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

Average Accuracy: 99.3% → 99.5%

Time: 12.5s

Precision: binary64

Cost: 27008

Math

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

↓

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

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 (* (sqrt (fma v (* v -6.0) 2.0)) PI))
  (/ (fma (* v v) -5.0 1.0) (* t (- 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 / (sqrt(fma(v, (v * -6.0), 2.0)) * ((double) M_PI))) * (fma((v * v), -5.0, 1.0) / (t * (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(1.0 / Float64(sqrt(fma(v, Float64(v * -6.0), 2.0)) * pi)) * Float64(fma(Float64(v * v), -5.0, 1.0) / Float64(t * 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[(1.0 / N[(N[Sqrt[N[(v * N[(v * -6.0), $MachinePrecision] + 2.0), $MachinePrecision]], $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(v * v), $MachinePrecision] * -5.0 + 1.0), $MachinePrecision] / N[(t * N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

2. Simplified 99.3%

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

   [Start] 99.3	\[ \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
   cancel-sign-sub-inv [=>] 99.3	\[ \frac{\color{blue}{1 + \left(-5\right) \cdot \left(v \cdot v\right)}}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
   +-commutative [=>] 99.3	\[ \frac{\color{blue}{\left(-5\right) \cdot \left(v \cdot v\right) + 1}}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
   *-commutative [=>] 99.3	\[ \frac{\color{blue}{\left(v \cdot v\right) \cdot \left(-5\right)} + 1}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
   fma-def [=>] 99.3	\[ \frac{\color{blue}{\mathsf{fma}\left(v \cdot v, -5, 1\right)}}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
   metadata-eval [=>] 99.3	\[ \frac{\mathsf{fma}\left(v \cdot v, \color{blue}{-5}, 1\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
   associate-*l* [=>] 99.3	\[ \frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\color{blue}{\left(\pi \cdot \left(t \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right)\right)} \cdot \left(1 - v \cdot v\right)} \]
   associate-*l* [=>] 99.3	\[ \frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\color{blue}{\pi \cdot \left(\left(t \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)\right)}} \]
   *-commutative [=>] 99.3	\[ \frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\pi \cdot \left(\color{blue}{\left(\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)} \cdot t\right)} \cdot \left(1 - v \cdot v\right)\right)} \]
   associate-*l* [=>] 99.3	\[ \frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\pi \cdot \color{blue}{\left(\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)} \cdot \left(t \cdot \left(1 - v \cdot v\right)\right)\right)}} \]

3. Applied egg-rr 99.5%

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

   [Start] 99.3	\[ \frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\pi \cdot \left(\sqrt{2 + \left(v \cdot v\right) \cdot -6} \cdot \left(t \cdot \left(1 - v \cdot v\right)\right)\right)} \]
   *-un-lft-identity [=>] 99.3	\[ \frac{\color{blue}{1 \cdot \mathsf{fma}\left(v \cdot v, -5, 1\right)}}{\pi \cdot \left(\sqrt{2 + \left(v \cdot v\right) \cdot -6} \cdot \left(t \cdot \left(1 - v \cdot v\right)\right)\right)} \]
   associate-*r* [=>] 99.4	\[ \frac{1 \cdot \mathsf{fma}\left(v \cdot v, -5, 1\right)}{\color{blue}{\left(\pi \cdot \sqrt{2 + \left(v \cdot v\right) \cdot -6}\right) \cdot \left(t \cdot \left(1 - v \cdot v\right)\right)}} \]
   times-frac [=>] 99.5	\[ \color{blue}{\frac{1}{\pi \cdot \sqrt{2 + \left(v \cdot v\right) \cdot -6}} \cdot \frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{t \cdot \left(1 - v \cdot v\right)}} \]
   *-commutative [=>] 99.5	\[ \frac{1}{\color{blue}{\sqrt{2 + \left(v \cdot v\right) \cdot -6} \cdot \pi}} \cdot \frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{t \cdot \left(1 - v \cdot v\right)} \]
   +-commutative [=>] 99.5	\[ \frac{1}{\sqrt{\color{blue}{\left(v \cdot v\right) \cdot -6 + 2}} \cdot \pi} \cdot \frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{t \cdot \left(1 - v \cdot v\right)} \]
   associate-*l* [=>] 99.5	\[ \frac{1}{\sqrt{\color{blue}{v \cdot \left(v \cdot -6\right)} + 2} \cdot \pi} \cdot \frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{t \cdot \left(1 - v \cdot v\right)} \]
   fma-def [=>] 99.5	\[ \frac{1}{\sqrt{\color{blue}{\mathsf{fma}\left(v, v \cdot -6, 2\right)}} \cdot \pi} \cdot \frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{t \cdot \left(1 - v \cdot v\right)} \]

4. Final simplification 99.5%

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

Alternatives

Alternative 1
Accuracy	99.4%
Cost	14592

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

Alternative 2
Accuracy	99.3%
Cost	14464

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

Alternative 3
Accuracy	99.3%
Cost	14336

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

Alternative 4
Accuracy	99.3%
Cost	14336

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

Alternative 5
Accuracy	98.5%
Cost	13312

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

Alternative 6
Accuracy	97.9%
Cost	13184

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

Alternative 7
Accuracy	98.3%
Cost	13184

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

Alternative 8
Accuracy	98.3%
Cost	13184

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

Alternative 9
Accuracy	98.5%
Cost	13184

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

Alternative 10
Accuracy	98.4%
Cost	13184

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

Alternative 11
Accuracy	98.6%
Cost	13184

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

Alternative 12
Accuracy	97.9%
Cost	13056

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

Alternative 13
Accuracy	97.9%
Cost	13056

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

Reproduce

herbie shell --seed 2023135 
(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)))))