Average Error: 0.5 → 0.5
Time: 5.0s
Precision: binary64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\cos^{-1} \left(\sqrt[3]{\frac{\left(1 - 5 \cdot \left(v \cdot v\right)\right) \cdot \left(\left(1 - 5 \cdot \left(v \cdot v\right)\right) \cdot \left(1 - 5 \cdot \left(v \cdot v\right)\right)\right)}{\left(v \cdot v - 1\right) \cdot \left(\left(v \cdot v - 1\right) \cdot \left(v \cdot v - 1\right)\right)}}\right)\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\cos^{-1} \left(\sqrt[3]{\frac{\left(1 - 5 \cdot \left(v \cdot v\right)\right) \cdot \left(\left(1 - 5 \cdot \left(v \cdot v\right)\right) \cdot \left(1 - 5 \cdot \left(v \cdot v\right)\right)\right)}{\left(v \cdot v - 1\right) \cdot \left(\left(v \cdot v - 1\right) \cdot \left(v \cdot v - 1\right)\right)}}\right)
(FPCore (v)
 :precision binary64
 (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))
↓
(FPCore (v)
 :precision binary64
 (acos
  (cbrt
   (/
    (*
     (- 1.0 (* 5.0 (* v v)))
     (* (- 1.0 (* 5.0 (* v v))) (- 1.0 (* 5.0 (* v v)))))
    (* (- (* v v) 1.0) (* (- (* v v) 1.0) (- (* v v) 1.0)))))))
double code(double v) {
	return acos((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0));
}
↓
double code(double v) {
	return acos(cbrt(((1.0 - (5.0 * (v * v))) * ((1.0 - (5.0 * (v * v))) * (1.0 - (5.0 * (v * v))))) / (((v * v) - 1.0) * (((v * v) - 1.0) * ((v * v) - 1.0)))));
}

Error

Bits error versus v

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.5

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

  3. Applied add-cbrt-cube_binary640.5

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

  4. Applied add-cbrt-cube_binary640.5

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

  5. Applied cbrt-undiv_binary640.5

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

  6. Final simplification0.5

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

Reproduce

herbie shell --seed 2021175 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))