Average Error: 0.5 → 0.5
Time: 5.2s
Precision: binary64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[{e}^{\left(\log \left(\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right)\right)}\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
{e}^{\left(\log \left(\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{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
 (pow E (log (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))))
double code(double v) {
	return ((double) acos((((double) (1.0 - ((double) (5.0 * ((double) (v * v)))))) / ((double) (((double) (v * v)) - 1.0)))));
}
↓
double code(double v) {
	return ((double) pow(((double) M_E), ((double) log(((double) acos((((double) (1.0 - ((double) (5.0 * ((double) (v * v)))))) / ((double) (((double) (v * v)) - 1.0)))))))));
}

Error

Bits error versus v

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Your Program's Arguments

Results

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Derivation

  1. Initial program Error: 0.5 bits

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

  3. Applied add-exp-logError: 0.5 bits

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

  5. Applied pow1Error: 0.5 bits

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

  6. Applied log-powError: 0.5 bits

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

  7. Applied exp-prodError: 0.5 bits

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

  8. SimplifiedError: 0.5 bits

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

  9. Final simplificationError: 0.5 bits

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

Reproduce

herbie shell --seed 2020203 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))