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

Average Error: 0.4 → 0.4

Time: 10.7s

Precision: binary64

Cost: 20608

Math

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

↓

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

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

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

   [Start] 0.4	\[ \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
   cancel-sign-sub-inv [=>] 0.4	\[ \frac{\color{blue}{1 + \left(-5\right) \cdot \left(v \cdot v\right)}}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
   +-commutative [=>] 0.4	\[ \frac{\color{blue}{\left(-5\right) \cdot \left(v \cdot v\right) + 1}}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
   *-commutative [=>] 0.4	\[ \frac{\color{blue}{\left(v \cdot v\right) \cdot \left(-5\right)} + 1}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
   fma-def [=>] 0.4	\[ \frac{\color{blue}{\mathsf{fma}\left(v \cdot v, -5, 1\right)}}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
   metadata-eval [=>] 0.4	\[ \frac{\mathsf{fma}\left(v \cdot v, \color{blue}{-5}, 1\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
   associate-*l* [=>] 0.4	\[ \frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\color{blue}{\left(\pi \cdot \left(t \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right)\right)} \cdot \left(1 - v \cdot v\right)} \]
   associate-*l* [=>] 0.4	\[ \frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\color{blue}{\pi \cdot \left(\left(t \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)\right)}} \]
   *-commutative [=>] 0.4	\[ \frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\pi \cdot \left(\color{blue}{\left(\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)} \cdot t\right)} \cdot \left(1 - v \cdot v\right)\right)} \]
   associate-*l* [=>] 0.4	\[ \frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\pi \cdot \color{blue}{\left(\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)} \cdot \left(t \cdot \left(1 - v \cdot v\right)\right)\right)}} \]

3. Applied egg-rr 0.4

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

4. Simplified 0.4

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

   [Start] 0.4	\[ \left(-1 - v \cdot \left(v \cdot -5\right)\right) \cdot \frac{1}{\left(\sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)} \cdot \left(t \cdot \left(1 - v \cdot v\right)\right)\right) \cdot \left(-\pi\right)} \]
   *-commutative [=>] 0.4	\[ \color{blue}{\frac{1}{\left(\sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)} \cdot \left(t \cdot \left(1 - v \cdot v\right)\right)\right) \cdot \left(-\pi\right)} \cdot \left(-1 - v \cdot \left(v \cdot -5\right)\right)} \]
   associate-*l/ [=>] 0.4	\[ \color{blue}{\frac{1 \cdot \left(-1 - v \cdot \left(v \cdot -5\right)\right)}{\left(\sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)} \cdot \left(t \cdot \left(1 - v \cdot v\right)\right)\right) \cdot \left(-\pi\right)}} \]
   associate-*l* [=>] 0.4	\[ \frac{1 \cdot \left(-1 - v \cdot \left(v \cdot -5\right)\right)}{\color{blue}{\sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)} \cdot \left(\left(t \cdot \left(1 - v \cdot v\right)\right) \cdot \left(-\pi\right)\right)}} \]
   associate-/r* [=>] 0.4	\[ \color{blue}{\frac{\frac{1 \cdot \left(-1 - v \cdot \left(v \cdot -5\right)\right)}{\sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)}}}{\left(t \cdot \left(1 - v \cdot v\right)\right) \cdot \left(-\pi\right)}} \]
   *-lft-identity [=>] 0.4	\[ \frac{\frac{\color{blue}{-1 - v \cdot \left(v \cdot -5\right)}}{\sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)}}}{\left(t \cdot \left(1 - v \cdot v\right)\right) \cdot \left(-\pi\right)} \]
   associate-*r* [=>] 0.4	\[ \frac{\frac{-1 - \color{blue}{\left(v \cdot v\right) \cdot -5}}{\sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)}}}{\left(t \cdot \left(1 - v \cdot v\right)\right) \cdot \left(-\pi\right)} \]
   *-commutative [<=] 0.4	\[ \frac{\frac{-1 - \color{blue}{-5 \cdot \left(v \cdot v\right)}}{\sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)}}}{\left(t \cdot \left(1 - v \cdot v\right)\right) \cdot \left(-\pi\right)} \]
   cancel-sign-sub-inv [=>] 0.4	\[ \frac{\frac{\color{blue}{-1 + \left(--5\right) \cdot \left(v \cdot v\right)}}{\sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)}}}{\left(t \cdot \left(1 - v \cdot v\right)\right) \cdot \left(-\pi\right)} \]
   metadata-eval [=>] 0.4	\[ \frac{\frac{-1 + \color{blue}{5} \cdot \left(v \cdot v\right)}{\sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)}}}{\left(t \cdot \left(1 - v \cdot v\right)\right) \cdot \left(-\pi\right)} \]
   associate-*l* [=>] 0.4	\[ \frac{\frac{-1 + 5 \cdot \left(v \cdot v\right)}{\sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)}}}{\color{blue}{t \cdot \left(\left(1 - v \cdot v\right) \cdot \left(-\pi\right)\right)}} \]

5. Taylor expanded in v around 0 0.4

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

6. Simplified 0.4

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

   [Start] 0.4	\[ \frac{\frac{-1 + 5 \cdot \left(v \cdot v\right)}{\sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)}}}{{v}^{2} \cdot \left(t \cdot \pi\right) + -1 \cdot \left(t \cdot \pi\right)} \]
   distribute-rgt-out [=>] 0.4	\[ \frac{\frac{-1 + 5 \cdot \left(v \cdot v\right)}{\sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)}}}{\color{blue}{\left(t \cdot \pi\right) \cdot \left({v}^{2} + -1\right)}} \]
   unpow2 [=>] 0.4	\[ \frac{\frac{-1 + 5 \cdot \left(v \cdot v\right)}{\sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)}}}{\left(t \cdot \pi\right) \cdot \left(\color{blue}{v \cdot v} + -1\right)} \]

7. Final simplification 0.4

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

Alternatives

Alternative 1
Error	0.4
Cost	14464

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

Alternative 2
Error	1.0
Cost	13312

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

Alternative 3
Error	1.1
Cost	13184

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

Alternative 4
Error	1.0
Cost	13184

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

Alternative 5
Error	1.4
Cost	13056

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

Alternative 6
Error	1.3
Cost	13056

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

Reproduce

herbie shell --seed 2023012 
(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)))))