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

Average Error: 0.5 → 0.3

Time: 10.6s

Precision: binary64

Cost: 20672

Math

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

↓

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

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)) PI)
  (* (* t (+ (* v v) -1.0)) (sqrt (+ 2.0 (* (* v v) -6.0))))))

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) / ((double) M_PI)) / ((t * ((v * v) + -1.0)) * sqrt((2.0 + ((v * v) * -6.0))));
}

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

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

   [Start] 0.5	\[ \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
   cancel-sign-sub-inv [=>] 0.5	\[ \frac{\color{blue}{1 + \left(-5\right) \cdot \left(v \cdot v\right)}}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
   metadata-eval [=>] 0.5	\[ \frac{1 + \color{blue}{-5} \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
   distribute-rgt-out-- [<=] 0.5	\[ \frac{1 + -5 \cdot \left(v \cdot v\right)}{\color{blue}{1 \cdot \left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) - \left(v \cdot v\right) \cdot \left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right)}} \]
   *-lft-identity [=>] 0.5	\[ \frac{1 + -5 \cdot \left(v \cdot v\right)}{\color{blue}{\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}} - \left(v \cdot v\right) \cdot \left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right)} \]
   *-commutative [=>] 0.5	\[ \frac{1 + -5 \cdot \left(v \cdot v\right)}{\color{blue}{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)} \cdot \left(\pi \cdot t\right)} - \left(v \cdot v\right) \cdot \left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right)} \]
   *-commutative [=>] 0.5	\[ \frac{1 + -5 \cdot \left(v \cdot v\right)}{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)} \cdot \left(\pi \cdot t\right) - \left(v \cdot v\right) \cdot \color{blue}{\left(\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)} \cdot \left(\pi \cdot t\right)\right)}} \]
   associate-*r* [=>] 0.5	\[ \frac{1 + -5 \cdot \left(v \cdot v\right)}{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)} \cdot \left(\pi \cdot t\right) - \color{blue}{\left(\left(v \cdot v\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(\pi \cdot t\right)}} \]
   distribute-rgt-out-- [=>] 0.5	\[ \frac{1 + -5 \cdot \left(v \cdot v\right)}{\color{blue}{\left(\pi \cdot t\right) \cdot \left(\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)} - \left(v \cdot v\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right)}} \]

3. Applied egg-rr 0.5

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

4. Simplified 0.3

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

   [Start] 0.5	\[ \left(-1 - v \cdot \left(v \cdot -5\right)\right) \cdot \frac{1}{\pi \cdot \left(-t \cdot \left(\sqrt{2 + 2 \cdot \left(\left(v \cdot v\right) \cdot -3\right)} \cdot \left(1 - v \cdot v\right)\right)\right)} \]
   associate-*r/ [=>] 0.5	\[ \color{blue}{\frac{\left(-1 - v \cdot \left(v \cdot -5\right)\right) \cdot 1}{\pi \cdot \left(-t \cdot \left(\sqrt{2 + 2 \cdot \left(\left(v \cdot v\right) \cdot -3\right)} \cdot \left(1 - v \cdot v\right)\right)\right)}} \]
   *-rgt-identity [=>] 0.5	\[ \frac{\color{blue}{-1 - v \cdot \left(v \cdot -5\right)}}{\pi \cdot \left(-t \cdot \left(\sqrt{2 + 2 \cdot \left(\left(v \cdot v\right) \cdot -3\right)} \cdot \left(1 - v \cdot v\right)\right)\right)} \]
   sub-neg [=>] 0.5	\[ \frac{\color{blue}{-1 + \left(-v \cdot \left(v \cdot -5\right)\right)}}{\pi \cdot \left(-t \cdot \left(\sqrt{2 + 2 \cdot \left(\left(v \cdot v\right) \cdot -3\right)} \cdot \left(1 - v \cdot v\right)\right)\right)} \]
   metadata-eval [<=] 0.5	\[ \frac{\color{blue}{\left(-1\right)} + \left(-v \cdot \left(v \cdot -5\right)\right)}{\pi \cdot \left(-t \cdot \left(\sqrt{2 + 2 \cdot \left(\left(v \cdot v\right) \cdot -3\right)} \cdot \left(1 - v \cdot v\right)\right)\right)} \]
   distribute-neg-in [<=] 0.5	\[ \frac{\color{blue}{-\left(1 + v \cdot \left(v \cdot -5\right)\right)}}{\pi \cdot \left(-t \cdot \left(\sqrt{2 + 2 \cdot \left(\left(v \cdot v\right) \cdot -3\right)} \cdot \left(1 - v \cdot v\right)\right)\right)} \]
   +-commutative [=>] 0.5	\[ \frac{-\color{blue}{\left(v \cdot \left(v \cdot -5\right) + 1\right)}}{\pi \cdot \left(-t \cdot \left(\sqrt{2 + 2 \cdot \left(\left(v \cdot v\right) \cdot -3\right)} \cdot \left(1 - v \cdot v\right)\right)\right)} \]
   associate-*r* [=>] 0.5	\[ \frac{-\left(\color{blue}{\left(v \cdot v\right) \cdot -5} + 1\right)}{\pi \cdot \left(-t \cdot \left(\sqrt{2 + 2 \cdot \left(\left(v \cdot v\right) \cdot -3\right)} \cdot \left(1 - v \cdot v\right)\right)\right)} \]
   unpow2 [<=] 0.5	\[ \frac{-\left(\color{blue}{{v}^{2}} \cdot -5 + 1\right)}{\pi \cdot \left(-t \cdot \left(\sqrt{2 + 2 \cdot \left(\left(v \cdot v\right) \cdot -3\right)} \cdot \left(1 - v \cdot v\right)\right)\right)} \]
   fma-udef [<=] 0.5	\[ \frac{-\color{blue}{\mathsf{fma}\left({v}^{2}, -5, 1\right)}}{\pi \cdot \left(-t \cdot \left(\sqrt{2 + 2 \cdot \left(\left(v \cdot v\right) \cdot -3\right)} \cdot \left(1 - v \cdot v\right)\right)\right)} \]
   unpow2 [=>] 0.5	\[ \frac{-\mathsf{fma}\left(\color{blue}{v \cdot v}, -5, 1\right)}{\pi \cdot \left(-t \cdot \left(\sqrt{2 + 2 \cdot \left(\left(v \cdot v\right) \cdot -3\right)} \cdot \left(1 - v \cdot v\right)\right)\right)} \]
   distribute-frac-neg [=>] 0.5	\[ \color{blue}{-\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\pi \cdot \left(-t \cdot \left(\sqrt{2 + 2 \cdot \left(\left(v \cdot v\right) \cdot -3\right)} \cdot \left(1 - v \cdot v\right)\right)\right)}} \]
   associate-/r* [=>] 0.3	\[ -\color{blue}{\frac{\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\pi}}{-t \cdot \left(\sqrt{2 + 2 \cdot \left(\left(v \cdot v\right) \cdot -3\right)} \cdot \left(1 - v \cdot v\right)\right)}} \]
   distribute-neg-frac [=>] 0.3	\[ \color{blue}{\frac{-\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\pi}}{-t \cdot \left(\sqrt{2 + 2 \cdot \left(\left(v \cdot v\right) \cdot -3\right)} \cdot \left(1 - v \cdot v\right)\right)}} \]

5. Final simplification 0.3

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

Alternatives

Alternative 1
Error	0.4
Cost	14336

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

Alternative 2
Error	1.1
Cost	13184

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

Alternative 3
Error	1.0
Cost	13184

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

Alternative 4
Error	0.7
Cost	13184

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

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 2023046 
(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)))))