Falkner and Boettcher, Equation (22+)

Average Error: 1.54% → 0.02%

Time: 5.7s

Precision: binary64

Cost: 20096

Math

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

↓

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

FPCore

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

↓

(FPCore (v)
 :precision binary64
 (/
  (/ 1.3333333333333333 PI)
  (* (sqrt (fma v (* v -6.0) 2.0)) (- 1.0 (* 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)))));
}

↓

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

↓

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

↓

code[v_] := N[(N[(1.3333333333333333 / Pi), $MachinePrecision] / N[(N[Sqrt[N[(v * N[(v * -6.0), $MachinePrecision] + 2.0), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 1.54

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

2. Simplified 0.02

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

   [Start] 1.54	\[ \frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]
   associate-/r* [=>] 0.02	\[ \color{blue}{\frac{\frac{4}{\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}} \]
   associate-*l* [=>] 0.02	\[ \frac{\frac{4}{\color{blue}{3 \cdot \left(\pi \cdot \left(1 - v \cdot v\right)\right)}}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]
   associate-/r* [=>] 0.02	\[ \frac{\color{blue}{\frac{\frac{4}{3}}{\pi \cdot \left(1 - v \cdot v\right)}}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]
   metadata-eval [=>] 0.02	\[ \frac{\frac{\color{blue}{1.3333333333333333}}{\pi \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]
   cancel-sign-sub-inv [=>] 0.02	\[ \frac{\frac{1.3333333333333333}{\pi \cdot \left(1 - v \cdot v\right)}}{\sqrt{\color{blue}{2 + \left(-6\right) \cdot \left(v \cdot v\right)}}} \]
   *-commutative [=>] 0.02	\[ \frac{\frac{1.3333333333333333}{\pi \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 + \color{blue}{\left(v \cdot v\right) \cdot \left(-6\right)}}} \]
   metadata-eval [=>] 0.02	\[ \frac{\frac{1.3333333333333333}{\pi \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 + \left(v \cdot v\right) \cdot \color{blue}{-6}}} \]

3. Applied egg-rr 1.56

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{\frac{1.3333333333333333}{\pi}}{1 - v \cdot v}}{\sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)}}\right)} - 1} \]

4. Simplified 0.02

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

   [Start] 1.56	\[ e^{\mathsf{log1p}\left(\frac{\frac{\frac{1.3333333333333333}{\pi}}{1 - v \cdot v}}{\sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)}}\right)} - 1 \]
   expm1-def [=>] 0.02	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{\frac{1.3333333333333333}{\pi}}{1 - v \cdot v}}{\sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)}}\right)\right)} \]
   expm1-log1p [=>] 0.02	\[ \color{blue}{\frac{\frac{\frac{1.3333333333333333}{\pi}}{1 - v \cdot v}}{\sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)}}} \]
   associate-/l/ [=>] 0.02	\[ \color{blue}{\frac{\frac{1.3333333333333333}{\pi}}{\sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)} \cdot \left(1 - v \cdot v\right)}} \]

5. Final simplification 0.02

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

Alternatives

Alternative 1
Error	0.02%
Cost	13824

\[\frac{\frac{1.3333333333333333}{\pi}}{\left(1 - v \cdot v\right) \cdot \sqrt{2 + v \cdot \left(v \cdot -6\right)}} \]

Alternative 2
Error	1.07%
Cost	13568

\[\frac{1.3333333333333333 \cdot \frac{1}{\pi}}{\sqrt{2 + v \cdot \left(v \cdot -6\right)}} \]

Alternative 3
Error	1.07%
Cost	13440

\[\frac{\frac{1.3333333333333333}{\pi}}{\sqrt{2 + v \cdot \left(v \cdot -6\right)}} \]

Alternative 4
Error	2.64%
Cost	13056

\[1.3333333333333333 \cdot \frac{\sqrt{0.5}}{\pi} \]

Alternative 5
Error	1.13%
Cost	13056

\[\frac{1.3333333333333333}{\frac{\pi}{\sqrt{0.5}}} \]

Reproduce

herbie shell --seed 2023102 
(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)))))))