Falkner and Boettcher, Appendix B, 2

Average Accuracy: 100.0% → 100.0%

Time: 6.0s

Precision: binary64

Cost: 13632

Math

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

↓

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

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
 (* (sqrt (* (fma (* v v) -3.0 1.0) 0.125)) (- 1.0 (* v v))))

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 sqrt((fma((v * v), -3.0, 1.0) * 0.125)) * (1.0 - (v * v));
}

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(sqrt(Float64(fma(Float64(v * v), -3.0, 1.0) * 0.125)) * Float64(1.0 - Float64(v * v)))
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[Sqrt[N[(N[(N[(v * v), $MachinePrecision] * -3.0 + 1.0), $MachinePrecision] * 0.125), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(v * v), $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. Applied egg-rr 100.0%

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

   [Start] 100.0	\[ \left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right) \]
   add-sqr-sqrt [=>] 98.5	\[ \color{blue}{\left(\sqrt{\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}} \cdot \sqrt{\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}}\right)} \cdot \left(1 - v \cdot v\right) \]
   sqrt-unprod [=>] 100.0	\[ \color{blue}{\sqrt{\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right)}} \cdot \left(1 - v \cdot v\right) \]
   *-commutative [=>] 100.0	\[ \sqrt{\color{blue}{\left(\sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot \frac{\sqrt{2}}{4}\right)} \cdot \left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right)} \cdot \left(1 - v \cdot v\right) \]
   *-commutative [=>] 100.0	\[ \sqrt{\left(\sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot \frac{\sqrt{2}}{4}\right) \cdot \color{blue}{\left(\sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot \frac{\sqrt{2}}{4}\right)}} \cdot \left(1 - v \cdot v\right) \]
   swap-sqr [=>] 100.0	\[ \sqrt{\color{blue}{\left(\sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(\frac{\sqrt{2}}{4} \cdot \frac{\sqrt{2}}{4}\right)}} \cdot \left(1 - v \cdot v\right) \]
   add-sqr-sqrt [<=] 100.0	\[ \sqrt{\color{blue}{\left(1 - 3 \cdot \left(v \cdot v\right)\right)} \cdot \left(\frac{\sqrt{2}}{4} \cdot \frac{\sqrt{2}}{4}\right)} \cdot \left(1 - v \cdot v\right) \]
   sub-neg [=>] 100.0	\[ \sqrt{\color{blue}{\left(1 + \left(-3 \cdot \left(v \cdot v\right)\right)\right)} \cdot \left(\frac{\sqrt{2}}{4} \cdot \frac{\sqrt{2}}{4}\right)} \cdot \left(1 - v \cdot v\right) \]
   +-commutative [=>] 100.0	\[ \sqrt{\color{blue}{\left(\left(-3 \cdot \left(v \cdot v\right)\right) + 1\right)} \cdot \left(\frac{\sqrt{2}}{4} \cdot \frac{\sqrt{2}}{4}\right)} \cdot \left(1 - v \cdot v\right) \]
   *-commutative [=>] 100.0	\[ \sqrt{\left(\left(-\color{blue}{\left(v \cdot v\right) \cdot 3}\right) + 1\right) \cdot \left(\frac{\sqrt{2}}{4} \cdot \frac{\sqrt{2}}{4}\right)} \cdot \left(1 - v \cdot v\right) \]
   distribute-rgt-neg-in [=>] 100.0	\[ \sqrt{\left(\color{blue}{\left(v \cdot v\right) \cdot \left(-3\right)} + 1\right) \cdot \left(\frac{\sqrt{2}}{4} \cdot \frac{\sqrt{2}}{4}\right)} \cdot \left(1 - v \cdot v\right) \]
   fma-def [=>] 100.0	\[ \sqrt{\color{blue}{\mathsf{fma}\left(v \cdot v, -3, 1\right)} \cdot \left(\frac{\sqrt{2}}{4} \cdot \frac{\sqrt{2}}{4}\right)} \cdot \left(1 - v \cdot v\right) \]
   metadata-eval [=>] 100.0	\[ \sqrt{\mathsf{fma}\left(v \cdot v, \color{blue}{-3}, 1\right) \cdot \left(\frac{\sqrt{2}}{4} \cdot \frac{\sqrt{2}}{4}\right)} \cdot \left(1 - v \cdot v\right) \]
   frac-times [=>] 100.0	\[ \sqrt{\mathsf{fma}\left(v \cdot v, -3, 1\right) \cdot \color{blue}{\frac{\sqrt{2} \cdot \sqrt{2}}{4 \cdot 4}}} \cdot \left(1 - v \cdot v\right) \]
   add-sqr-sqrt [<=] 100.0	\[ \sqrt{\mathsf{fma}\left(v \cdot v, -3, 1\right) \cdot \frac{\color{blue}{2}}{4 \cdot 4}} \cdot \left(1 - v \cdot v\right) \]
   metadata-eval [=>] 100.0	\[ \sqrt{\mathsf{fma}\left(v \cdot v, -3, 1\right) \cdot \frac{2}{\color{blue}{16}}} \cdot \left(1 - v \cdot v\right) \]
   metadata-eval [=>] 100.0	\[ \sqrt{\mathsf{fma}\left(v \cdot v, -3, 1\right) \cdot \color{blue}{0.125}} \cdot \left(1 - v \cdot v\right) \]

3. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	99.6%
Cost	13376

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

Alternative 2
Accuracy	99.6%
Cost	6976

\[\sqrt{2} \cdot \left(0.25 + \left(v \cdot v\right) \cdot -0.625\right) \]

Alternative 3
Accuracy	99.0%
Cost	6848

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

Alternative 4
Accuracy	99.0%
Cost	6464

\[\sqrt{0.125} \]

Reproduce

herbie shell --seed 2023147 
(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))))