Average Error: 0.5 → 0.6
Time: 29.0s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[e^{\sqrt{\log \left(\cos^{-1} \left(\frac{1 - \log \left(e^{5 \cdot \left(v \cdot v\right)}\right)}{v \cdot v - 1}\right)\right)} \cdot \sqrt{\log \left(\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right)}}\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
e^{\sqrt{\log \left(\cos^{-1} \left(\frac{1 - \log \left(e^{5 \cdot \left(v \cdot v\right)}\right)}{v \cdot v - 1}\right)\right)} \cdot \sqrt{\log \left(\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right)}}
double f(double v) {
        double r111139 = 1.0;
        double r111140 = 5.0;
        double r111141 = v;
        double r111142 = r111141 * r111141;
        double r111143 = r111140 * r111142;
        double r111144 = r111139 - r111143;
        double r111145 = r111142 - r111139;
        double r111146 = r111144 / r111145;
        double r111147 = acos(r111146);
        return r111147;
}
↓
double f(double v) {
        double r111148 = 1.0;
        double r111149 = 5.0;
        double r111150 = v;
        double r111151 = r111150 * r111150;
        double r111152 = r111149 * r111151;
        double r111153 = exp(r111152);
        double r111154 = log(r111153);
        double r111155 = r111148 - r111154;
        double r111156 = r111151 - r111148;
        double r111157 = r111155 / r111156;
        double r111158 = acos(r111157);
        double r111159 = log(r111158);
        double r111160 = sqrt(r111159);
        double r111161 = r111148 - r111152;
        double r111162 = r111161 / r111156;
        double r111163 = acos(r111162);
        double r111164 = log(r111163);
        double r111165 = sqrt(r111164);
        double r111166 = r111160 * r111165;
        double r111167 = exp(r111166);
        return r111167;
}

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-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 add-sqr-sqrt0.6

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

  7. Applied add-log-exp0.6

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

  8. Final simplification0.6

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

Reproduce

herbie shell --seed 2019323 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))