Falkner and Boettcher, Appendix B, 1

Average Accuracy: 99.2% → 99.2%

Time: 22.5s

Precision: binary64

Cost: 32896

Math

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

↓

\[-1 + {\left(\sqrt{1 + \cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)}^{2} \]

FPCore

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

↓

(FPCore (v)
 :precision binary64
 (+
  -1.0
  (pow (sqrt (+ 1.0 (acos (/ (fma v (* v -5.0) 1.0) (fma v v -1.0))))) 2.0)))

C

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

↓

double code(double v) {
	return -1.0 + pow(sqrt((1.0 + acos((fma(v, (v * -5.0), 1.0) / fma(v, v, -1.0))))), 2.0);
}

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 Float64(-1.0 + (sqrt(Float64(1.0 + acos(Float64(fma(v, Float64(v * -5.0), 1.0) / fma(v, v, -1.0))))) ^ 2.0))
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[(-1.0 + N[Power[N[Sqrt[N[(1.0 + N[ArcCos[N[(N[(v * N[(v * -5.0), $MachinePrecision] + 1.0), $MachinePrecision] / N[(v * v + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $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%

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

3. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{{\left(\sqrt{1 + \cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)}^{2}} - 1 \]
   Proof

   [Start] 99.2	\[ e^{\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 + \left(v \cdot v\right) \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)} - 1 \]
   add-sqr-sqrt [=>] 99.2	\[ \color{blue}{\sqrt{e^{\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 + \left(v \cdot v\right) \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}} \cdot \sqrt{e^{\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 + \left(v \cdot v\right) \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}}} - 1 \]
   pow2 [=>] 99.2	\[ \color{blue}{{\left(\sqrt{e^{\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 + \left(v \cdot v\right) \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}}\right)}^{2}} - 1 \]
   log1p-udef [=>] 99.2	\[ {\left(\sqrt{e^{\color{blue}{\log \left(1 + \cos^{-1} \left(\frac{1 + \left(v \cdot v\right) \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}}}\right)}^{2} - 1 \]
   add-exp-log [<=] 99.2	\[ {\left(\sqrt{\color{blue}{1 + \cos^{-1} \left(\frac{1 + \left(v \cdot v\right) \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)}}\right)}^{2} - 1 \]
   +-commutative [=>] 99.2	\[ {\left(\sqrt{1 + \cos^{-1} \left(\frac{\color{blue}{\left(v \cdot v\right) \cdot -5 + 1}}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)}^{2} - 1 \]
   associate-*l* [=>] 99.2	\[ {\left(\sqrt{1 + \cos^{-1} \left(\frac{\color{blue}{v \cdot \left(v \cdot -5\right)} + 1}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)}^{2} - 1 \]
   fma-def [=>] 99.2	\[ {\left(\sqrt{1 + \cos^{-1} \left(\frac{\color{blue}{\mathsf{fma}\left(v, v \cdot -5, 1\right)}}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)}^{2} - 1 \]

4. Final simplification 99.2%

   \[\leadsto -1 + {\left(\sqrt{1 + \cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)}^{2} \]

Alternatives

Alternative 1
Accuracy	99.2%
Cost	7488

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

Alternative 2
Accuracy	99.2%
Cost	7232

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

Alternative 3
Accuracy	98.1%
Cost	6464

\[\cos^{-1} -1 \]

Reproduce

herbie shell --seed 2023131 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))