Falkner and Boettcher, Appendix B, 1

Average Accuracy: 99.2% → 99.2%

Time: 15.6s

Precision: binary64

Cost: 26304

Math

\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \]

↓

\[\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 - \left(v \cdot v\right) \cdot 5}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right) \]

FPCore

(FPCore (v)
 :precision binary64
 (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))

↓

(FPCore (v)
 :precision binary64
 (expm1 (log1p (acos (/ (- 1.0 (* (* v v) 5.0)) (fma v v -1.0))))))

C

double code(double v) {
	return acos(((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0)));
}

↓

double code(double v) {
	return expm1(log1p(acos(((1.0 - ((v * v) * 5.0)) / fma(v, v, -1.0)))));
}

Julia

function code(v)
	return acos(Float64(Float64(1.0 - Float64(5.0 * Float64(v * v))) / Float64(Float64(v * v) - 1.0)))
end

↓

function code(v)
	return expm1(log1p(acos(Float64(Float64(1.0 - Float64(Float64(v * v) * 5.0)) / fma(v, v, -1.0)))))
end

Wolfram

code[v_] := N[ArcCos[N[(N[(1.0 - N[(5.0 * N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(v * v), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

↓

code[v_] := N[(Exp[N[Log[1 + N[ArcCos[N[(N[(1.0 - N[(N[(v * v), $MachinePrecision] * 5.0), $MachinePrecision]), $MachinePrecision] / N[(v * v + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]

Derivation

1. Initial program 99.2%

   \[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \]

2. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 + \left(v \cdot v\right) \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right)} \]
   Proof

   [Start] 99.2	\[ \cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \]
   expm1-log1p-u [=>] 99.2	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right)\right)} \]
   cancel-sign-sub-inv [=>] 99.2	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{\color{blue}{1 + \left(-5\right) \cdot \left(v \cdot v\right)}}{v \cdot v - 1}\right)\right)\right) \]
   *-commutative [=>] 99.2	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 + \color{blue}{\left(v \cdot v\right) \cdot \left(-5\right)}}{v \cdot v - 1}\right)\right)\right) \]
   metadata-eval [=>] 99.2	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 + \left(v \cdot v\right) \cdot \color{blue}{-5}}{v \cdot v - 1}\right)\right)\right) \]
   fma-neg [=>] 99.2	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 + \left(v \cdot v\right) \cdot -5}{\color{blue}{\mathsf{fma}\left(v, v, -1\right)}}\right)\right)\right) \]
   metadata-eval [=>] 99.2	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 + \left(v \cdot v\right) \cdot -5}{\mathsf{fma}\left(v, v, \color{blue}{-1}\right)}\right)\right)\right) \]

3. Final simplification 99.2%

   \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 - \left(v \cdot v\right) \cdot 5}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right) \]

Alternatives

Alternative 1
Accuracy	99.2%
Cost	20096

\[\pi \cdot 0.5 - \sin^{-1} \left(\frac{1 - \left(v \cdot v\right) \cdot 5}{\mathsf{fma}\left(v, v, -1\right)}\right) \]

Alternative 2
Accuracy	99.2%
Cost	7232

\[\cos^{-1} \left(\frac{1 - \left(v \cdot v\right) \cdot 5}{v \cdot v + -1}\right) \]

Reproduce

herbie shell --seed 2023139 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))