Falkner and Boettcher, Equation (22+)

Percentage Accurate: 98.5% → 100.0%

Time: 8.4s

Alternatives: 4

Speedup: 2.1×

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, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)}\\ \frac{4}{\mathsf{fma}\left(3 \cdot t\_0, \pi, t\_0 \cdot \left(\pi \cdot \left(3 \cdot \left(v \cdot \left(-v\right)\right)\right)\right)\right)} \end{array} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (let* ((t_0 (sqrt (fma v (* v -6.0) 2.0))))
   (/ 4.0 (fma (* 3.0 t_0) PI (* t_0 (* PI (* 3.0 (* v (- v)))))))))

C

double code(double v) {
	double t_0 = sqrt(fma(v, (v * -6.0), 2.0));
	return 4.0 / fma((3.0 * t_0), ((double) M_PI), (t_0 * (((double) M_PI) * (3.0 * (v * -v)))));
}

Julia

function code(v)
	t_0 = sqrt(fma(v, Float64(v * -6.0), 2.0))
	return Float64(4.0 / fma(Float64(3.0 * t_0), pi, Float64(t_0 * Float64(pi * Float64(3.0 * Float64(v * Float64(-v)))))))
end

Wolfram

code[v_] := Block[{t$95$0 = N[Sqrt[N[(v * N[(v * -6.0), $MachinePrecision] + 2.0), $MachinePrecision]], $MachinePrecision]}, N[(4.0 / N[(N[(3.0 * t$95$0), $MachinePrecision] * Pi + N[(t$95$0 * N[(Pi * N[(3.0 * 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. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift--.f64 N/A

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

   5. sub-neg N/A

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

   6. distribute-lft-in N/A

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

   7. *-rgt-identity N/A

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

   8. distribute-rgt-in N/A

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

   9. *-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. associate-*r* N/A

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

4. Applied rewrites 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 99.1% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.5%

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

4. Applied rewrites 98.8%

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

5. Taylor expanded in v around 0

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

   1. lower-PI.f64 98.9

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

7. Applied rewrites 98.9%

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

Alternative 3: 99.0% accurate, 2.1× 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. Add Preprocessing

4. Applied rewrites 98.8%

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

5. Taylor expanded in v around 0

   \[\leadsto \frac{\frac{4}{3}}{\color{blue}{\mathsf{PI}\left(\right) \cdot \sqrt{2}}} \]
6. 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-PI.f64 N/A

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

   3. lower-sqrt.f64 98.8

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

7. Applied rewrites 98.8%

   \[\leadsto \frac{1.3333333333333333}{\color{blue}{\pi \cdot \sqrt{2}}} \]
8. 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 97.3

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

5. Applied rewrites 97.3%

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

Reproduce

herbie shell --seed 2024226 
(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)))))))