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

Average Error: 0.4 → 0.4

Time: 11.4s

Precision: binary64

Cost: 20608

Math

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

↓

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

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 (* (sqrt (+ 2.0 (* (* v v) -6.0))) (* 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 fma((v * v), -5.0, 1.0) / (((double) M_PI) * (sqrt((2.0 + ((v * v) * -6.0))) * (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(fma(Float64(v * v), -5.0, 1.0) / Float64(pi * Float64(sqrt(Float64(2.0 + Float64(Float64(v * v) * -6.0))) * 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[(N[(v * v), $MachinePrecision] * -5.0 + 1.0), $MachinePrecision] / N[(Pi * N[(N[Sqrt[N[(2.0 + N[(N[(v * v), $MachinePrecision] * -6.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(t * N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision]), $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. Final simplification 0.4

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

Alternatives

Alternative 1
Error	0.4
Cost	14464

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

Alternative 2
Error	0.4
Cost	14336

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

Alternative 3
Error	1.1
Cost	13184

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

Alternative 4
Error	0.8
Cost	13184

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

Alternative 5
Error	1.4
Cost	13056

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

Reproduce

herbie shell --seed 2023187 
(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)))))