Average Error: 0.6 → 0.6
Time: 4.6s
Precision: binary64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\sqrt[3]{{\left({\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)}^{\left(\sqrt{3}\right)}\right)}^{\left(\sqrt{3}\right)}}\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\sqrt[3]{{\left({\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)}^{\left(\sqrt{3}\right)}\right)}^{\left(\sqrt{3}\right)}}
(FPCore (v)
 :precision binary64
 (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))
↓
(FPCore (v)
 :precision binary64
 (cbrt
  (pow
   (pow (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))) (sqrt 3.0))
   (sqrt 3.0))))
double code(double v) {
	return acos((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0));
}
↓
double code(double v) {
	return cbrt(pow(pow(acos((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0)), sqrt(3.0)), sqrt(3.0)));
}

Error

Bits error versus v

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.6

    \[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
  2. Using strategy rm

  3. Applied add-cbrt-cube_binary641.5

    \[\leadsto \color{blue}{\sqrt[3]{\left(\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \cdot \cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right) \cdot \cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)}}\]

  4. Simplified1.5

    \[\leadsto \sqrt[3]{\color{blue}{{\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)}^{3}}}\]
  5. Using strategy rm

  6. Applied add-sqr-sqrt_binary640.6

    \[\leadsto \sqrt[3]{{\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)}^{\color{blue}{\left(\sqrt{3} \cdot \sqrt{3}\right)}}}\]

  7. Applied pow-unpow_binary640.6

    \[\leadsto \sqrt[3]{\color{blue}{{\left({\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)}^{\left(\sqrt{3}\right)}\right)}^{\left(\sqrt{3}\right)}}}\]

  8. Final simplification0.6

    \[\leadsto \sqrt[3]{{\left({\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)}^{\left(\sqrt{3}\right)}\right)}^{\left(\sqrt{3}\right)}}\]

Reproduce

herbie shell --seed 2020220 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))