Average Error: 0.5 → 0.5
Time: 16.4s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[e^{\log \left(\cos^{-1} \left(\frac{1 - \left(v \cdot v\right) \cdot 5}{v \cdot v - 1}\right)\right)}\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
e^{\log \left(\cos^{-1} \left(\frac{1 - \left(v \cdot v\right) \cdot 5}{v \cdot v - 1}\right)\right)}
double f(double v) {
        double r5420099 = 1.0;
        double r5420100 = 5.0;
        double r5420101 = v;
        double r5420102 = r5420101 * r5420101;
        double r5420103 = r5420100 * r5420102;
        double r5420104 = r5420099 - r5420103;
        double r5420105 = r5420102 - r5420099;
        double r5420106 = r5420104 / r5420105;
        double r5420107 = acos(r5420106);
        return r5420107;
}
↓
double f(double v) {
        double r5420108 = 1.0;
        double r5420109 = v;
        double r5420110 = r5420109 * r5420109;
        double r5420111 = 5.0;
        double r5420112 = r5420110 * r5420111;
        double r5420113 = r5420108 - r5420112;
        double r5420114 = r5420110 - r5420108;
        double r5420115 = r5420113 / r5420114;
        double r5420116 = acos(r5420115);
        double r5420117 = log(r5420116);
        double r5420118 = exp(r5420117);
        return r5420118;
}

Error

Bits error versus v

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Results

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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-exp-log0.5

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

  5. Applied pow10.5

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

  6. Applied log-pow0.5

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

  7. Final simplification0.5

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

Reproduce

herbie shell --seed 2019153 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))