}}{\mathsf{fma}\left(v, v, -1\right)}}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}} \]

   9. metadata-eval 100.0%

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

   10. lift-sqrt.f64 N/A

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

   11. pow1/2 N/A

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

   12. lift-fma.f64 N/A

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

   13. *-commutative N/A

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

   14. lift-fma.f64 N/A

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

   15. pow1/2 N/A

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

   16. lift-sqrt.f64 100.0%

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

8. Applied rewrites 100.0%

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

Alternative 4: 100.0% accurate, 1.3× speedup

Math

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

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

Wolfram

code[v_] := N[(-0.4244131815783876 / N[(N[(v * v + -1.0), $MachinePrecision] * N[Sqrt[N[(N[(v * v), $MachinePrecision] * -6.0 + 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. 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. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. associate-*l* N/A

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

   8. associate-/r* N/A

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

   9. lower-/.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

3. Applied rewrites 100.0%

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

4. Evaluated real constant 100.0%

   \[\leadsto \frac{\color{blue}{1.2732395447351628}}{\left(\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)} \cdot \left(1 - v \cdot v\right)\right) \cdot 3} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{5734161139222659}{4503599627370496}}{\left(\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)} \cdot \left(1 - v \cdot v\right)\right) \cdot 3}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\frac{5734161139222659}{4503599627370496}}{\color{blue}{\left(\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)} \cdot \left(1 - v \cdot v\right)\right) \cdot 3}} \]

   3. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{\frac{5734161139222659}{4503599627370496}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)} \cdot \left(1 - v \cdot v\right)}}{3}} \]

   4. mult-flip N/A

      \[\leadsto \color{blue}{\frac{\frac{5734161139222659}{4503599627370496}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)} \cdot \left(1 - v \cdot v\right)} \cdot \frac{1}{3}} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\frac{5734161139222659}{4503599627370496}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)} \cdot \left(1 - v \cdot v\right)} \cdot \color{blue}{\frac{1}{3}} \]

   6. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{5734161139222659}{4503599627370496}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)} \cdot \left(1 - v \cdot v\right)} \cdot \frac{1}{3}} \]

6. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{-1.2732395447351628}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)} \cdot \mathsf{fma}\left(v, v, -1\right)} \cdot 0.3333333333333333} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{-5734161139222659}{4503599627370496}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)} \cdot \mathsf{fma}\left(v, v, -1\right)} \cdot \frac{1}{3}} \]

   2. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{-5734161139222659}{4503599627370496}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)} \cdot \mathsf{fma}\left(v, v, -1\right)}} \cdot \frac{1}{3} \]

   3. associate-*l/ N/A

      \[\leadsto \color{blue}{\frac{\frac{-5734161139222659}{4503599627370496} \cdot \frac{1}{3}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)} \cdot \mathsf{fma}\left(v, v, -1\right)}} \]

   4. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{-5734161139222659}{4503599627370496} \cdot \frac{1}{3}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)} \cdot \mathsf{fma}\left(v, v, -1\right)}} \]

   5. metadata-eval 100.0%

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 100.0%

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

   9. lift-sqrt.f64 N/A

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

   10. pow1/2 N/A

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

   11. lift-fma.f64 N/A

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

   12. *-commutative N/A

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

   13. lift-fma.f64 N/A

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

   14. pow1/2 N/A

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

   15. lift-sqrt.f64 100.0%

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

8. Applied rewrites 100.0%

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

Alternative 5: 98.9% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

3. Applied rewrites 100.0%

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

4. Taylor expanded in v around 0

   \[\leadsto \frac{\color{blue}{\frac{\frac{4}{3}}{\pi}}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}} \]
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. lower-PI.f64 98.9%

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

6. Applied rewrites 98.9%

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

7. Evaluated real constant 98.9%

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

   1. lift-fma.f64 N/A

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

   2. metadata-eval N/A

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

   3. associate-+r+ N/A

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

   4. add-flip N/A

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

   5. *-commutative N/A

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

   6. metadata-eval N/A

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

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

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. metadata-eval N/A

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

   11. distribute-neg-out N/A

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

   12. add-flip-rev N/A

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

   13. metadata-eval N/A

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

   14. sub-negate-rev N/A

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

   15. lift-*.f64 N/A

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

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

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

   17. metadata-eval N/A

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

   18. metadata-eval N/A

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

   19. lower--.f64 N/A

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

9. Applied rewrites 98.9%

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

Alternative 6: 98.9% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[v_] := N[(0.4244131815783876 / 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. 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

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

3. Applied rewrites 100.0%

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

4. Taylor expanded in v around 0

   \[\leadsto \frac{\color{blue}{\frac{\frac{4}{3}}{\pi}}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}} \]
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. lower-PI.f64 98.9%

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

6. Applied rewrites 98.9%

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

7. Evaluated real constant 98.9%

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

Alternative 7: 98.9% accurate, 29.1× 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 98.9%

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

4. Applied rewrites 98.9%

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

5. Evaluated real constant 98.9%

   \[\leadsto 0.30010543871903533 \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2025189 
(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)))))))