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

Average Accuracy: 99.3% → 99.6%

Time: 13.6s

Precision: binary64

Cost: 27008

Math

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

↓

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

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
 (*
  (/ (fma v (* v -5.0) 1.0) t)
  (/ (/ (/ 1.0 PI) (- 1.0 (* v v))) (sqrt (fma v (* v -6.0) 2.0)))))

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 (fma(v, (v * -5.0), 1.0) / t) * (((1.0 / ((double) M_PI)) / (1.0 - (v * v))) / sqrt(fma(v, (v * -6.0), 2.0)));
}

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

Derivation

1. Initial program 99.4%

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

2. Simplified 99.6%

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

   [Start] 99.4	\[ \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
   *-commutative [=>] 99.4	\[ \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\left(1 - v \cdot v\right) \cdot \left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right)}} \]
   associate-*l* [=>] 99.4	\[ \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(1 - v \cdot v\right) \cdot \color{blue}{\left(\pi \cdot \left(t \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right)\right)}} \]
   associate-*r* [=>] 99.4	\[ \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\left(\left(1 - v \cdot v\right) \cdot \pi\right) \cdot \left(t \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right)}} \]
   associate-/r* [=>] 99.6	\[ \color{blue}{\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(1 - v \cdot v\right) \cdot \pi}}{t \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}}} \]

3. Applied egg-rr 99.6%

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

   [Start] 99.6	\[ \frac{\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\pi \cdot \left(1 - v \cdot v\right)}}{t \cdot \sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}} \]
   div-inv [=>] 99.6	\[ \frac{\color{blue}{\mathsf{fma}\left(v, v \cdot -5, 1\right) \cdot \frac{1}{\pi \cdot \left(1 - v \cdot v\right)}}}{t \cdot \sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}} \]
   times-frac [=>] 99.6	\[ \color{blue}{\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{t} \cdot \frac{\frac{1}{\pi \cdot \left(1 - v \cdot v\right)}}{\sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}}} \]
   associate-/r* [=>] 99.6	\[ \frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{t} \cdot \frac{\color{blue}{\frac{\frac{1}{\pi}}{1 - v \cdot v}}}{\sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}} \]
   fma-udef [=>] 99.6	\[ \frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{t} \cdot \frac{\frac{\frac{1}{\pi}}{1 - v \cdot v}}{\sqrt{\color{blue}{\left(v \cdot v\right) \cdot -6 + 2}}} \]
   associate-*l* [=>] 99.6	\[ \frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{t} \cdot \frac{\frac{\frac{1}{\pi}}{1 - v \cdot v}}{\sqrt{\color{blue}{v \cdot \left(v \cdot -6\right)} + 2}} \]
   fma-def [=>] 99.6	\[ \frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{t} \cdot \frac{\frac{\frac{1}{\pi}}{1 - v \cdot v}}{\sqrt{\color{blue}{\mathsf{fma}\left(v, v \cdot -6, 2\right)}}} \]

4. Final simplification 99.6%

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

Alternatives

Alternative 1
Accuracy	99.6%
Cost	14464

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

Alternative 2
Accuracy	99.3%
Cost	14400

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

Alternative 3
Accuracy	98.4%
Cost	13184

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

Alternative 4
Accuracy	98.6%
Cost	13184

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

Alternative 5
Accuracy	98.9%
Cost	13184

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

Alternative 6
Accuracy	97.9%
Cost	13056

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

Alternative 7
Accuracy	98.0%
Cost	13056

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

Reproduce

herbie shell --seed 2023157 
(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)))))