Falkner and Boettcher, Equation (22+)

Percentage Accurate: 98.5% → 100.0%

Time: 7.0s

Alternatives: 8

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.5%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   8. sqr-neg 100.0%

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

   9. *-commutative 100.0%

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

   10. distribute-rgt-neg-in 100.0%

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

   11. sqr-neg 100.0%

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

   12. 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. Step-by-step derivation

   1. clear-num 100.0%

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

   2. associate-/r/ 100.0%

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

   3. pow2 100.0%

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

5. Applied egg-rr 100.0%

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

6. Final simplification 100.0%

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

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 98.5%

   \[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]
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. sub-neg 100.0%

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

   8. sqr-neg 100.0%

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

   9. *-commutative 100.0%

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

   10. distribute-rgt-neg-in 100.0%

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

   11. sqr-neg 100.0%

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

   12. 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: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   8. sqr-neg 100.0%

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

   9. *-commutative 100.0%

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

   10. distribute-rgt-neg-in 100.0%

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

   11. sqr-neg 100.0%

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

   12. 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. Step-by-step derivation

   1. clear-num 100.0%

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

   2. associate-/r/ 100.0%

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

   3. pow2 100.0%

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

5. Applied egg-rr 100.0%

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

6. Taylor expanded in v around 0 98.5%

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

7. Final simplification 98.5%

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

Alternative 5: 99.0% 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 98.5%

   \[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]
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. sub-neg 100.0%

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

   8. sqr-neg 100.0%

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

   9. *-commutative 100.0%

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

   10. distribute-rgt-neg-in 100.0%

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

   11. sqr-neg 100.0%

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

   12. 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 98.5%

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

5. Final simplification 98.5%

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

Alternative 6: 98.9% accurate, 1.1× speedup

Math

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

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. sub-neg 100.0%

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

   8. sqr-neg 100.0%

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

   9. *-commutative 100.0%

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

   10. distribute-rgt-neg-in 100.0%

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

   11. sqr-neg 100.0%

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

   12. 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 98.5%

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

5. Taylor expanded in v around 0 98.4%

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

6. Final simplification 98.4%

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

Alternative 7: 97.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{\sqrt{0.8888888888888888}}{\pi} \end{array} \]

FPCore

(FPCore (v) :precision binary64 (/ (sqrt 0.8888888888888888) PI))

C

double code(double v) {
	return sqrt(0.8888888888888888) / ((double) M_PI);
}

Java

public static double code(double v) {
	return Math.sqrt(0.8888888888888888) / Math.PI;
}

Python

def code(v):
	return math.sqrt(0.8888888888888888) / math.pi

Julia

function code(v)
	return Float64(sqrt(0.8888888888888888) / pi)
end

MATLAB

function tmp = code(v)
	tmp = sqrt(0.8888888888888888) / pi;
end

Wolfram

code[v_] := N[(N[Sqrt[0.8888888888888888], $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 96.9%

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

   1. associate-*r/ 96.9%

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

4. Simplified 96.9%

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

6. Applied egg-rr 96.9%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\sqrt{0.8888888888888888}}{\pi}\right)} - 1} \]
7. Step-by-step derivation

   1. expm1-def 96.9%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{0.8888888888888888}}{\pi}\right)\right)} \]

   2. expm1-log1p 96.9%

      \[\leadsto \color{blue}{\frac{\sqrt{0.8888888888888888}}{\pi}} \]

8. Simplified 96.9%

   \[\leadsto \color{blue}{\frac{\sqrt{0.8888888888888888}}{\pi}} \]

9. Final simplification 96.9%

   \[\leadsto \frac{\sqrt{0.8888888888888888}}{\pi} \]

Alternative 8: 23.3% accurate, 218.0× speedup

Math

\[\begin{array}{l} \\ 0.3333333333333333 \end{array} \]

FPCore

(FPCore (v) :precision binary64 0.3333333333333333)

C

double code(double v) {
	return 0.3333333333333333;
}

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = 0.3333333333333333d0
end function

Java

public static double code(double v) {
	return 0.3333333333333333;
}

Python

def code(v):
	return 0.3333333333333333

Julia

function code(v)
	return 0.3333333333333333
end

MATLAB

function tmp = code(v)
	tmp = 0.3333333333333333;
end

Wolfram

code[v_] := 0.3333333333333333

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. sub-neg 100.0%

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

   8. sqr-neg 100.0%

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

   9. *-commutative 100.0%

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

   10. distribute-rgt-neg-in 100.0%

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

   11. sqr-neg 100.0%

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

   12. 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. Step-by-step derivation

   1. clear-num 100.0%

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

   2. associate-/r/ 100.0%

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

   3. pow2 100.0%

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

5. Applied egg-rr 100.0%

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

7. Applied egg-rr 0.0%

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

   1. *-inverses 17.6%

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

9. Simplified 17.6%

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

10. Taylor expanded in v around 0 17.6%

    \[\leadsto \frac{\frac{1}{1} \cdot 1.3333333333333333}{\color{blue}{\sqrt{2}}} \]

12. Applied egg-rr 23.3%

    \[\leadsto \color{blue}{0.3333333333333333} \]

13. Final simplification 23.3%

    \[\leadsto 0.3333333333333333 \]

Reproduce

herbie shell --seed 2023313 
(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)))))))