Falkner and Boettcher, Equation (22+)

Percentage Accurate: 98.5% → 100.0%

Time: 5.0s

Alternatives: 5

Speedup: 1.0×

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, 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 98.5%

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

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

Alternative 2: 98.9% 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 98.5%

   \[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]
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. 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 99.4%

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

   1. clear-num 99.4%

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

   2. associate-/r/ 99.4%

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

6. Applied egg-rr 99.4%

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

7. Final simplification 99.4%

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

Alternative 3: 98.9% 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 98.5%

   \[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]
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. 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 99.4%

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

5. Final simplification 99.4%

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

Alternative 4: 97.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\sqrt{0.8888888888888888}}{\pi \cdot \left(1 - v \cdot v\right)} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (/ (sqrt 0.8888888888888888) (* PI (- 1.0 (* v v)))))

C

double code(double v) {
	return sqrt(0.8888888888888888) / (((double) M_PI) * (1.0 - (v * v)));
}

Java

public static double code(double v) {
	return Math.sqrt(0.8888888888888888) / (Math.PI * (1.0 - (v * v)));
}

Python

def code(v):
	return math.sqrt(0.8888888888888888) / (math.pi * (1.0 - (v * v)))

Julia

function code(v)
	return Float64(sqrt(0.8888888888888888) / Float64(pi * Float64(1.0 - Float64(v * v))))
end

MATLAB

function tmp = code(v)
	tmp = sqrt(0.8888888888888888) / (pi * (1.0 - (v * v)));
end

Wolfram

code[v_] := N[(N[Sqrt[0.8888888888888888], $MachinePrecision] / N[(Pi * N[(1.0 - N[(v * v), $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. 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. Step-by-step derivation

   1. div-inv 100.0%

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

   2. div-inv 100.0%

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

   3. associate-*l* 100.0%

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

   4. pow1/2 100.0%

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

   5. pow-flip 98.5%

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

   6. +-commutative 98.5%

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

   7. associate-*l* 98.5%

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

   8. fma-def 98.5%

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

   9. metadata-eval 98.5%

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

5. Applied egg-rr 98.5%

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

   1. associate-*r* 98.5%

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

   2. associate-*r/ 100.0%

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

   3. metadata-eval 100.0%

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

   4. associate-*l/ 98.5%

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

7. Simplified 98.5%

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

   1. add-sqr-sqrt 98.5%

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

   2. sqrt-unprod 98.5%

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

   3. swap-sqr 98.5%

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

   4. metadata-eval 98.5%

      \[\leadsto \frac{\sqrt{\color{blue}{1.7777777777777777} \cdot \left({\left(\mathsf{fma}\left(v, v \cdot -6, 2\right)\right)}^{-0.5} \cdot {\left(\mathsf{fma}\left(v, v \cdot -6, 2\right)\right)}^{-0.5}\right)}}{\pi \cdot \left(1 - v \cdot v\right)} \]

   5. fma-udef 98.5%

      \[\leadsto \frac{\sqrt{1.7777777777777777 \cdot \left({\color{blue}{\left(v \cdot \left(v \cdot -6\right) + 2\right)}}^{-0.5} \cdot {\left(\mathsf{fma}\left(v, v \cdot -6, 2\right)\right)}^{-0.5}\right)}}{\pi \cdot \left(1 - v \cdot v\right)} \]

   6. associate-*l* 98.5%

      \[\leadsto \frac{\sqrt{1.7777777777777777 \cdot \left({\left(\color{blue}{\left(v \cdot v\right) \cdot -6} + 2\right)}^{-0.5} \cdot {\left(\mathsf{fma}\left(v, v \cdot -6, 2\right)\right)}^{-0.5}\right)}}{\pi \cdot \left(1 - v \cdot v\right)} \]

   7. +-commutative 98.5%

      \[\leadsto \frac{\sqrt{1.7777777777777777 \cdot \left({\color{blue}{\left(2 + \left(v \cdot v\right) \cdot -6\right)}}^{-0.5} \cdot {\left(\mathsf{fma}\left(v, v \cdot -6, 2\right)\right)}^{-0.5}\right)}}{\pi \cdot \left(1 - v \cdot v\right)} \]

   8. fma-udef 98.5%

      \[\leadsto \frac{\sqrt{1.7777777777777777 \cdot \left({\left(2 + \left(v \cdot v\right) \cdot -6\right)}^{-0.5} \cdot {\color{blue}{\left(v \cdot \left(v \cdot -6\right) + 2\right)}}^{-0.5}\right)}}{\pi \cdot \left(1 - v \cdot v\right)} \]

   9. associate-*l* 98.5%

      \[\leadsto \frac{\sqrt{1.7777777777777777 \cdot \left({\left(2 + \left(v \cdot v\right) \cdot -6\right)}^{-0.5} \cdot {\left(\color{blue}{\left(v \cdot v\right) \cdot -6} + 2\right)}^{-0.5}\right)}}{\pi \cdot \left(1 - v \cdot v\right)} \]

   10. +-commutative 98.5%

       \[\leadsto \frac{\sqrt{1.7777777777777777 \cdot \left({\left(2 + \left(v \cdot v\right) \cdot -6\right)}^{-0.5} \cdot {\color{blue}{\left(2 + \left(v \cdot v\right) \cdot -6\right)}}^{-0.5}\right)}}{\pi \cdot \left(1 - v \cdot v\right)} \]

   11. pow-prod-up 98.5%

       \[\leadsto \frac{\sqrt{1.7777777777777777 \cdot \color{blue}{{\left(2 + \left(v \cdot v\right) \cdot -6\right)}^{\left(-0.5 + -0.5\right)}}}}{\pi \cdot \left(1 - v \cdot v\right)} \]

   12. +-commutative 98.5%

       \[\leadsto \frac{\sqrt{1.7777777777777777 \cdot {\color{blue}{\left(\left(v \cdot v\right) \cdot -6 + 2\right)}}^{\left(-0.5 + -0.5\right)}}}{\pi \cdot \left(1 - v \cdot v\right)} \]

   13. associate-*l* 98.5%

       \[\leadsto \frac{\sqrt{1.7777777777777777 \cdot {\left(\color{blue}{v \cdot \left(v \cdot -6\right)} + 2\right)}^{\left(-0.5 + -0.5\right)}}}{\pi \cdot \left(1 - v \cdot v\right)} \]

   14. fma-udef 98.5%

       \[\leadsto \frac{\sqrt{1.7777777777777777 \cdot {\color{blue}{\left(\mathsf{fma}\left(v, v \cdot -6, 2\right)\right)}}^{\left(-0.5 + -0.5\right)}}}{\pi \cdot \left(1 - v \cdot v\right)} \]

   15. metadata-eval 98.5%

       \[\leadsto \frac{\sqrt{1.7777777777777777 \cdot {\left(\mathsf{fma}\left(v, v \cdot -6, 2\right)\right)}^{\color{blue}{-1}}}}{\pi \cdot \left(1 - v \cdot v\right)} \]

9. Applied egg-rr 98.5%

   \[\leadsto \frac{\color{blue}{\sqrt{1.7777777777777777 \cdot {\left(\mathsf{fma}\left(v, v \cdot -6, 2\right)\right)}^{-1}}}}{\pi \cdot \left(1 - v \cdot v\right)} \]
10. Step-by-step derivation

    1. unpow-1 98.5%

       \[\leadsto \frac{\sqrt{1.7777777777777777 \cdot \color{blue}{\frac{1}{\mathsf{fma}\left(v, v \cdot -6, 2\right)}}}}{\pi \cdot \left(1 - v \cdot v\right)} \]

    2. associate-*r/ 98.5%

       \[\leadsto \frac{\sqrt{\color{blue}{\frac{1.7777777777777777 \cdot 1}{\mathsf{fma}\left(v, v \cdot -6, 2\right)}}}}{\pi \cdot \left(1 - v \cdot v\right)} \]

    3. metadata-eval 98.5%

       \[\leadsto \frac{\sqrt{\frac{\color{blue}{1.7777777777777777}}{\mathsf{fma}\left(v, v \cdot -6, 2\right)}}}{\pi \cdot \left(1 - v \cdot v\right)} \]

11. Simplified 98.5%

    \[\leadsto \frac{\color{blue}{\sqrt{\frac{1.7777777777777777}{\mathsf{fma}\left(v, v \cdot -6, 2\right)}}}}{\pi \cdot \left(1 - v \cdot v\right)} \]

12. Taylor expanded in v around 0 97.8%

    \[\leadsto \frac{\sqrt{\color{blue}{0.8888888888888888}}}{\pi \cdot \left(1 - v \cdot v\right)} \]

13. Final simplification 97.8%

    \[\leadsto \frac{\sqrt{0.8888888888888888}}{\pi \cdot \left(1 - v \cdot v\right)} \]

Alternative 5: 97.3% 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. 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. Step-by-step derivation

   1. div-inv 100.0%

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

   2. div-inv 100.0%

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

   3. associate-*l* 100.0%

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

   4. pow1/2 100.0%

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

   5. pow-flip 98.5%

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

   6. +-commutative 98.5%

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

   7. associate-*l* 98.5%

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

   8. fma-def 98.5%

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

   9. metadata-eval 98.5%

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

5. Applied egg-rr 98.5%

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

   1. associate-*r* 98.5%

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

   2. associate-*r/ 100.0%

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

   3. metadata-eval 100.0%

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

   4. associate-*l/ 98.5%

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

7. Simplified 98.5%

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

   1. add-sqr-sqrt 98.5%

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

   2. sqrt-unprod 98.5%

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

   3. swap-sqr 98.5%

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

   4. metadata-eval 98.5%

      \[\leadsto \frac{\sqrt{\color{blue}{1.7777777777777777} \cdot \left({\left(\mathsf{fma}\left(v, v \cdot -6, 2\right)\right)}^{-0.5} \cdot {\left(\mathsf{fma}\left(v, v \cdot -6, 2\right)\right)}^{-0.5}\right)}}{\pi \cdot \left(1 - v \cdot v\right)} \]

   5. fma-udef 98.5%

      \[\leadsto \frac{\sqrt{1.7777777777777777 \cdot \left({\color{blue}{\left(v \cdot \left(v \cdot -6\right) + 2\right)}}^{-0.5} \cdot {\left(\mathsf{fma}\left(v, v \cdot -6, 2\right)\right)}^{-0.5}\right)}}{\pi \cdot \left(1 - v \cdot v\right)} \]

   6. associate-*l* 98.5%

      \[\leadsto \frac{\sqrt{1.7777777777777777 \cdot \left({\left(\color{blue}{\left(v \cdot v\right) \cdot -6} + 2\right)}^{-0.5} \cdot {\left(\mathsf{fma}\left(v, v \cdot -6, 2\right)\right)}^{-0.5}\right)}}{\pi \cdot \left(1 - v \cdot v\right)} \]

   7. +-commutative 98.5%

      \[\leadsto \frac{\sqrt{1.7777777777777777 \cdot \left({\color{blue}{\left(2 + \left(v \cdot v\right) \cdot -6\right)}}^{-0.5} \cdot {\left(\mathsf{fma}\left(v, v \cdot -6, 2\right)\right)}^{-0.5}\right)}}{\pi \cdot \left(1 - v \cdot v\right)} \]

   8. fma-udef 98.5%

      \[\leadsto \frac{\sqrt{1.7777777777777777 \cdot \left({\left(2 + \left(v \cdot v\right) \cdot -6\right)}^{-0.5} \cdot {\color{blue}{\left(v \cdot \left(v \cdot -6\right) + 2\right)}}^{-0.5}\right)}}{\pi \cdot \left(1 - v \cdot v\right)} \]

   9. associate-*l* 98.5%

      \[\leadsto \frac{\sqrt{1.7777777777777777 \cdot \left({\left(2 + \left(v \cdot v\right) \cdot -6\right)}^{-0.5} \cdot {\left(\color{blue}{\left(v \cdot v\right) \cdot -6} + 2\right)}^{-0.5}\right)}}{\pi \cdot \left(1 - v \cdot v\right)} \]

   10. +-commutative 98.5%

       \[\leadsto \frac{\sqrt{1.7777777777777777 \cdot \left({\left(2 + \left(v \cdot v\right) \cdot -6\right)}^{-0.5} \cdot {\color{blue}{\left(2 + \left(v \cdot v\right) \cdot -6\right)}}^{-0.5}\right)}}{\pi \cdot \left(1 - v \cdot v\right)} \]

   11. pow-prod-up 98.5%

       \[\leadsto \frac{\sqrt{1.7777777777777777 \cdot \color{blue}{{\left(2 + \left(v \cdot v\right) \cdot -6\right)}^{\left(-0.5 + -0.5\right)}}}}{\pi \cdot \left(1 - v \cdot v\right)} \]

   12. +-commutative 98.5%

       \[\leadsto \frac{\sqrt{1.7777777777777777 \cdot {\color{blue}{\left(\left(v \cdot v\right) \cdot -6 + 2\right)}}^{\left(-0.5 + -0.5\right)}}}{\pi \cdot \left(1 - v \cdot v\right)} \]

   13. associate-*l* 98.5%

       \[\leadsto \frac{\sqrt{1.7777777777777777 \cdot {\left(\color{blue}{v \cdot \left(v \cdot -6\right)} + 2\right)}^{\left(-0.5 + -0.5\right)}}}{\pi \cdot \left(1 - v \cdot v\right)} \]

   14. fma-udef 98.5%

       \[\leadsto \frac{\sqrt{1.7777777777777777 \cdot {\color{blue}{\left(\mathsf{fma}\left(v, v \cdot -6, 2\right)\right)}}^{\left(-0.5 + -0.5\right)}}}{\pi \cdot \left(1 - v \cdot v\right)} \]

   15. metadata-eval 98.5%

       \[\leadsto \frac{\sqrt{1.7777777777777777 \cdot {\left(\mathsf{fma}\left(v, v \cdot -6, 2\right)\right)}^{\color{blue}{-1}}}}{\pi \cdot \left(1 - v \cdot v\right)} \]

9. Applied egg-rr 98.5%

   \[\leadsto \frac{\color{blue}{\sqrt{1.7777777777777777 \cdot {\left(\mathsf{fma}\left(v, v \cdot -6, 2\right)\right)}^{-1}}}}{\pi \cdot \left(1 - v \cdot v\right)} \]
10. Step-by-step derivation

    1. unpow-1 98.5%

       \[\leadsto \frac{\sqrt{1.7777777777777777 \cdot \color{blue}{\frac{1}{\mathsf{fma}\left(v, v \cdot -6, 2\right)}}}}{\pi \cdot \left(1 - v \cdot v\right)} \]

    2. associate-*r/ 98.5%

       \[\leadsto \frac{\sqrt{\color{blue}{\frac{1.7777777777777777 \cdot 1}{\mathsf{fma}\left(v, v \cdot -6, 2\right)}}}}{\pi \cdot \left(1 - v \cdot v\right)} \]

    3. metadata-eval 98.5%

       \[\leadsto \frac{\sqrt{\frac{\color{blue}{1.7777777777777777}}{\mathsf{fma}\left(v, v \cdot -6, 2\right)}}}{\pi \cdot \left(1 - v \cdot v\right)} \]

11. Simplified 98.5%

    \[\leadsto \frac{\color{blue}{\sqrt{\frac{1.7777777777777777}{\mathsf{fma}\left(v, v \cdot -6, 2\right)}}}}{\pi \cdot \left(1 - v \cdot v\right)} \]

12. Taylor expanded in v around 0 97.8%

    \[\leadsto \frac{\sqrt{\color{blue}{0.8888888888888888}}}{\pi \cdot \left(1 - v \cdot v\right)} \]

13. Taylor expanded in v around 0 97.8%

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

14. Final simplification 97.8%

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

Reproduce

herbie shell --seed 2023217 
(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)))))))