Falkner and Boettcher, Appendix B, 2

Average Accuracy: 100.0% → 100.0%

Time: 8.7s

Precision: binary64

Cost: 13632

Math

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

↓

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

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

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

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(0.125 * fma(v, Float64(v * -3.0), 1.0))))
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[(0.125 * N[(v * N[(v * -3.0), $MachinePrecision] + 1.0), $MachinePrecision]), $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%

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

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

3. Applied egg-rr 100.0%

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

   [Start] 100.0	\[ \frac{\sqrt{2}}{4} \cdot \left(\sqrt{1 - \left(3 \cdot v\right) \cdot v} \cdot \left(1 - v \cdot v\right)\right) \]
   associate-*r* [=>] 100.0	\[ \color{blue}{\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - \left(3 \cdot v\right) \cdot v}\right) \cdot \left(1 - v \cdot v\right)} \]
   cancel-sign-sub-inv [=>] 100.0	\[ \left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - \left(3 \cdot v\right) \cdot v}\right) \cdot \color{blue}{\left(1 + \left(-v\right) \cdot v\right)} \]
   distribute-lft-in [=>] 100.0	\[ \color{blue}{\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - \left(3 \cdot v\right) \cdot v}\right) \cdot 1 + \left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - \left(3 \cdot v\right) \cdot v}\right) \cdot \left(\left(-v\right) \cdot v\right)} \]

4. Simplified 100.0%

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

   [Start] 100.0	\[ \sqrt{1 + v \cdot \left(v \cdot -3\right)} \cdot \sqrt{0.125} + \left(\sqrt{1 + v \cdot \left(v \cdot -3\right)} \cdot \sqrt{0.125}\right) \cdot \left(v \cdot \left(-v\right)\right) \]
   *-commutative [=>] 100.0	\[ \sqrt{1 + v \cdot \left(v \cdot -3\right)} \cdot \sqrt{0.125} + \color{blue}{\left(v \cdot \left(-v\right)\right) \cdot \left(\sqrt{1 + v \cdot \left(v \cdot -3\right)} \cdot \sqrt{0.125}\right)} \]
   *-lft-identity [<=] 100.0	\[ \color{blue}{\left(1 \cdot \sqrt{1 + v \cdot \left(v \cdot -3\right)}\right)} \cdot \sqrt{0.125} + \left(v \cdot \left(-v\right)\right) \cdot \left(\sqrt{1 + v \cdot \left(v \cdot -3\right)} \cdot \sqrt{0.125}\right) \]
   associate-*l* [=>] 100.0	\[ \color{blue}{1 \cdot \left(\sqrt{1 + v \cdot \left(v \cdot -3\right)} \cdot \sqrt{0.125}\right)} + \left(v \cdot \left(-v\right)\right) \cdot \left(\sqrt{1 + v \cdot \left(v \cdot -3\right)} \cdot \sqrt{0.125}\right) \]
   distribute-rgt-in [<=] 100.0	\[ \color{blue}{\left(\sqrt{1 + v \cdot \left(v \cdot -3\right)} \cdot \sqrt{0.125}\right) \cdot \left(1 + v \cdot \left(-v\right)\right)} \]
   *-commutative [=>] 100.0	\[ \left(\sqrt{1 + v \cdot \left(v \cdot -3\right)} \cdot \sqrt{0.125}\right) \cdot \left(1 + \color{blue}{\left(-v\right) \cdot v}\right) \]
   cancel-sign-sub-inv [<=] 100.0	\[ \left(\sqrt{1 + v \cdot \left(v \cdot -3\right)} \cdot \sqrt{0.125}\right) \cdot \color{blue}{\left(1 - v \cdot v\right)} \]
   associate-*r* [<=] 100.0	\[ \color{blue}{\sqrt{1 + v \cdot \left(v \cdot -3\right)} \cdot \left(\sqrt{0.125} \cdot \left(1 - v \cdot v\right)\right)} \]
   associate-*r* [=>] 100.0	\[ \sqrt{1 + \color{blue}{\left(v \cdot v\right) \cdot -3}} \cdot \left(\sqrt{0.125} \cdot \left(1 - v \cdot v\right)\right) \]

5. Applied egg-rr 100.0%

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

   [Start] 100.0	\[ \sqrt{1 + \left(v \cdot v\right) \cdot -3} \cdot \left(\sqrt{0.125} \cdot \left(1 - v \cdot v\right)\right) \]
   associate-*r* [=>] 100.0	\[ \color{blue}{\left(\sqrt{1 + \left(v \cdot v\right) \cdot -3} \cdot \sqrt{0.125}\right) \cdot \left(1 - v \cdot v\right)} \]
   cancel-sign-sub-inv [=>] 100.0	\[ \left(\sqrt{1 + \left(v \cdot v\right) \cdot -3} \cdot \sqrt{0.125}\right) \cdot \color{blue}{\left(1 + \left(-v\right) \cdot v\right)} \]
   distribute-rgt-in [=>] 100.0	\[ \color{blue}{1 \cdot \left(\sqrt{1 + \left(v \cdot v\right) \cdot -3} \cdot \sqrt{0.125}\right) + \left(\left(-v\right) \cdot v\right) \cdot \left(\sqrt{1 + \left(v \cdot v\right) \cdot -3} \cdot \sqrt{0.125}\right)} \]
   *-un-lft-identity [<=] 100.0	\[ \color{blue}{\sqrt{1 + \left(v \cdot v\right) \cdot -3} \cdot \sqrt{0.125}} + \left(\left(-v\right) \cdot v\right) \cdot \left(\sqrt{1 + \left(v \cdot v\right) \cdot -3} \cdot \sqrt{0.125}\right) \]
   sqrt-unprod [=>] 100.0	\[ \color{blue}{\sqrt{\left(1 + \left(v \cdot v\right) \cdot -3\right) \cdot 0.125}} + \left(\left(-v\right) \cdot v\right) \cdot \left(\sqrt{1 + \left(v \cdot v\right) \cdot -3} \cdot \sqrt{0.125}\right) \]
   +-commutative [=>] 100.0	\[ \sqrt{\color{blue}{\left(\left(v \cdot v\right) \cdot -3 + 1\right)} \cdot 0.125} + \left(\left(-v\right) \cdot v\right) \cdot \left(\sqrt{1 + \left(v \cdot v\right) \cdot -3} \cdot \sqrt{0.125}\right) \]
   associate-*l* [=>] 100.0	\[ \sqrt{\left(\color{blue}{v \cdot \left(v \cdot -3\right)} + 1\right) \cdot 0.125} + \left(\left(-v\right) \cdot v\right) \cdot \left(\sqrt{1 + \left(v \cdot v\right) \cdot -3} \cdot \sqrt{0.125}\right) \]
   fma-def [=>] 100.0	\[ \sqrt{\color{blue}{\mathsf{fma}\left(v, v \cdot -3, 1\right)} \cdot 0.125} + \left(\left(-v\right) \cdot v\right) \cdot \left(\sqrt{1 + \left(v \cdot v\right) \cdot -3} \cdot \sqrt{0.125}\right) \]
   distribute-lft-neg-in [<=] 100.0	\[ \sqrt{\mathsf{fma}\left(v, v \cdot -3, 1\right) \cdot 0.125} + \color{blue}{\left(-v \cdot v\right)} \cdot \left(\sqrt{1 + \left(v \cdot v\right) \cdot -3} \cdot \sqrt{0.125}\right) \]
   sqrt-unprod [=>] 100.0	\[ \sqrt{\mathsf{fma}\left(v, v \cdot -3, 1\right) \cdot 0.125} + \left(-v \cdot v\right) \cdot \color{blue}{\sqrt{\left(1 + \left(v \cdot v\right) \cdot -3\right) \cdot 0.125}} \]
   +-commutative [=>] 100.0	\[ \sqrt{\mathsf{fma}\left(v, v \cdot -3, 1\right) \cdot 0.125} + \left(-v \cdot v\right) \cdot \sqrt{\color{blue}{\left(\left(v \cdot v\right) \cdot -3 + 1\right)} \cdot 0.125} \]
   associate-*l* [=>] 100.0	\[ \sqrt{\mathsf{fma}\left(v, v \cdot -3, 1\right) \cdot 0.125} + \left(-v \cdot v\right) \cdot \sqrt{\left(\color{blue}{v \cdot \left(v \cdot -3\right)} + 1\right) \cdot 0.125} \]
   fma-def [=>] 100.0	\[ \sqrt{\mathsf{fma}\left(v, v \cdot -3, 1\right) \cdot 0.125} + \left(-v \cdot v\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(v, v \cdot -3, 1\right)} \cdot 0.125} \]

6. Simplified 100.0%

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

   [Start] 100.0	\[ \sqrt{\mathsf{fma}\left(v, v \cdot -3, 1\right) \cdot 0.125} + \left(-v \cdot v\right) \cdot \sqrt{\mathsf{fma}\left(v, v \cdot -3, 1\right) \cdot 0.125} \]
   distribute-rgt1-in [=>] 100.0	\[ \color{blue}{\left(\left(-v \cdot v\right) + 1\right) \cdot \sqrt{\mathsf{fma}\left(v, v \cdot -3, 1\right) \cdot 0.125}} \]
   +-commutative [<=] 100.0	\[ \color{blue}{\left(1 + \left(-v \cdot v\right)\right)} \cdot \sqrt{\mathsf{fma}\left(v, v \cdot -3, 1\right) \cdot 0.125} \]
   sub-neg [<=] 100.0	\[ \color{blue}{\left(1 - v \cdot v\right)} \cdot \sqrt{\mathsf{fma}\left(v, v \cdot -3, 1\right) \cdot 0.125} \]
   *-commutative [=>] 100.0	\[ \left(1 - v \cdot v\right) \cdot \sqrt{\color{blue}{0.125 \cdot \mathsf{fma}\left(v, v \cdot -3, 1\right)}} \]

7. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	100.0%
Cost	7744

\[\begin{array}{l} t_0 := 1 - v \cdot v\\ \sqrt{0.125 \cdot \left(\left(1 + v \cdot \left(v \cdot -3\right)\right) \cdot \left(t_0 \cdot t_0\right)\right)} \end{array} \]

Alternative 2
Accuracy	99.4%
Cost	6976

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

Alternative 3
Accuracy	99.5%
Cost	6976

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

Alternative 4
Accuracy	98.9%
Cost	6464

\[\sqrt{0.125} \]

Reproduce

herbie shell --seed 2023138 
(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))))