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

Average Error: 0.5 → 0.3

Time: 10.2s

Precision: binary64

Cost: 20736

Math

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

↓

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

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

Derivation

1. Initial program 0.5

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

2. Simplified 0.5

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

   [Start] 0.5	\[ \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
   associate-/r* [=>] 0.5	\[ \color{blue}{\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}}}{1 - v \cdot v}} \]
   associate-*l* [=>] 0.5	\[ \frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\pi \cdot \left(t \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right)}}}{1 - v \cdot v} \]
   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 0.3

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

4. Final simplification 0.3

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

Alternatives

Alternative 1
Error	0.5
Cost	20608

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

Alternative 2
Error	0.4
Cost	20608

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

Alternative 3
Error	0.5
Cost	14464

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

Alternative 4
Error	0.5
Cost	14464

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

Alternative 5
Error	1.1
Cost	13184

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

Alternative 6
Error	0.9
Cost	13184

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

Alternative 7
Error	0.8
Cost	13184

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

Alternative 8
Error	1.4
Cost	13056

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

Reproduce

herbie shell --seed 2023018 
(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)))))