Falkner and Boettcher, Equation (20:1,3)

Average Error: 0.4 → 0.3

Time: 11.2s

Precision: binary64

Cost: 33408

Math

\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]

↓

\[\frac{\frac{\frac{-1 + \sqrt[3]{{v}^{6} \cdot 125}}{\pi}}{\left(-t\right) \cdot \sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}}}{1 - v \cdot v} \]

FPCore

(FPCore (v t)
 :precision binary64
 (/
  (- 1.0 (* 5.0 (* v v)))
  (* (* (* PI t) (sqrt (* 2.0 (- 1.0 (* 3.0 (* v v)))))) (- 1.0 (* v v)))))

↓

(FPCore (v t)
 :precision binary64
 (/
  (/
   (/ (+ -1.0 (cbrt (* (pow v 6.0) 125.0))) PI)
   (* (- t) (sqrt (fma (* v v) -6.0 2.0))))
  (- 1.0 (* v v))))

C

double code(double v, double t) {
	return (1.0 - (5.0 * (v * v))) / (((((double) M_PI) * t) * sqrt((2.0 * (1.0 - (3.0 * (v * v)))))) * (1.0 - (v * v)));
}

↓

double code(double v, double t) {
	return (((-1.0 + cbrt((pow(v, 6.0) * 125.0))) / ((double) M_PI)) / (-t * sqrt(fma((v * v), -6.0, 2.0)))) / (1.0 - (v * v));
}

Julia

function code(v, t)
	return Float64(Float64(1.0 - Float64(5.0 * Float64(v * v))) / Float64(Float64(Float64(pi * t) * sqrt(Float64(2.0 * Float64(1.0 - Float64(3.0 * Float64(v * v)))))) * Float64(1.0 - Float64(v * v))))
end

↓

function code(v, t)
	return Float64(Float64(Float64(Float64(-1.0 + cbrt(Float64((v ^ 6.0) * 125.0))) / pi) / Float64(Float64(-t) * sqrt(fma(Float64(v * v), -6.0, 2.0)))) / Float64(1.0 - Float64(v * v)))
end

Wolfram

code[v_, t_] := N[(N[(1.0 - N[(5.0 * N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(Pi * t), $MachinePrecision] * N[Sqrt[N[(2.0 * N[(1.0 - N[(3.0 * N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[v_, t_] := N[(N[(N[(N[(-1.0 + N[Power[N[(N[Power[v, 6.0], $MachinePrecision] * 125.0), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision] / N[((-t) * N[Sqrt[N[(N[(v * v), $MachinePrecision] * -6.0 + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.4

   \[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]

2. Simplified 0.4

   \[\leadsto \color{blue}{\frac{\frac{\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\pi \cdot t}}{\sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}}}{1 - v \cdot v}} \]
   Proof

   [Start] 0.4	\[ \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
   associate-/r* [=>] 0.4	\[ \color{blue}{\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}}}{1 - v \cdot v}} \]
   associate-*l* [=>] 0.4	\[ \frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\pi \cdot \left(t \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right)}}}{1 - v \cdot v} \]
   associate-/r* [=>] 0.3	\[ \frac{\color{blue}{\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\pi}}{t \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}}}}{1 - v \cdot v} \]

3. Applied egg-rr 0.4

   \[\leadsto \frac{\color{blue}{-\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\left(\pi \cdot \left(-t\right)\right) \cdot \sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}}}}{1 - v \cdot v} \]

4. Simplified 0.3

   \[\leadsto \frac{\color{blue}{\frac{\frac{-1 + \left(v \cdot v\right) \cdot 5}{\pi}}{\left(-t\right) \cdot \sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}}}}{1 - v \cdot v} \]
   Proof

   [Start] 0.4	\[ \frac{-\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\left(\pi \cdot \left(-t\right)\right) \cdot \sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}}}{1 - v \cdot v} \]
   distribute-neg-frac [=>] 0.4	\[ \frac{\color{blue}{\frac{-\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\left(\pi \cdot \left(-t\right)\right) \cdot \sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}}}}{1 - v \cdot v} \]
   associate-*l* [=>] 0.4	\[ \frac{\frac{-\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\color{blue}{\pi \cdot \left(\left(-t\right) \cdot \sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}\right)}}}{1 - v \cdot v} \]
   associate-/r* [=>] 0.3	\[ \frac{\color{blue}{\frac{\frac{-\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\pi}}{\left(-t\right) \cdot \sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}}}}{1 - v \cdot v} \]
   unpow2 [<=] 0.3	\[ \frac{\frac{\frac{-\mathsf{fma}\left(\color{blue}{{v}^{2}}, -5, 1\right)}{\pi}}{\left(-t\right) \cdot \sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}}}{1 - v \cdot v} \]
   fma-udef [=>] 0.3	\[ \frac{\frac{\frac{-\color{blue}{\left({v}^{2} \cdot -5 + 1\right)}}{\pi}}{\left(-t\right) \cdot \sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}}}{1 - v \cdot v} \]
   unpow2 [=>] 0.3	\[ \frac{\frac{\frac{-\left(\color{blue}{\left(v \cdot v\right)} \cdot -5 + 1\right)}{\pi}}{\left(-t\right) \cdot \sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}}}{1 - v \cdot v} \]
   associate-*r* [<=] 0.3	\[ \frac{\frac{\frac{-\left(\color{blue}{v \cdot \left(v \cdot -5\right)} + 1\right)}{\pi}}{\left(-t\right) \cdot \sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}}}{1 - v \cdot v} \]
   +-commutative [<=] 0.3	\[ \frac{\frac{\frac{-\color{blue}{\left(1 + v \cdot \left(v \cdot -5\right)\right)}}{\pi}}{\left(-t\right) \cdot \sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}}}{1 - v \cdot v} \]
   distribute-neg-in [=>] 0.3	\[ \frac{\frac{\frac{\color{blue}{\left(-1\right) + \left(-v \cdot \left(v \cdot -5\right)\right)}}{\pi}}{\left(-t\right) \cdot \sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}}}{1 - v \cdot v} \]
   metadata-eval [=>] 0.3	\[ \frac{\frac{\frac{\color{blue}{-1} + \left(-v \cdot \left(v \cdot -5\right)\right)}{\pi}}{\left(-t\right) \cdot \sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}}}{1 - v \cdot v} \]
   associate-*r* [=>] 0.3	\[ \frac{\frac{\frac{-1 + \left(-\color{blue}{\left(v \cdot v\right) \cdot -5}\right)}{\pi}}{\left(-t\right) \cdot \sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}}}{1 - v \cdot v} \]
   unpow2 [<=] 0.3	\[ \frac{\frac{\frac{-1 + \left(-\color{blue}{{v}^{2}} \cdot -5\right)}{\pi}}{\left(-t\right) \cdot \sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}}}{1 - v \cdot v} \]
   distribute-rgt-neg-in [=>] 0.3	\[ \frac{\frac{\frac{-1 + \color{blue}{{v}^{2} \cdot \left(--5\right)}}{\pi}}{\left(-t\right) \cdot \sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}}}{1 - v \cdot v} \]
   unpow2 [=>] 0.3	\[ \frac{\frac{\frac{-1 + \color{blue}{\left(v \cdot v\right)} \cdot \left(--5\right)}{\pi}}{\left(-t\right) \cdot \sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}}}{1 - v \cdot v} \]
   metadata-eval [=>] 0.3	\[ \frac{\frac{\frac{-1 + \left(v \cdot v\right) \cdot \color{blue}{5}}{\pi}}{\left(-t\right) \cdot \sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}}}{1 - v \cdot v} \]

5. Applied egg-rr 0.3

   \[\leadsto \frac{\frac{\frac{-1 + \color{blue}{{\left(125 \cdot {\left(v \cdot v\right)}^{3}\right)}^{0.3333333333333333}}}{\pi}}{\left(-t\right) \cdot \sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}}}{1 - v \cdot v} \]

6. Simplified 0.3

   \[\leadsto \frac{\frac{\frac{-1 + \color{blue}{\sqrt[3]{{v}^{6} \cdot 125}}}{\pi}}{\left(-t\right) \cdot \sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}}}{1 - v \cdot v} \]
   Proof

   [Start] 0.3	\[ \frac{\frac{\frac{-1 + {\left(125 \cdot {\left(v \cdot v\right)}^{3}\right)}^{0.3333333333333333}}{\pi}}{\left(-t\right) \cdot \sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}}}{1 - v \cdot v} \]
   unpow1/3 [=>] 0.3	\[ \frac{\frac{\frac{-1 + \color{blue}{\sqrt[3]{125 \cdot {\left(v \cdot v\right)}^{3}}}}{\pi}}{\left(-t\right) \cdot \sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}}}{1 - v \cdot v} \]
   *-commutative [=>] 0.3	\[ \frac{\frac{\frac{-1 + \sqrt[3]{\color{blue}{{\left(v \cdot v\right)}^{3} \cdot 125}}}{\pi}}{\left(-t\right) \cdot \sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}}}{1 - v \cdot v} \]
   cube-prod [=>] 0.3	\[ \frac{\frac{\frac{-1 + \sqrt[3]{\color{blue}{\left({v}^{3} \cdot {v}^{3}\right)} \cdot 125}}{\pi}}{\left(-t\right) \cdot \sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}}}{1 - v \cdot v} \]
   pow-sqr [=>] 0.3	\[ \frac{\frac{\frac{-1 + \sqrt[3]{\color{blue}{{v}^{\left(2 \cdot 3\right)}} \cdot 125}}{\pi}}{\left(-t\right) \cdot \sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}}}{1 - v \cdot v} \]
   metadata-eval [=>] 0.3	\[ \frac{\frac{\frac{-1 + \sqrt[3]{{v}^{\color{blue}{6}} \cdot 125}}{\pi}}{\left(-t\right) \cdot \sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}}}{1 - v \cdot v} \]

7. Final simplification 0.3

   \[\leadsto \frac{\frac{\frac{-1 + \sqrt[3]{{v}^{6} \cdot 125}}{\pi}}{\left(-t\right) \cdot \sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}}}{1 - v \cdot v} \]

Alternatives

Alternative 1
Error	0.3
Cost	20672

\[\frac{\frac{\frac{-1 + \left(v \cdot v\right) \cdot 5}{\pi}}{\left(-t\right) \cdot \sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}}}{1 - v \cdot v} \]

Alternative 2
Error	0.9
Cost	14016

\[\frac{\frac{\frac{-1 + \left(v \cdot v\right) \cdot 5}{\pi}}{\left(-t\right) \cdot \sqrt{2}}}{1 - v \cdot v} \]

Alternative 3
Error	1.3
Cost	13184

\[\frac{1}{t} \cdot \frac{\sqrt{0.5}}{\pi} \]

Alternative 4
Error	1.1
Cost	13184

\[\frac{\frac{\frac{1}{t}}{\pi}}{\sqrt{2}} \]

Alternative 5
Error	1.3
Cost	13056

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

Alternative 6
Error	1.3
Cost	13056

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

Reproduce

herbie shell --seed 2023066 
(FPCore (v t)
  :name "Falkner and Boettcher, Equation (20:1,3)"
  :precision binary64
  (/ (- 1.0 (* 5.0 (* v v))) (* (* (* PI t) (sqrt (* 2.0 (- 1.0 (* 3.0 (* v v)))))) (- 1.0 (* v v)))))