Falkner and Boettcher, Equation (22+)

Percentage Accurate: 98.5% → 98.9%

Time: 7.7s

Alternatives: 3

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: 98.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{\pi \cdot \sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)}}}{0.75} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (/ (/ 1.0 (* PI (sqrt (fma v (* v -6.0) 2.0)))) 0.75))

C

double code(double v) {
	return (1.0 / (((double) M_PI) * sqrt(fma(v, (v * -6.0), 2.0)))) / 0.75;
}

Julia

function code(v)
	return Float64(Float64(1.0 / Float64(pi * sqrt(fma(v, Float64(v * -6.0), 2.0)))) / 0.75)
end

Wolfram

code[v_] := N[(N[(1.0 / N[(Pi * N[Sqrt[N[(v * N[(v * -6.0), $MachinePrecision] + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 0.75), $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 egg-rr 99.4%

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

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. PI-lowering-PI.f64 99.4

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

7. Simplified 99.4%

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

   1. clear-num N/A

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

   2. div-inv N/A

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

   3. associate-/r* N/A

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

   4. /-lowering-/.f64 N/A

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

   5. /-lowering-/.f64 N/A

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

   6. *-commutative N/A

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

   7. *-lowering-*.f64 N/A

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

   8. PI-lowering-PI.f64 N/A

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

   9. sqrt-lowering-sqrt.f64 N/A

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

   10. accelerator-lowering-fma.f64 N/A

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

   11. *-lowering-*.f64 N/A

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

   12. metadata-eval 99.4

       \[\leadsto \frac{\frac{1}{\pi \cdot \sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)}}}{\color{blue}{0.75}} \]

9. Applied egg-rr 99.4%

   \[\leadsto \color{blue}{\frac{\frac{1}{\pi \cdot \sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)}}}{0.75}} \]
10. Add Preprocessing

Alternative 2: 98.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

4. Applied egg-rr 99.4%

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

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. PI-lowering-PI.f64 99.4

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

7. Simplified 99.4%

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

8. Final simplification 99.4%

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

Alternative 3: 98.9% 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 egg-rr 99.4%

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

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. *-lowering-*.f64 N/A

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

   2. PI-lowering-PI.f64 N/A

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

   3. sqrt-lowering-sqrt.f64 99.4

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

7. Simplified 99.4%

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

Reproduce

herbie shell --seed 2024204 
(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)))))))