Average Error: 0.5 → 0.6
Time: 24.5s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\cos^{-1} \left(\frac{1 - \log \left(e^{5 \cdot \left(v \cdot v\right)}\right)}{v \cdot v - 1}\right)\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\cos^{-1} \left(\frac{1 - \log \left(e^{5 \cdot \left(v \cdot v\right)}\right)}{v \cdot v - 1}\right)
double f(double v) {
        double r146118 = 1.0;
        double r146119 = 5.0;
        double r146120 = v;
        double r146121 = r146120 * r146120;
        double r146122 = r146119 * r146121;
        double r146123 = r146118 - r146122;
        double r146124 = r146121 - r146118;
        double r146125 = r146123 / r146124;
        double r146126 = acos(r146125);
        return r146126;
}
↓
double f(double v) {
        double r146127 = 1.0;
        double r146128 = 5.0;
        double r146129 = v;
        double r146130 = r146129 * r146129;
        double r146131 = r146128 * r146130;
        double r146132 = exp(r146131);
        double r146133 = log(r146132);
        double r146134 = r146127 - r146133;
        double r146135 = r146130 - r146127;
        double r146136 = r146134 / r146135;
        double r146137 = acos(r146136);
        return r146137;
}

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-log-exp0.6

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

  4. Final simplification0.6

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

Reproduce

herbie shell --seed 2019303 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))