Falkner and Boettcher, Equation (22+)

Percentage Accurate: 98.5% → 100.0%

Time: 5.6s

Alternatives: 6

Speedup: 1.1×

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, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\frac{1}{\pi \cdot 0.75}}{1 - v \cdot v}}{\sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (/ (/ (/ 1.0 (* PI 0.75)) (- 1.0 (* v v))) (sqrt (fma (* v v) -6.0 2.0))))

C

double code(double v) {
	return ((1.0 / (((double) M_PI) * 0.75)) / (1.0 - (v * v))) / sqrt(fma((v * v), -6.0, 2.0));
}

Julia

function code(v)
	return Float64(Float64(Float64(1.0 / Float64(pi * 0.75)) / Float64(1.0 - Float64(v * v))) / sqrt(fma(Float64(v * v), -6.0, 2.0)))
end

Wolfram

code[v_] := N[(N[(N[(1.0 / N[(Pi * 0.75), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(v * v), $MachinePrecision]), $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. associate-/r* 100.0%

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

   2. *-commutative 100.0%

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

   3. sqr-neg 100.0%

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

   4. associate-/l/ 100.0%

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

   5. associate-/r* 100.0%

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

   6. metadata-eval 100.0%

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

   7. sqr-neg 100.0%

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

   8. sub-neg 100.0%

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

   9. sqr-neg 100.0%

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

   10. +-commutative 100.0%

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

   11. *-commutative 100.0%

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

   12. distribute-rgt-neg-in 100.0%

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

   13. fma-def 100.0%

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

3. Simplified 100.0%

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

   1. clear-num 100.0%

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

   2. inv-pow 100.0%

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

   3. div-inv 100.0%

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

   4. metadata-eval 100.0%

      \[\leadsto \frac{\frac{{\left(\pi \cdot \color{blue}{0.75}\right)}^{-1}}{1 - v \cdot v}}{\sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}} \]

5. Applied egg-rr 100.0%

   \[\leadsto \frac{\frac{\color{blue}{{\left(\pi \cdot 0.75\right)}^{-1}}}{1 - v \cdot v}}{\sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}} \]
6. Step-by-step derivation

   1. unpow-1 100.0%

      \[\leadsto \frac{\frac{\color{blue}{\frac{1}{\pi \cdot 0.75}}}{1 - v \cdot v}}{\sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}} \]

7. Simplified 100.0%

   \[\leadsto \frac{\frac{\color{blue}{\frac{1}{\pi \cdot 0.75}}}{1 - v \cdot v}}{\sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}} \]

8. Final simplification 100.0%

   \[\leadsto \frac{\frac{\frac{1}{\pi \cdot 0.75}}{1 - v \cdot v}}{\sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}} \]

Alternative 2: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double v) {
	return (1.3333333333333333 / (Math.PI * (1.0 - (v * v)))) / Math.sqrt((2.0 - (v * (v * 6.0))));
}

Python

def code(v):
	return (1.3333333333333333 / (math.pi * (1.0 - (v * v)))) / math.sqrt((2.0 - (v * (v * 6.0))))

Julia

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

MATLAB

function tmp = code(v)
	tmp = (1.3333333333333333 / (pi * (1.0 - (v * v)))) / sqrt((2.0 - (v * (v * 6.0))));
end

Wolfram

code[v_] := N[(N[(1.3333333333333333 / N[(Pi * N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(2.0 - N[(v * N[(v * 6.0), $MachinePrecision]), $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. associate-/r* 100.0%

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

   2. associate-*l* 100.0%

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

   3. sqr-neg 100.0%

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

   4. associate-/r* 100.0%

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

   5. metadata-eval 100.0%

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

   6. sqr-neg 100.0%

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

   7. sqr-neg 100.0%

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

   8. *-commutative 100.0%

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

   9. sqr-neg 100.0%

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

   10. associate-*l* 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 3: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{\pi \cdot 0.75}}{\sqrt{2 - v \cdot \left(v \cdot 6\right)}} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (/ (/ 1.0 (* PI 0.75)) (sqrt (- 2.0 (* v (* v 6.0))))))

C

double code(double v) {
	return (1.0 / (((double) M_PI) * 0.75)) / sqrt((2.0 - (v * (v * 6.0))));
}

Java

public static double code(double v) {
	return (1.0 / (Math.PI * 0.75)) / Math.sqrt((2.0 - (v * (v * 6.0))));
}

Python

def code(v):
	return (1.0 / (math.pi * 0.75)) / math.sqrt((2.0 - (v * (v * 6.0))))

Julia

function code(v)
	return Float64(Float64(1.0 / Float64(pi * 0.75)) / sqrt(Float64(2.0 - Float64(v * Float64(v * 6.0)))))
end

MATLAB

function tmp = code(v)
	tmp = (1.0 / (pi * 0.75)) / sqrt((2.0 - (v * (v * 6.0))));
end

Wolfram

code[v_] := N[(N[(1.0 / N[(Pi * 0.75), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(2.0 - N[(v * N[(v * 6.0), $MachinePrecision]), $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. associate-/r* 100.0%

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

   2. associate-*l* 100.0%

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

   3. sqr-neg 100.0%

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

   4. associate-/r* 100.0%

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

   5. metadata-eval 100.0%

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

   6. sqr-neg 100.0%

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

   7. sqr-neg 100.0%

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

   8. *-commutative 100.0%

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

   9. sqr-neg 100.0%

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

   10. associate-*l* 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in v around 0 98.8%

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

   1. clear-num 100.0%

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

   2. inv-pow 100.0%

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

   3. div-inv 100.0%

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

   4. metadata-eval 100.0%

      \[\leadsto \frac{\frac{{\left(\pi \cdot \color{blue}{0.75}\right)}^{-1}}{1 - v \cdot v}}{\sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}} \]

6. Applied egg-rr 98.8%

   \[\leadsto \frac{\color{blue}{{\left(\pi \cdot 0.75\right)}^{-1}}}{\sqrt{2 - v \cdot \left(v \cdot 6\right)}} \]
7. Step-by-step derivation

   1. unpow-1 100.0%

      \[\leadsto \frac{\frac{\color{blue}{\frac{1}{\pi \cdot 0.75}}}{1 - v \cdot v}}{\sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}} \]

8. Simplified 98.8%

   \[\leadsto \frac{\color{blue}{\frac{1}{\pi \cdot 0.75}}}{\sqrt{2 - v \cdot \left(v \cdot 6\right)}} \]

9. Final simplification 98.8%

   \[\leadsto \frac{\frac{1}{\pi \cdot 0.75}}{\sqrt{2 - v \cdot \left(v \cdot 6\right)}} \]

Alternative 4: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double v) {
	return (1.3333333333333333 / Math.PI) / Math.sqrt((2.0 - (v * (v * 6.0))));
}

Python

def code(v):
	return (1.3333333333333333 / math.pi) / math.sqrt((2.0 - (v * (v * 6.0))))

Julia

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

MATLAB

function tmp = code(v)
	tmp = (1.3333333333333333 / pi) / sqrt((2.0 - (v * (v * 6.0))));
end

Wolfram

code[v_] := N[(N[(1.3333333333333333 / Pi), $MachinePrecision] / N[Sqrt[N[(2.0 - N[(v * N[(v * 6.0), $MachinePrecision]), $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. associate-/r* 100.0%

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

   2. associate-*l* 100.0%

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

   3. sqr-neg 100.0%

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

   4. associate-/r* 100.0%

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

   5. metadata-eval 100.0%

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

   6. sqr-neg 100.0%

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

   7. sqr-neg 100.0%

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

   8. *-commutative 100.0%

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

   9. sqr-neg 100.0%

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

   10. associate-*l* 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in v around 0 98.8%

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

5. Final simplification 98.8%

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

Alternative 5: 97.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[v_] := N[(1.3333333333333333 * N[(N[Sqrt[0.5], $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. Taylor expanded in v around 0 97.3%

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

3. Final simplification 97.3%

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

Alternative 6: 99.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[v_] := N[(N[(1.3333333333333333 / Pi), $MachinePrecision] * N[Sqrt[0.5], $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 97.3%

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

   1. *-commutative 97.3%

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

4. Simplified 97.3%

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

5. Taylor expanded in v around 0 97.3%

   \[\leadsto \color{blue}{1.3333333333333333 \cdot \frac{\sqrt{0.5}}{\pi}} \]
6. Step-by-step derivation

   1. *-commutative 97.3%

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

   2. *-rgt-identity 97.3%

      \[\leadsto \frac{\color{blue}{\sqrt{0.5} \cdot 1}}{\pi} \cdot 1.3333333333333333 \]

   3. associate-*r/ 97.3%

      \[\leadsto \color{blue}{\left(\sqrt{0.5} \cdot \frac{1}{\pi}\right)} \cdot 1.3333333333333333 \]

   4. associate-*l* 97.3%

      \[\leadsto \color{blue}{\sqrt{0.5} \cdot \left(\frac{1}{\pi} \cdot 1.3333333333333333\right)} \]

   5. associate-*l/ 98.8%

      \[\leadsto \sqrt{0.5} \cdot \color{blue}{\frac{1 \cdot 1.3333333333333333}{\pi}} \]

   6. metadata-eval 98.8%

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

7. Simplified 98.8%

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

8. Final simplification 98.8%

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

Reproduce

herbie shell --seed 2023331 
(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)))))))