Falkner and Boettcher, Appendix B, 1

Average Accuracy: 99.2% → 99.2%

Time: 23.8s

Precision: binary64

Cost: 20032

Math

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

↓

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

FPCore

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

↓

(FPCore (v)
 :precision binary64
 (+ -1.0 (+ 1.0 (acos (/ (fma v (* v -5.0) 1.0) (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 -1.0 + (1.0 + acos((fma(v, (v * -5.0), 1.0) / 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 Float64(-1.0 + Float64(1.0 + acos(Float64(fma(v, Float64(v * -5.0), 1.0) / 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[(-1.0 + 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]

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%

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

   [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	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right)\right)} \]
   sub-neg [=>] 99.2	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{\color{blue}{1 + \left(-5 \cdot \left(v \cdot v\right)\right)}}{v \cdot v - 1}\right)\right)\right) \]
   +-commutative [=>] 99.2	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{\color{blue}{\left(-5 \cdot \left(v \cdot v\right)\right) + 1}}{v \cdot v - 1}\right)\right)\right) \]
   *-commutative [=>] 99.2	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{\left(-\color{blue}{\left(v \cdot v\right) \cdot 5}\right) + 1}{v \cdot v - 1}\right)\right)\right) \]
   distribute-rgt-neg-in [=>] 99.2	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{\color{blue}{\left(v \cdot v\right) \cdot \left(-5\right)} + 1}{v \cdot v - 1}\right)\right)\right) \]
   fma-def [=>] 99.2	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{\color{blue}{\mathsf{fma}\left(v \cdot v, -5, 1\right)}}{v \cdot v - 1}\right)\right)\right) \]
   metadata-eval [=>] 99.2	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{\mathsf{fma}\left(v \cdot v, \color{blue}{-5}, 1\right)}{v \cdot v - 1}\right)\right)\right) \]
   fma-neg [=>] 99.2	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\color{blue}{\mathsf{fma}\left(v, v, -1\right)}}\right)\right)\right) \]
   metadata-eval [=>] 99.2	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\mathsf{fma}\left(v, v, \color{blue}{-1}\right)}\right)\right)\right) \]

3. Applied egg-rr 95.0%

   \[\leadsto \mathsf{expm1}\left(\color{blue}{{\left(\sqrt[3]{\mathsf{log1p}\left(\cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}\right)}^{3}}\right) \]
   Step-by-step derivation

   [Start] 99.2	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right) \]
   add-cube-cbrt [=>] 95.0	\[ \mathsf{expm1}\left(\color{blue}{\left(\sqrt[3]{\mathsf{log1p}\left(\cos^{-1} \left(\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)} \cdot \sqrt[3]{\mathsf{log1p}\left(\cos^{-1} \left(\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}\right) \cdot \sqrt[3]{\mathsf{log1p}\left(\cos^{-1} \left(\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}}\right) \]
   pow3 [=>] 95.0	\[ \mathsf{expm1}\left(\color{blue}{{\left(\sqrt[3]{\mathsf{log1p}\left(\cos^{-1} \left(\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}\right)}^{3}}\right) \]
   fma-udef [=>] 95.0	\[ \mathsf{expm1}\left({\left(\sqrt[3]{\mathsf{log1p}\left(\cos^{-1} \left(\frac{\color{blue}{\left(v \cdot v\right) \cdot -5 + 1}}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}\right)}^{3}\right) \]
   associate-*l* [=>] 95.0	\[ \mathsf{expm1}\left({\left(\sqrt[3]{\mathsf{log1p}\left(\cos^{-1} \left(\frac{\color{blue}{v \cdot \left(v \cdot -5\right)} + 1}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}\right)}^{3}\right) \]
   fma-def [=>] 95.0	\[ \mathsf{expm1}\left({\left(\sqrt[3]{\mathsf{log1p}\left(\cos^{-1} \left(\frac{\color{blue}{\mathsf{fma}\left(v, v \cdot -5, 1\right)}}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}\right)}^{3}\right) \]

4. Applied egg-rr 96.8%

   \[\leadsto \mathsf{expm1}\left({\color{blue}{\left({\left({\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right)}^{0.16666666666666666}\right)}^{2}\right)}}^{3}\right) \]
   Step-by-step derivation

   [Start] 95.0	\[ \mathsf{expm1}\left({\left(\sqrt[3]{\mathsf{log1p}\left(\cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}\right)}^{3}\right) \]
   add-sqr-sqrt [=>] 96.8	\[ \mathsf{expm1}\left({\color{blue}{\left(\sqrt{\sqrt[3]{\mathsf{log1p}\left(\cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}} \cdot \sqrt{\sqrt[3]{\mathsf{log1p}\left(\cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}}\right)}}^{3}\right) \]
   pow2 [=>] 96.8	\[ \mathsf{expm1}\left({\color{blue}{\left({\left(\sqrt{\sqrt[3]{\mathsf{log1p}\left(\cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}}\right)}^{2}\right)}}^{3}\right) \]
   pow1/3 [=>] 96.8	\[ \mathsf{expm1}\left({\left({\left(\sqrt{\color{blue}{{\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right)}^{0.3333333333333333}}}\right)}^{2}\right)}^{3}\right) \]
   sqrt-pow1 [=>] 96.8	\[ \mathsf{expm1}\left({\left({\color{blue}{\left({\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right)}^{\left(\frac{0.3333333333333333}{2}\right)}\right)}}^{2}\right)}^{3}\right) \]
   metadata-eval [=>] 96.8	\[ \mathsf{expm1}\left({\left({\left({\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right)}^{\color{blue}{0.16666666666666666}}\right)}^{2}\right)}^{3}\right) \]

5. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{\left(1 + \cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right) - 1} \]
   Step-by-step derivation

   [Start] 96.8	\[ \mathsf{expm1}\left({\left({\left({\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right)}^{0.16666666666666666}\right)}^{2}\right)}^{3}\right) \]
   pow-pow [=>] 96.8	\[ \mathsf{expm1}\left({\color{blue}{\left({\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right)}^{\left(0.16666666666666666 \cdot 2\right)}\right)}}^{3}\right) \]
   metadata-eval [=>] 96.8	\[ \mathsf{expm1}\left({\left({\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right)}^{\color{blue}{0.3333333333333333}}\right)}^{3}\right) \]
   pow1/3 [<=] 95.0	\[ \mathsf{expm1}\left({\color{blue}{\left(\sqrt[3]{\mathsf{log1p}\left(\cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}\right)}}^{3}\right) \]
   rem-cube-cbrt [=>] 99.2	\[ \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(\cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}\right) \]
   expm1-udef [=>] 99.2	\[ \color{blue}{e^{\mathsf{log1p}\left(\cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)} - 1} \]
   log1p-udef [=>] 99.2	\[ e^{\color{blue}{\log \left(1 + \cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}} - 1 \]
   add-exp-log [<=] 99.2	\[ \color{blue}{\left(1 + \cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)} - 1 \]

6. Final simplification 99.2%

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

Alternatives

Alternative 1
Accuracy	99.2%
Cost	7232

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

Alternative 2
Accuracy	97.9%
Cost	6464

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

Reproduce

herbie shell --seed 2023159 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))