Falkner and Boettcher, Equation (22+)

Percentage Accurate: 98.5% → 100.0%

Time: 4.6s

Alternatives: 4

Speedup: 2.1×

Specification

Math

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

FPCore

(FPCore (v)
  :precision binary64
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* 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 / N[(N[(N[(3 * Pi), $MachinePrecision] * N[(1 - N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(2 - N[(6 * N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 98.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (v)
  :precision binary64
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* 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 / N[(N[(N[(3 * Pi), $MachinePrecision] * N[(1 - N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(2 - N[(6 * N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (v)
  :precision binary64
  (/ 4/3 (* (* (- 1 (* v v)) PI) (sqrt (- 2 (* 6 (* v v)))))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[v_] := N[(4/3 / N[(N[(N[(1 - N[(v * v), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision] * N[Sqrt[N[(2 - N[(6 * N[(v * v), $MachinePrecision]), $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. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. associate-*l* N/A

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

   7. associate-/r* N/A

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

   8. lower-/.f64 N/A

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

   9. metadata-eval N/A

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

   10. lower-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 100.0%

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

3. Applied rewrites 100.0%

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

Alternative 2: 99.0% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (v)
  :precision binary64
  (/ 4/3 (* PI (sqrt (- (- 1 (* (* v v) 6)) -1)))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[v_] := N[(4/3 / N[(Pi * N[Sqrt[N[(N[(1 - N[(N[(v * v), $MachinePrecision] * 6), $MachinePrecision]), $MachinePrecision] - -1), $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. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. associate-*l* N/A

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

   7. associate-/r* N/A

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

   8. lower-/.f64 N/A

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

   9. metadata-eval N/A

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

   10. lower-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 100.0%

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

3. Applied rewrites 100.0%

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

4. Taylor expanded in v around 0

   \[\leadsto \frac{\frac{4}{3}}{\color{blue}{\pi} \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]
5. Step-by-step derivation

   1. lower-PI.f64 99.0%

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

6. Applied rewrites 99.0%

   \[\leadsto \frac{\frac{4}{3}}{\color{blue}{\pi} \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]
7. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. metadata-eval N/A

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

   3. associate--l- N/A

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

   4. +-commutative N/A

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

   5. associate--l- N/A

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

   6. lift--.f64 N/A

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

   7. lift--.f64 99.0%

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 99.0%

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

8. Applied rewrites 99.0%

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

Alternative 3: 99.0% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (v)
  :precision binary64
  (/ 4/3 (* PI (sqrt (- 2 (* 6 (* 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(1.3333333333333333 / Float64(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[(4/3 / N[(Pi * N[Sqrt[N[(2 - N[(6 * N[(v * v), $MachinePrecision]), $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. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. associate-*l* N/A

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

   7. associate-/r* N/A

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

   8. lower-/.f64 N/A

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

   9. metadata-eval N/A

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

   10. lower-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 100.0%

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

3. Applied rewrites 100.0%

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

4. Taylor expanded in v around 0

   \[\leadsto \frac{\frac{4}{3}}{\color{blue}{\pi} \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]
5. Step-by-step derivation

   1. lower-PI.f64 99.0%

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

6. Applied rewrites 99.0%

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

Alternative 4: 98.9% accurate, 2.1× speedup

Math

\[\frac{\frac{4}{3}}{\pi \cdot \sqrt{2}} \]

FPCore

(FPCore (v)
  :precision binary64
  (/ 4/3 (* PI (sqrt 2))))

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[(4/3 / N[(Pi * N[Sqrt[2], $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. Taylor expanded in v around 0

   \[\leadsto \color{blue}{\frac{\frac{4}{3}}{\pi \cdot \sqrt{2}}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\frac{4}{3}}{\color{blue}{\mathsf{PI}\left(\right) \cdot \sqrt{2}}} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\frac{4}{3}}{\mathsf{PI}\left(\right) \cdot \color{blue}{\sqrt{2}}} \]

   3. lower-PI.f64 N/A

      \[\leadsto \frac{\frac{4}{3}}{\pi \cdot \sqrt{\color{blue}{2}}} \]

   4. lower-sqrt.f64 98.9%

      \[\leadsto \frac{\frac{4}{3}}{\pi \cdot \sqrt{2}} \]

4. Applied rewrites 98.9%

   \[\leadsto \color{blue}{\frac{\frac{4}{3}}{\pi \cdot \sqrt{2}}} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025271 -o generate:evaluate
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  :precision binary64
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))