Falkner and Boettcher, Equation (22+)

Percentage Accurate: 98.5% → 100.0%

Time: 6.2s

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

\[\begin{array}{l} \\ \frac{1.3333333333333333}{\pi} \cdot \frac{{\left(\mathsf{fma}\left(v \cdot v, -6, 2\right)\right)}^{-0.5}}{1 - v \cdot v} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (*
  (/ 1.3333333333333333 PI)
  (/ (pow (fma (* v v) -6.0 2.0) -0.5) (- 1.0 (* v v)))))

C

double code(double v) {
	return (1.3333333333333333 / ((double) M_PI)) * (pow(fma((v * v), -6.0, 2.0), -0.5) / (1.0 - (v * v)));
}

Julia

function code(v)
	return Float64(Float64(1.3333333333333333 / pi) * Float64((fma(Float64(v * v), -6.0, 2.0) ^ -0.5) / Float64(1.0 - Float64(v * v))))
end

Wolfram

code[v_] := N[(N[(1.3333333333333333 / Pi), $MachinePrecision] * N[(N[Power[N[(N[(v * v), $MachinePrecision] * -6.0 + 2.0), $MachinePrecision], -0.5], $MachinePrecision] / N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.4%

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

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

3. Simplified 100.0%

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

   1. div-inv 100.0%

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

   2. div-inv 100.0%

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

   3. associate-*l* 100.0%

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

   4. pow1/2 100.0%

      \[\leadsto 1.3333333333333333 \cdot \left(\frac{1}{\pi \cdot \left(1 - v \cdot v\right)} \cdot \frac{1}{\color{blue}{{\left(2 - 6 \cdot \left(v \cdot v\right)\right)}^{0.5}}}\right) \]

   5. pow-flip 98.4%

      \[\leadsto 1.3333333333333333 \cdot \left(\frac{1}{\pi \cdot \left(1 - v \cdot v\right)} \cdot \color{blue}{{\left(2 - 6 \cdot \left(v \cdot v\right)\right)}^{\left(-0.5\right)}}\right) \]

   6. sub-neg 98.4%

      \[\leadsto 1.3333333333333333 \cdot \left(\frac{1}{\pi \cdot \left(1 - v \cdot v\right)} \cdot {\color{blue}{\left(2 + \left(-6 \cdot \left(v \cdot v\right)\right)\right)}}^{\left(-0.5\right)}\right) \]

   7. +-commutative 98.4%

      \[\leadsto 1.3333333333333333 \cdot \left(\frac{1}{\pi \cdot \left(1 - v \cdot v\right)} \cdot {\color{blue}{\left(\left(-6 \cdot \left(v \cdot v\right)\right) + 2\right)}}^{\left(-0.5\right)}\right) \]

   8. *-commutative 98.4%

      \[\leadsto 1.3333333333333333 \cdot \left(\frac{1}{\pi \cdot \left(1 - v \cdot v\right)} \cdot {\left(\left(-\color{blue}{\left(v \cdot v\right) \cdot 6}\right) + 2\right)}^{\left(-0.5\right)}\right) \]

   9. distribute-rgt-neg-in 98.4%

      \[\leadsto 1.3333333333333333 \cdot \left(\frac{1}{\pi \cdot \left(1 - v \cdot v\right)} \cdot {\left(\color{blue}{\left(v \cdot v\right) \cdot \left(-6\right)} + 2\right)}^{\left(-0.5\right)}\right) \]

   10. fma-def 98.4%

       \[\leadsto 1.3333333333333333 \cdot \left(\frac{1}{\pi \cdot \left(1 - v \cdot v\right)} \cdot {\color{blue}{\left(\mathsf{fma}\left(v \cdot v, -6, 2\right)\right)}}^{\left(-0.5\right)}\right) \]

   11. metadata-eval 98.4%

       \[\leadsto 1.3333333333333333 \cdot \left(\frac{1}{\pi \cdot \left(1 - v \cdot v\right)} \cdot {\left(\mathsf{fma}\left(v \cdot v, \color{blue}{-6}, 2\right)\right)}^{\left(-0.5\right)}\right) \]

   12. metadata-eval 98.4%

       \[\leadsto 1.3333333333333333 \cdot \left(\frac{1}{\pi \cdot \left(1 - v \cdot v\right)} \cdot {\left(\mathsf{fma}\left(v \cdot v, -6, 2\right)\right)}^{\color{blue}{-0.5}}\right) \]

5. Applied egg-rr 98.4%

   \[\leadsto \color{blue}{1.3333333333333333 \cdot \left(\frac{1}{\pi \cdot \left(1 - v \cdot v\right)} \cdot {\left(\mathsf{fma}\left(v \cdot v, -6, 2\right)\right)}^{-0.5}\right)} \]
6. Step-by-step derivation

   1. associate-*r* 98.4%

      \[\leadsto \color{blue}{\left(1.3333333333333333 \cdot \frac{1}{\pi \cdot \left(1 - v \cdot v\right)}\right) \cdot {\left(\mathsf{fma}\left(v \cdot v, -6, 2\right)\right)}^{-0.5}} \]

   2. associate-*r/ 100.0%

      \[\leadsto \color{blue}{\frac{1.3333333333333333 \cdot 1}{\pi \cdot \left(1 - v \cdot v\right)}} \cdot {\left(\mathsf{fma}\left(v \cdot v, -6, 2\right)\right)}^{-0.5} \]

   3. metadata-eval 100.0%

      \[\leadsto \frac{\color{blue}{1.3333333333333333}}{\pi \cdot \left(1 - v \cdot v\right)} \cdot {\left(\mathsf{fma}\left(v \cdot v, -6, 2\right)\right)}^{-0.5} \]

   4. associate-*l/ 98.4%

      \[\leadsto \color{blue}{\frac{1.3333333333333333 \cdot {\left(\mathsf{fma}\left(v \cdot v, -6, 2\right)\right)}^{-0.5}}{\pi \cdot \left(1 - v \cdot v\right)}} \]

   5. times-frac 100.0%

      \[\leadsto \color{blue}{\frac{1.3333333333333333}{\pi} \cdot \frac{{\left(\mathsf{fma}\left(v \cdot v, -6, 2\right)\right)}^{-0.5}}{1 - v \cdot v}} \]

7. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{1.3333333333333333}{\pi} \cdot \frac{{\left(\mathsf{fma}\left(v \cdot v, -6, 2\right)\right)}^{-0.5}}{1 - v \cdot v}} \]

8. Final simplification 100.0%

   \[\leadsto \frac{1.3333333333333333}{\pi} \cdot \frac{{\left(\mathsf{fma}\left(v \cdot v, -6, 2\right)\right)}^{-0.5}}{1 - v \cdot v} \]

Alternative 2: 100.0% accurate, 1.0× speedup

Math

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

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(Float64(v * 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[(N[(v * v), $MachinePrecision] * 6.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.4%

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

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 3: 99.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.4%

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

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

3. Simplified 100.0%

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

4. Taylor expanded in v around 0 99.5%

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

   1. clear-num 99.5%

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

   2. associate-/r/ 99.5%

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

6. Applied egg-rr 99.5%

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

7. Final simplification 99.5%

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

Alternative 4: 99.1% accurate, 1.0× speedup

Math

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

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(Float64(v * 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[(N[(v * v), $MachinePrecision] * 6.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.4%

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

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

3. Simplified 100.0%

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

4. Taylor expanded in v around 0 99.5%

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

5. Final simplification 99.5%

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

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 98.4%

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

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

3. Simplified 100.0%

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

4. Taylor expanded in v around 0 98.0%

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

5. Final simplification 98.0%

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

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 98.4%

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

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

3. Simplified 100.0%

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

   1. div-inv 100.0%

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

   2. div-inv 100.0%

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

   3. associate-*l* 100.0%

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

   4. pow1/2 100.0%

      \[\leadsto 1.3333333333333333 \cdot \left(\frac{1}{\pi \cdot \left(1 - v \cdot v\right)} \cdot \frac{1}{\color{blue}{{\left(2 - 6 \cdot \left(v \cdot v\right)\right)}^{0.5}}}\right) \]

   5. pow-flip 98.4%

      \[\leadsto 1.3333333333333333 \cdot \left(\frac{1}{\pi \cdot \left(1 - v \cdot v\right)} \cdot \color{blue}{{\left(2 - 6 \cdot \left(v \cdot v\right)\right)}^{\left(-0.5\right)}}\right) \]

   6. sub-neg 98.4%

      \[\leadsto 1.3333333333333333 \cdot \left(\frac{1}{\pi \cdot \left(1 - v \cdot v\right)} \cdot {\color{blue}{\left(2 + \left(-6 \cdot \left(v \cdot v\right)\right)\right)}}^{\left(-0.5\right)}\right) \]

   7. +-commutative 98.4%

      \[\leadsto 1.3333333333333333 \cdot \left(\frac{1}{\pi \cdot \left(1 - v \cdot v\right)} \cdot {\color{blue}{\left(\left(-6 \cdot \left(v \cdot v\right)\right) + 2\right)}}^{\left(-0.5\right)}\right) \]

   8. *-commutative 98.4%

      \[\leadsto 1.3333333333333333 \cdot \left(\frac{1}{\pi \cdot \left(1 - v \cdot v\right)} \cdot {\left(\left(-\color{blue}{\left(v \cdot v\right) \cdot 6}\right) + 2\right)}^{\left(-0.5\right)}\right) \]

   9. distribute-rgt-neg-in 98.4%

      \[\leadsto 1.3333333333333333 \cdot \left(\frac{1}{\pi \cdot \left(1 - v \cdot v\right)} \cdot {\left(\color{blue}{\left(v \cdot v\right) \cdot \left(-6\right)} + 2\right)}^{\left(-0.5\right)}\right) \]

   10. fma-def 98.4%

       \[\leadsto 1.3333333333333333 \cdot \left(\frac{1}{\pi \cdot \left(1 - v \cdot v\right)} \cdot {\color{blue}{\left(\mathsf{fma}\left(v \cdot v, -6, 2\right)\right)}}^{\left(-0.5\right)}\right) \]

   11. metadata-eval 98.4%

       \[\leadsto 1.3333333333333333 \cdot \left(\frac{1}{\pi \cdot \left(1 - v \cdot v\right)} \cdot {\left(\mathsf{fma}\left(v \cdot v, \color{blue}{-6}, 2\right)\right)}^{\left(-0.5\right)}\right) \]

   12. metadata-eval 98.4%

       \[\leadsto 1.3333333333333333 \cdot \left(\frac{1}{\pi \cdot \left(1 - v \cdot v\right)} \cdot {\left(\mathsf{fma}\left(v \cdot v, -6, 2\right)\right)}^{\color{blue}{-0.5}}\right) \]

5. Applied egg-rr 98.4%

   \[\leadsto \color{blue}{1.3333333333333333 \cdot \left(\frac{1}{\pi \cdot \left(1 - v \cdot v\right)} \cdot {\left(\mathsf{fma}\left(v \cdot v, -6, 2\right)\right)}^{-0.5}\right)} \]
6. Step-by-step derivation

   1. associate-*r* 98.4%

      \[\leadsto \color{blue}{\left(1.3333333333333333 \cdot \frac{1}{\pi \cdot \left(1 - v \cdot v\right)}\right) \cdot {\left(\mathsf{fma}\left(v \cdot v, -6, 2\right)\right)}^{-0.5}} \]

   2. associate-*r/ 100.0%

      \[\leadsto \color{blue}{\frac{1.3333333333333333 \cdot 1}{\pi \cdot \left(1 - v \cdot v\right)}} \cdot {\left(\mathsf{fma}\left(v \cdot v, -6, 2\right)\right)}^{-0.5} \]

   3. metadata-eval 100.0%

      \[\leadsto \frac{\color{blue}{1.3333333333333333}}{\pi \cdot \left(1 - v \cdot v\right)} \cdot {\left(\mathsf{fma}\left(v \cdot v, -6, 2\right)\right)}^{-0.5} \]

   4. associate-*l/ 98.4%

      \[\leadsto \color{blue}{\frac{1.3333333333333333 \cdot {\left(\mathsf{fma}\left(v \cdot v, -6, 2\right)\right)}^{-0.5}}{\pi \cdot \left(1 - v \cdot v\right)}} \]

   5. times-frac 100.0%

      \[\leadsto \color{blue}{\frac{1.3333333333333333}{\pi} \cdot \frac{{\left(\mathsf{fma}\left(v \cdot v, -6, 2\right)\right)}^{-0.5}}{1 - v \cdot v}} \]

7. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{1.3333333333333333}{\pi} \cdot \frac{{\left(\mathsf{fma}\left(v \cdot v, -6, 2\right)\right)}^{-0.5}}{1 - v \cdot v}} \]

8. Taylor expanded in v around 0 99.5%

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

9. Final simplification 99.5%

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

Reproduce

herbie shell --seed 2023285 
(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)))))))