Falkner and Boettcher, Equation (22+)

Percentage Accurate: 98.5% → 100.0%

Time: 8.0s

Alternatives: 5

Speedup: 3.2×

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

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

FPCore

(FPCore (v)
 :precision binary64
 (/ (/ 4.0 (* (sqrt (- 2.0 (* 6.0 (* v v)))) 3.0)) (- PI (* (* v v) PI))))

C

double code(double v) {
	return (4.0 / (sqrt((2.0 - (6.0 * (v * v)))) * 3.0)) / (((double) M_PI) - ((v * v) * ((double) M_PI)));
}

Java

public static double code(double v) {
	return (4.0 / (Math.sqrt((2.0 - (6.0 * (v * v)))) * 3.0)) / (Math.PI - ((v * v) * Math.PI));
}

Python

def code(v):
	return (4.0 / (math.sqrt((2.0 - (6.0 * (v * v)))) * 3.0)) / (math.pi - ((v * v) * math.pi))

Julia

function code(v)
	return Float64(Float64(4.0 / Float64(sqrt(Float64(2.0 - Float64(6.0 * Float64(v * v)))) * 3.0)) / Float64(pi - Float64(Float64(v * v) * pi)))
end

MATLAB

function tmp = code(v)
	tmp = (4.0 / (sqrt((2.0 - (6.0 * (v * v)))) * 3.0)) / (pi - ((v * v) * pi));
end

Wolfram

code[v_] := N[(N[(4.0 / N[(N[Sqrt[N[(2.0 - N[(6.0 * N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision] / N[(Pi - N[(N[(v * v), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.5%

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

3. Applied rewrites 100.0%

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

Alternative 2: 99.0% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (v)
 :precision binary64
 (/ (/ 4.0 (* (sqrt (- 2.0 (* 6.0 (* v v)))) PI)) 3.0))

C

double code(double v) {
	return (4.0 / (sqrt((2.0 - (6.0 * (v * v)))) * ((double) M_PI))) / 3.0;
}

Java

public static double code(double v) {
	return (4.0 / (Math.sqrt((2.0 - (6.0 * (v * v)))) * Math.PI)) / 3.0;
}

Python

def code(v):
	return (4.0 / (math.sqrt((2.0 - (6.0 * (v * v)))) * math.pi)) / 3.0

Julia

function code(v)
	return Float64(Float64(4.0 / Float64(sqrt(Float64(2.0 - Float64(6.0 * Float64(v * v)))) * pi)) / 3.0)
end

MATLAB

function tmp = code(v)
	tmp = (4.0 / (sqrt((2.0 - (6.0 * (v * v)))) * pi)) / 3.0;
end

Wolfram

code[v_] := N[(N[(4.0 / N[(N[Sqrt[N[(2.0 - N[(6.0 * N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision] / 3.0), $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. Applied rewrites 100.0%

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

5. Applied rewrites 100.0%

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

6. Taylor expanded in v around 0

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

8. Applied rewrites 99.0%

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

Alternative 3: 98.9% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.5%

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

3. Applied rewrites 100.0%

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

4. Taylor expanded in v around 0

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

6. Applied rewrites 98.9%

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

Alternative 4: 98.9% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.5%

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

3. Applied rewrites 100.0%

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

4. Taylor expanded in v around 0

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

6. Applied rewrites 98.9%

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

8. Applied rewrites 98.9%

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

Alternative 5: 98.9% accurate, 3.2× 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 98.5%

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

2. Taylor expanded in v around 0

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

4. Applied rewrites 98.9%

   \[\leadsto \color{blue}{\frac{1.3333333333333333}{\pi \cdot \sqrt{2}}} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025153 
(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)))))))