Falkner and Boettcher, Equation (22+)

Percentage Accurate: 98.5% → 100.0%

Time: 5.3s

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, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\frac{1.3333333333333333}{\pi}}{1 - {v}^{6}} \cdot \left(\left(1 + {v}^{4}\right) + {v}^{2}\right)}{\sqrt{2 - v \cdot \left(v \cdot 6\right)}} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (/
  (*
   (/ (/ 1.3333333333333333 PI) (- 1.0 (pow v 6.0)))
   (+ (+ 1.0 (pow v 4.0)) (pow v 2.0)))
  (sqrt (- 2.0 (* v (* v 6.0))))))

C

double code(double v) {
	return (((1.3333333333333333 / ((double) M_PI)) / (1.0 - pow(v, 6.0))) * ((1.0 + pow(v, 4.0)) + pow(v, 2.0))) / sqrt((2.0 - (v * (v * 6.0))));
}

Java

public static double code(double v) {
	return (((1.3333333333333333 / Math.PI) / (1.0 - Math.pow(v, 6.0))) * ((1.0 + Math.pow(v, 4.0)) + Math.pow(v, 2.0))) / Math.sqrt((2.0 - (v * (v * 6.0))));
}

Python

def code(v):
	return (((1.3333333333333333 / math.pi) / (1.0 - math.pow(v, 6.0))) * ((1.0 + math.pow(v, 4.0)) + math.pow(v, 2.0))) / math.sqrt((2.0 - (v * (v * 6.0))))

Julia

function code(v)
	return Float64(Float64(Float64(Float64(1.3333333333333333 / pi) / Float64(1.0 - (v ^ 6.0))) * Float64(Float64(1.0 + (v ^ 4.0)) + (v ^ 2.0))) / sqrt(Float64(2.0 - Float64(v * Float64(v * 6.0)))))
end

MATLAB

function tmp = code(v)
	tmp = (((1.3333333333333333 / pi) / (1.0 - (v ^ 6.0))) * ((1.0 + (v ^ 4.0)) + (v ^ 2.0))) / sqrt((2.0 - (v * (v * 6.0))));
end

Wolfram

code[v_] := N[(N[(N[(N[(1.3333333333333333 / Pi), $MachinePrecision] / N[(1.0 - N[Power[v, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[v, 4.0], $MachinePrecision]), $MachinePrecision] + N[Power[v, 2.0], $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. Step-by-step derivation

   1. associate-/r* 100.0%

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

   2. flip3-- 100.0%

      \[\leadsto \frac{\frac{\frac{1.3333333333333333}{\pi}}{\color{blue}{\frac{{1}^{3} - {\left(v \cdot v\right)}^{3}}{1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)}}}}{\sqrt{2 - v \cdot \left(v \cdot 6\right)}} \]

   3. associate-/r/ 100.0%

      \[\leadsto \frac{\color{blue}{\frac{\frac{1.3333333333333333}{\pi}}{{1}^{3} - {\left(v \cdot v\right)}^{3}} \cdot \left(1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)\right)}}{\sqrt{2 - v \cdot \left(v \cdot 6\right)}} \]

   4. metadata-eval 100.0%

      \[\leadsto \frac{\frac{\frac{1.3333333333333333}{\pi}}{\color{blue}{1} - {\left(v \cdot v\right)}^{3}} \cdot \left(1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)\right)}{\sqrt{2 - v \cdot \left(v \cdot 6\right)}} \]

   5. pow2 100.0%

      \[\leadsto \frac{\frac{\frac{1.3333333333333333}{\pi}}{1 - {\color{blue}{\left({v}^{2}\right)}}^{3}} \cdot \left(1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)\right)}{\sqrt{2 - v \cdot \left(v \cdot 6\right)}} \]

   6. pow-pow 100.0%

      \[\leadsto \frac{\frac{\frac{1.3333333333333333}{\pi}}{1 - \color{blue}{{v}^{\left(2 \cdot 3\right)}}} \cdot \left(1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)\right)}{\sqrt{2 - v \cdot \left(v \cdot 6\right)}} \]

   7. metadata-eval 100.0%

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

   8. metadata-eval 100.0%

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

   9. *-un-lft-identity 100.0%

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

   10. associate-+r+ 100.0%

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

   11. pow2 100.0%

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

   12. pow2 100.0%

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

   13. pow-prod-up 100.0%

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

   14. metadata-eval 100.0%

       \[\leadsto \frac{\frac{\frac{1.3333333333333333}{\pi}}{1 - {v}^{6}} \cdot \left(\left(1 + {v}^{\color{blue}{4}}\right) + v \cdot v\right)}{\sqrt{2 - v \cdot \left(v \cdot 6\right)}} \]

   15. pow2 100.0%

       \[\leadsto \frac{\frac{\frac{1.3333333333333333}{\pi}}{1 - {v}^{6}} \cdot \left(\left(1 + {v}^{4}\right) + \color{blue}{{v}^{2}}\right)}{\sqrt{2 - v \cdot \left(v \cdot 6\right)}} \]

5. Applied egg-rr 100.0%

   \[\leadsto \frac{\color{blue}{\frac{\frac{1.3333333333333333}{\pi}}{1 - {v}^{6}} \cdot \left(\left(1 + {v}^{4}\right) + {v}^{2}\right)}}{\sqrt{2 - v \cdot \left(v \cdot 6\right)}} \]

6. Final simplification 100.0%

   \[\leadsto \frac{\frac{\frac{1.3333333333333333}{\pi}}{1 - {v}^{6}} \cdot \left(\left(1 + {v}^{4}\right) + {v}^{2}\right)}{\sqrt{2 - v \cdot \left(v \cdot 6\right)}} \]

Alternative 2: 100.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   3. sqr-neg 100.0%

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

   4. associate-/r* 100.0%

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

   5. associate-/l/ 100.0%

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

   6. associate-/r* 100.0%

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

3. Simplified 100.0%

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

   1. clear-num 100.0%

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

   2. associate-/r/ 100.0%

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

5. Applied egg-rr 100.0%

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

6. Final simplification 100.0%

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

Alternative 3: 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 4: 99.1% 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 99.1%

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

5. Final simplification 99.1%

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

Alternative 5: 97.5% accurate, 1.1× speedup

Math

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

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(Float64(1.3333333333333333 * sqrt(0.5)) / pi)
end

MATLAB

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

Wolfram

code[v_] := N[(N[(1.3333333333333333 * N[Sqrt[0.5], $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. Taylor expanded in v around 0 97.6%

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

   1. associate-*r/ 97.6%

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

4. Simplified 97.6%

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

5. Final simplification 97.6%

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

Alternative 6: 99.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[v_] := N[(N[(1.3333333333333333 / Pi), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.5%

   \[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]
2. 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 99.1%

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

5. Taylor expanded in v around 0 99.1%

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

6. Final simplification 99.1%

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

Reproduce

herbie shell --seed 2023326 
(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)))))))