Falkner and Boettcher, Equation (22+)

Percentage Accurate: 98.5% → 98.5%

Time: 7.8s

Alternatives: 4

Speedup: 1.8×

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: 98.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (v)
 :precision binary64
 (/ 4.0 (* (* (* 3.0 PI) (- 1.0 (* v v))) (sqrt (- 2.0 (* (* v v) 6.0))))))

C

double code(double v) {
	return 4.0 / (((3.0 * ((double) M_PI)) * (1.0 - (v * v))) * sqrt((2.0 - ((v * v) * 6.0))));
}

Java

public static double code(double v) {
	return 4.0 / (((3.0 * Math.PI) * (1.0 - (v * v))) * Math.sqrt((2.0 - ((v * v) * 6.0))));
}

Python

def code(v):
	return 4.0 / (((3.0 * math.pi) * (1.0 - (v * v))) * math.sqrt((2.0 - ((v * v) * 6.0))))

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(Float64(v * v) * 6.0)))))
end

MATLAB

function tmp = code(v)
	tmp = 4.0 / (((3.0 * pi) * (1.0 - (v * v))) * sqrt((2.0 - ((v * v) * 6.0))));
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[(N[(v * v), $MachinePrecision] * 6.0), $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. Add Preprocessing

3. Final simplification 98.5%

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

Alternative 2: 98.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

3. Taylor expanded in v around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. distribute-rgt-out N/A

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

   4. lower-*.f64 N/A

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

   5. lower-PI.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 98.4

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

5. Simplified 98.4%

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

6. Taylor expanded in v around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 98.4

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

8. Simplified 98.4%

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

Alternative 3: 99.0% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[v_] := N[(4.0 / N[(Pi * N[(3.0 * N[Sqrt[2.0], $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. Add Preprocessing

3. Taylor expanded in v around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-PI.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-sqrt.f64 98.4

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

5. Simplified 98.4%

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

Alternative 4: 97.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \frac{1.3333333333333333 \cdot \sqrt{0.5}}{\pi} \end{array} \]

FPCore

(FPCore (v) :precision binary64 (/ (* 1.3333333333333333 (sqrt 0.5)) PI))

C

double code(double v) {
	return (1.3333333333333333 * sqrt(0.5)) / ((double) M_PI);
}

Java

public static double code(double v) {
	return (1.3333333333333333 * Math.sqrt(0.5)) / Math.PI;
}

Python

def code(v):
	return (1.3333333333333333 * math.sqrt(0.5)) / math.pi

Julia

function code(v)
	return Float64(Float64(1.3333333333333333 * sqrt(0.5)) / pi)
end

MATLAB

function tmp = code(v)
	tmp = (1.3333333333333333 * sqrt(0.5)) / pi;
end

Wolfram

code[v_] := N[(N[(1.3333333333333333 * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in v around 0

   \[\leadsto \color{blue}{\frac{4}{3} \cdot \frac{\sqrt{\frac{1}{2}}}{\mathsf{PI}\left(\right)}} \]
4. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{\frac{4}{3} \cdot \sqrt{\frac{1}{2}}}{\mathsf{PI}\left(\right)}} \]

   2. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{4}{3} \cdot \sqrt{\frac{1}{2}}}{\mathsf{PI}\left(\right)}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{4}{3} \cdot \sqrt{\frac{1}{2}}}}{\mathsf{PI}\left(\right)} \]

   4. lower-sqrt.f64 N/A

      \[\leadsto \frac{\frac{4}{3} \cdot \color{blue}{\sqrt{\frac{1}{2}}}}{\mathsf{PI}\left(\right)} \]

   5. lower-PI.f64 96.9

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

5. Simplified 96.9%

   \[\leadsto \color{blue}{\frac{1.3333333333333333 \cdot \sqrt{0.5}}{\pi}} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024215 
(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)))))))