Falkner and Boettcher, Appendix B, 2

Percentage Accurate: 100.0% → 100.0%

Time: 4.2s

Precision: binary64

Cost: 13632

Math

\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right) \]

↓

\[\begin{array}{l} \\ \left(1 - v \cdot v\right) \cdot \sqrt{\mathsf{fma}\left(v \cdot v, -3, 1\right) \cdot 0.125} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (* (* (/ (sqrt 2.0) 4.0) (sqrt (- 1.0 (* 3.0 (* v v))))) (- 1.0 (* v v))))

↓

(FPCore (v)
 :precision binary64
 (* (- 1.0 (* v v)) (sqrt (* (fma (* v v) -3.0 1.0) 0.125))))

C

double code(double v) {
	return ((sqrt(2.0) / 4.0) * sqrt((1.0 - (3.0 * (v * v))))) * (1.0 - (v * v));
}

↓

double code(double v) {
	return (1.0 - (v * v)) * sqrt((fma((v * v), -3.0, 1.0) * 0.125));
}

Julia

function code(v)
	return Float64(Float64(Float64(sqrt(2.0) / 4.0) * sqrt(Float64(1.0 - Float64(3.0 * Float64(v * v))))) * Float64(1.0 - Float64(v * v)))
end

↓

function code(v)
	return Float64(Float64(1.0 - Float64(v * v)) * sqrt(Float64(fma(Float64(v * v), -3.0, 1.0) * 0.125)))
end

Wolfram

code[v_] := N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] / 4.0), $MachinePrecision] * N[Sqrt[N[(1.0 - N[(3.0 * N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[v_] := N[(N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(N[(v * v), $MachinePrecision] * -3.0 + 1.0), $MachinePrecision] * 0.125), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right) \]

2. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{\sqrt{2}}{4} \cdot \left(\sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot \left(1 - v \cdot v\right)\right)} \]
   Step-by-step derivation

   [Start] 100.0%	\[ \left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right) \]
   associate-*l* [=>] 100.0%	\[ \color{blue}{\frac{\sqrt{2}}{4} \cdot \left(\sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot \left(1 - v \cdot v\right)\right)} \]

3. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\sqrt{0.125 \cdot \left(\mathsf{fma}\left(v \cdot v, -3, 1\right) \cdot {\left(1 - v \cdot v\right)}^{2}\right)}} \]
   Step-by-step derivation

   [Start] 100.0%	\[ \frac{\sqrt{2}}{4} \cdot \left(\sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot \left(1 - v \cdot v\right)\right) \]
   add-sqr-sqrt [=>] 98.4%	\[ \color{blue}{\sqrt{\frac{\sqrt{2}}{4} \cdot \left(\sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot \left(1 - v \cdot v\right)\right)} \cdot \sqrt{\frac{\sqrt{2}}{4} \cdot \left(\sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot \left(1 - v \cdot v\right)\right)}} \]
   sqrt-unprod [=>] 100.0%	\[ \color{blue}{\sqrt{\left(\frac{\sqrt{2}}{4} \cdot \left(\sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot \left(1 - v \cdot v\right)\right)\right) \cdot \left(\frac{\sqrt{2}}{4} \cdot \left(\sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot \left(1 - v \cdot v\right)\right)\right)}} \]
   swap-sqr [=>] 100.0%	\[ \sqrt{\color{blue}{\left(\frac{\sqrt{2}}{4} \cdot \frac{\sqrt{2}}{4}\right) \cdot \left(\left(\sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot \left(1 - v \cdot v\right)\right) \cdot \left(\sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot \left(1 - v \cdot v\right)\right)\right)}} \]
   frac-times [=>] 100.0%	\[ \sqrt{\color{blue}{\frac{\sqrt{2} \cdot \sqrt{2}}{4 \cdot 4}} \cdot \left(\left(\sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot \left(1 - v \cdot v\right)\right) \cdot \left(\sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot \left(1 - v \cdot v\right)\right)\right)} \]
   add-sqr-sqrt [<=] 100.0%	\[ \sqrt{\frac{\color{blue}{2}}{4 \cdot 4} \cdot \left(\left(\sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot \left(1 - v \cdot v\right)\right) \cdot \left(\sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot \left(1 - v \cdot v\right)\right)\right)} \]
   metadata-eval [=>] 100.0%	\[ \sqrt{\frac{2}{\color{blue}{16}} \cdot \left(\left(\sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot \left(1 - v \cdot v\right)\right) \cdot \left(\sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot \left(1 - v \cdot v\right)\right)\right)} \]
   metadata-eval [=>] 100.0%	\[ \sqrt{\color{blue}{0.125} \cdot \left(\left(\sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot \left(1 - v \cdot v\right)\right) \cdot \left(\sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot \left(1 - v \cdot v\right)\right)\right)} \]
   swap-sqr [=>] 100.0%	\[ \sqrt{0.125 \cdot \color{blue}{\left(\left(\sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(\left(1 - v \cdot v\right) \cdot \left(1 - v \cdot v\right)\right)\right)}} \]

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\left(1 - v \cdot v\right) \cdot \sqrt{\mathsf{fma}\left(v \cdot v, -3, 1\right) \cdot 0.125}} \]
   Step-by-step derivation

   [Start] 100.0%	\[ \sqrt{0.125 \cdot \left(\mathsf{fma}\left(v \cdot v, -3, 1\right) \cdot {\left(1 - v \cdot v\right)}^{2}\right)} \]
   associate-*r* [=>] 100.0%	\[ \sqrt{\color{blue}{\left(0.125 \cdot \mathsf{fma}\left(v \cdot v, -3, 1\right)\right) \cdot {\left(1 - v \cdot v\right)}^{2}}} \]
   sqrt-prod [=>] 100.0%	\[ \color{blue}{\sqrt{0.125 \cdot \mathsf{fma}\left(v \cdot v, -3, 1\right)} \cdot \sqrt{{\left(1 - v \cdot v\right)}^{2}}} \]
   sqrt-unprod [<=] 100.0%	\[ \color{blue}{\left(\sqrt{0.125} \cdot \sqrt{\mathsf{fma}\left(v \cdot v, -3, 1\right)}\right)} \cdot \sqrt{{\left(1 - v \cdot v\right)}^{2}} \]
   unpow2 [=>] 100.0%	\[ \left(\sqrt{0.125} \cdot \sqrt{\mathsf{fma}\left(v \cdot v, -3, 1\right)}\right) \cdot \sqrt{\color{blue}{\left(1 - v \cdot v\right) \cdot \left(1 - v \cdot v\right)}} \]
   sqrt-prod [=>] 100.0%	\[ \left(\sqrt{0.125} \cdot \sqrt{\mathsf{fma}\left(v \cdot v, -3, 1\right)}\right) \cdot \color{blue}{\left(\sqrt{1 - v \cdot v} \cdot \sqrt{1 - v \cdot v}\right)} \]
   add-sqr-sqrt [<=] 100.0%	\[ \left(\sqrt{0.125} \cdot \sqrt{\mathsf{fma}\left(v \cdot v, -3, 1\right)}\right) \cdot \color{blue}{\left(1 - v \cdot v\right)} \]
   *-commutative [=>] 100.0%	\[ \color{blue}{\left(1 - v \cdot v\right) \cdot \left(\sqrt{0.125} \cdot \sqrt{\mathsf{fma}\left(v \cdot v, -3, 1\right)}\right)} \]
   sqrt-unprod [=>] 100.0%	\[ \left(1 - v \cdot v\right) \cdot \color{blue}{\sqrt{0.125 \cdot \mathsf{fma}\left(v \cdot v, -3, 1\right)}} \]
   *-commutative [=>] 100.0%	\[ \left(1 - v \cdot v\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(v \cdot v, -3, 1\right) \cdot 0.125}} \]

5. Final simplification 100.0%

   \[\leadsto \left(1 - v \cdot v\right) \cdot \sqrt{\mathsf{fma}\left(v \cdot v, -3, 1\right) \cdot 0.125} \]

Alternatives

Alternative 1
Accuracy	100.0%
Cost	13632

\[\left(1 - v \cdot v\right) \cdot \sqrt{\mathsf{fma}\left(v \cdot v, -3, 1\right) \cdot 0.125} \]

Alternative 2
Accuracy	99.6%
Cost	7104

\[\frac{\sqrt{2}}{4} \cdot \left(1 + \left(v \cdot v\right) \cdot -2.5\right) \]

Alternative 3
Accuracy	99.6%
Cost	6976

\[\sqrt{0.125 \cdot \left(1 + \left(v \cdot v\right) \cdot -5\right)} \]

Alternative 4
Accuracy	99.1%
Cost	6848

\[\left(1 - v \cdot v\right) \cdot \sqrt{0.125} \]

Alternative 5
Accuracy	99.1%
Cost	6464

\[\sqrt{0.125} \]

Reproduce

herbie shell --seed 2023167 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  :precision binary64
  (* (* (/ (sqrt 2.0) 4.0) (sqrt (- 1.0 (* 3.0 (* v v))))) (- 1.0 (* v v))))