Falkner and Boettcher, Appendix B, 2

Percentage Accurate: 100.0% → 100.0%

Time: 3.7s

Precision: binary64

Cost: 26432

Math

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

↓

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

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
 (* (sqrt 2.0) (/ (sqrt (fma (* v v) -3.0 1.0)) (/ -4.0 (fma v v -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 sqrt(2.0) * (sqrt(fma((v * v), -3.0, 1.0)) / (-4.0 / fma(v, v, -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(sqrt(2.0) * Float64(sqrt(fma(Float64(v * v), -3.0, 1.0)) / Float64(-4.0 / fma(v, v, -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[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[N[(N[(v * v), $MachinePrecision] * -3.0 + 1.0), $MachinePrecision]], $MachinePrecision] / N[(-4.0 / N[(v * v + -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%

   \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{\sqrt{\mathsf{fma}\left(v \cdot v, -3, 1\right)}}{\frac{-4}{\mathsf{fma}\left(v, v, -1\right)}}} \]
   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%	\[ \color{blue}{\frac{\sqrt{2} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}}{4}} \cdot \left(1 - v \cdot v\right) \]
   associate-/r/ [<=] 100.0%	\[ \color{blue}{\frac{\sqrt{2} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}}{\frac{4}{1 - v \cdot v}}} \]
   associate-*r/ [<=] 100.0%	\[ \color{blue}{\sqrt{2} \cdot \frac{\sqrt{1 - 3 \cdot \left(v \cdot v\right)}}{\frac{4}{1 - v \cdot v}}} \]
   sub-neg [=>] 100.0%	\[ \sqrt{2} \cdot \frac{\sqrt{\color{blue}{1 + \left(-3 \cdot \left(v \cdot v\right)\right)}}}{\frac{4}{1 - v \cdot v}} \]
   +-commutative [=>] 100.0%	\[ \sqrt{2} \cdot \frac{\sqrt{\color{blue}{\left(-3 \cdot \left(v \cdot v\right)\right) + 1}}}{\frac{4}{1 - v \cdot v}} \]
   *-commutative [=>] 100.0%	\[ \sqrt{2} \cdot \frac{\sqrt{\left(-\color{blue}{\left(v \cdot v\right) \cdot 3}\right) + 1}}{\frac{4}{1 - v \cdot v}} \]
   distribute-rgt-neg-in [=>] 100.0%	\[ \sqrt{2} \cdot \frac{\sqrt{\color{blue}{\left(v \cdot v\right) \cdot \left(-3\right)} + 1}}{\frac{4}{1 - v \cdot v}} \]
   fma-def [=>] 100.0%	\[ \sqrt{2} \cdot \frac{\sqrt{\color{blue}{\mathsf{fma}\left(v \cdot v, -3, 1\right)}}}{\frac{4}{1 - v \cdot v}} \]
   metadata-eval [=>] 100.0%	\[ \sqrt{2} \cdot \frac{\sqrt{\mathsf{fma}\left(v \cdot v, \color{blue}{-3}, 1\right)}}{\frac{4}{1 - v \cdot v}} \]
   sub-neg [=>] 100.0%	\[ \sqrt{2} \cdot \frac{\sqrt{\mathsf{fma}\left(v \cdot v, -3, 1\right)}}{\frac{4}{\color{blue}{1 + \left(-v \cdot v\right)}}} \]
   +-commutative [=>] 100.0%	\[ \sqrt{2} \cdot \frac{\sqrt{\mathsf{fma}\left(v \cdot v, -3, 1\right)}}{\frac{4}{\color{blue}{\left(-v \cdot v\right) + 1}}} \]
   neg-sub0 [=>] 100.0%	\[ \sqrt{2} \cdot \frac{\sqrt{\mathsf{fma}\left(v \cdot v, -3, 1\right)}}{\frac{4}{\color{blue}{\left(0 - v \cdot v\right)} + 1}} \]
   associate-+l- [=>] 100.0%	\[ \sqrt{2} \cdot \frac{\sqrt{\mathsf{fma}\left(v \cdot v, -3, 1\right)}}{\frac{4}{\color{blue}{0 - \left(v \cdot v - 1\right)}}} \]
   sub0-neg [=>] 100.0%	\[ \sqrt{2} \cdot \frac{\sqrt{\mathsf{fma}\left(v \cdot v, -3, 1\right)}}{\frac{4}{\color{blue}{-\left(v \cdot v - 1\right)}}} \]
   neg-mul-1 [=>] 100.0%	\[ \sqrt{2} \cdot \frac{\sqrt{\mathsf{fma}\left(v \cdot v, -3, 1\right)}}{\frac{4}{\color{blue}{-1 \cdot \left(v \cdot v - 1\right)}}} \]
   associate-/r* [=>] 100.0%	\[ \sqrt{2} \cdot \frac{\sqrt{\mathsf{fma}\left(v \cdot v, -3, 1\right)}}{\color{blue}{\frac{\frac{4}{-1}}{v \cdot v - 1}}} \]
   metadata-eval [=>] 100.0%	\[ \sqrt{2} \cdot \frac{\sqrt{\mathsf{fma}\left(v \cdot v, -3, 1\right)}}{\frac{\color{blue}{-4}}{v \cdot v - 1}} \]
   fma-neg [=>] 100.0%	\[ \sqrt{2} \cdot \frac{\sqrt{\mathsf{fma}\left(v \cdot v, -3, 1\right)}}{\frac{-4}{\color{blue}{\mathsf{fma}\left(v, v, -1\right)}}} \]
   metadata-eval [=>] 100.0%	\[ \sqrt{2} \cdot \frac{\sqrt{\mathsf{fma}\left(v \cdot v, -3, 1\right)}}{\frac{-4}{\mathsf{fma}\left(v, v, \color{blue}{-1}\right)}} \]

3. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	100.0%
Cost	26432

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

Alternative 2
Accuracy	100.0%
Cost	13632

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

Alternative 3
Accuracy	99.5%
Cost	13248

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

Alternative 4
Accuracy	98.9%
Cost	6848

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

Alternative 5
Accuracy	98.8%
Cost	6464

\[\sqrt{0.125} \]

Reproduce

herbie shell --seed 2023165 
(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))))