Falkner and Boettcher, Appendix B, 1

Average Error: 0.5 → 0.5

Time: 12.5s

Precision: binary64

Cost: 20352

Math

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

↓

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

FPCore

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

↓

(FPCore (v)
 :precision binary64
 (/
  1.0
  (/ 1.0 (- (* PI 0.5) (asin (/ (fma v (* v 5.0) -1.0) (- 1.0 (* v v))))))))

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 / ((((double) M_PI) * 0.5) - asin((fma(v, (v * 5.0), -1.0) / (1.0 - (v * v))))));
}

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 / Float64(Float64(pi * 0.5) - asin(Float64(fma(v, Float64(v * 5.0), -1.0) / Float64(1.0 - Float64(v * v)))))))
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[(N[(Pi * 0.5), $MachinePrecision] - N[ArcSin[N[(N[(v * N[(v * 5.0), $MachinePrecision] + -1.0), $MachinePrecision] / N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.5

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

2. Applied egg-rr 0.5

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

3. Applied egg-rr 0.6

   \[\leadsto \pi \cdot 0.5 - \sin^{-1} \color{blue}{\left(\left(-1 - v \cdot \left(v \cdot -5\right)\right) \cdot \frac{1}{1 - v \cdot v}\right)} \]

4. Simplified 0.5

   \[\leadsto \pi \cdot 0.5 - \sin^{-1} \color{blue}{\left(\frac{-1 + \left(v \cdot v\right) \cdot 5}{1 - v \cdot v}\right)} \]
   Proof

   [Start] 0.6	\[ \pi \cdot 0.5 - \sin^{-1} \left(\left(-1 - v \cdot \left(v \cdot -5\right)\right) \cdot \frac{1}{1 - v \cdot v}\right) \]
   associate-*r/ [=>] 0.5	\[ \pi \cdot 0.5 - \sin^{-1} \color{blue}{\left(\frac{\left(-1 - v \cdot \left(v \cdot -5\right)\right) \cdot 1}{1 - v \cdot v}\right)} \]
   *-rgt-identity [=>] 0.5	\[ \pi \cdot 0.5 - \sin^{-1} \left(\frac{\color{blue}{-1 - v \cdot \left(v \cdot -5\right)}}{1 - v \cdot v}\right) \]
   associate-*r* [=>] 0.5	\[ \pi \cdot 0.5 - \sin^{-1} \left(\frac{-1 - \color{blue}{\left(v \cdot v\right) \cdot -5}}{1 - v \cdot v}\right) \]
   unpow2 [<=] 0.5	\[ \pi \cdot 0.5 - \sin^{-1} \left(\frac{-1 - \color{blue}{{v}^{2}} \cdot -5}{1 - v \cdot v}\right) \]
   *-commutative [<=] 0.5	\[ \pi \cdot 0.5 - \sin^{-1} \left(\frac{-1 - \color{blue}{-5 \cdot {v}^{2}}}{1 - v \cdot v}\right) \]
   cancel-sign-sub-inv [=>] 0.5	\[ \pi \cdot 0.5 - \sin^{-1} \left(\frac{\color{blue}{-1 + \left(--5\right) \cdot {v}^{2}}}{1 - v \cdot v}\right) \]
   metadata-eval [=>] 0.5	\[ \pi \cdot 0.5 - \sin^{-1} \left(\frac{-1 + \color{blue}{5} \cdot {v}^{2}}{1 - v \cdot v}\right) \]
   unpow2 [=>] 0.5	\[ \pi \cdot 0.5 - \sin^{-1} \left(\frac{-1 + 5 \cdot \color{blue}{\left(v \cdot v\right)}}{1 - v \cdot v}\right) \]
   *-commutative [=>] 0.5	\[ \pi \cdot 0.5 - \sin^{-1} \left(\frac{-1 + \color{blue}{\left(v \cdot v\right) \cdot 5}}{1 - v \cdot v}\right) \]

5. Applied egg-rr 0.5

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

6. Applied egg-rr 0.5

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

7. Simplified 0.5

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

   [Start] 0.5	\[ \frac{1}{\frac{1}{\pi \cdot 0.5 + \left(-\sin^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot 5, -1\right)}{1 - v \cdot v}\right)\right)}} \]
   sub-neg [<=] 0.5	\[ \frac{1}{\frac{1}{\color{blue}{\pi \cdot 0.5 - \sin^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot 5, -1\right)}{1 - v \cdot v}\right)}}} \]

8. Final simplification 0.5

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

Alternatives

Alternative 1
Error	0.5
Cost	13824

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

Alternative 2
Error	0.5
Cost	7488

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

Alternative 3
Error	0.9
Cost	7232

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

Alternative 4
Error	0.5
Cost	7232

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

Alternative 5
Error	1.2
Cost	6848

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

Alternative 6
Error	1.3
Cost	6464

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

Reproduce

herbie shell --seed 2023032 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))