Falkner and Boettcher, Equation (22+)

Percentage Accurate: 98.5% → 100.0%

Time: 4.3s

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, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   4. associate-/l/ 100.0%

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

   5. sub-neg 100.0%

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

   6. +-commutative 100.0%

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

   7. neg-sub0 100.0%

      \[\leadsto \frac{\frac{\frac{\frac{4}{3}}{\color{blue}{\left(0 - v \cdot v\right)} + 1}}{\pi}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]

   8. associate-+l- 100.0%

      \[\leadsto \frac{\frac{\frac{\frac{4}{3}}{\color{blue}{0 - \left(v \cdot v - 1\right)}}}{\pi}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]

   9. sub0-neg 100.0%

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

   10. neg-mul-1 100.0%

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

   11. associate-/r* 100.0%

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

   12. metadata-eval 100.0%

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

   13. metadata-eval 100.0%

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

   14. fma-neg 100.0%

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

   15. metadata-eval 100.0%

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

   16. sub-neg 100.0%

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

   17. +-commutative 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[v_] := N[(1.3333333333333333 / N[(N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision] * N[(Pi * N[Sqrt[N[(v * N[(v * -6.0), $MachinePrecision] + 2.0), $MachinePrecision]], $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. expm1-log1p-u 98.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\right)\right)} \]

   2. expm1-udef 98.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\right)} - 1} \]

3. Applied egg-rr 98.4%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{4}{\left(3 \cdot \pi\right) \cdot \left(\left(1 - v \cdot v\right) \cdot \sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)}\right)}\right)} - 1} \]
4. Step-by-step derivation

   1. expm1-def 98.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{4}{\left(3 \cdot \pi\right) \cdot \left(\left(1 - v \cdot v\right) \cdot \sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)}\right)}\right)\right)} \]

   2. expm1-log1p 98.4%

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

   3. associate-/r* 100.0%

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

   4. associate-/r* 100.0%

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

   5. associate-/r* 100.0%

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

   6. metadata-eval 100.0%

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

   7. associate-/r* 100.0%

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

   8. associate-/r* 100.0%

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

   9. *-commutative 100.0%

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

   10. associate-*l* 100.0%

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

5. Simplified 100.0%

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

6. Final simplification 100.0%

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

Alternative 3: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(v)
	tmp = (1.3333333333333333 / (pi * (1.0 - (v * v)))) / sqrt((2.0 + (-6.0 * (v * v))));
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[(-6.0 * N[(v * v), $MachinePrecision]), $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. associate-/r* 100.0%

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

   4. metadata-eval 100.0%

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

   5. sub-neg 100.0%

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

   6. *-commutative 100.0%

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

   7. distribute-rgt-neg-in 100.0%

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

   8. metadata-eval 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 4: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   4. metadata-eval 100.0%

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

   5. sub-neg 100.0%

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

   6. *-commutative 100.0%

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

   7. distribute-rgt-neg-in 100.0%

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

   8. metadata-eval 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in v around 0 99.2%

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

   1. clear-num 99.2%

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

   2. inv-pow 99.2%

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

6. Applied egg-rr 99.2%

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

   1. unpow-1 99.2%

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

8. Simplified 99.2%

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

9. Final simplification 99.2%

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

Alternative 5: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   4. metadata-eval 100.0%

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

   5. sub-neg 100.0%

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

   6. *-commutative 100.0%

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

   7. distribute-rgt-neg-in 100.0%

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

   8. metadata-eval 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in v around 0 99.2%

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

5. Final simplification 99.2%

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

Alternative 6: 98.9% accurate, 1.1× speedup

Math

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

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(1.3333333333333333 / Float64(pi * sqrt(2.0)))
end

MATLAB

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

Wolfram

code[v_] := N[(1.3333333333333333 / N[(Pi * N[Sqrt[2.0], $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. expm1-log1p-u 98.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\right)\right)} \]

   2. expm1-udef 98.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\right)} - 1} \]

3. Applied egg-rr 98.4%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{4}{\left(3 \cdot \pi\right) \cdot \left(\left(1 - v \cdot v\right) \cdot \sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)}\right)}\right)} - 1} \]
4. Step-by-step derivation

   1. expm1-def 98.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{4}{\left(3 \cdot \pi\right) \cdot \left(\left(1 - v \cdot v\right) \cdot \sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)}\right)}\right)\right)} \]

   2. expm1-log1p 98.4%

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

   3. associate-/r* 100.0%

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

   4. associate-/r* 100.0%

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

   5. associate-/r* 100.0%

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

   6. metadata-eval 100.0%

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

   7. associate-/r* 100.0%

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

   8. associate-/r* 100.0%

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

   9. *-commutative 100.0%

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

   10. associate-*l* 100.0%

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

5. Simplified 100.0%

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

6. Taylor expanded in v around 0 99.1%

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

7. Final simplification 99.1%

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

Reproduce

herbie shell --seed 2023257 
(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)))))))