Average Error: 0.6 → 1.1
Time: 6.1s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[e^{\log \left(\cos^{-1} \left(\frac{\frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{\sqrt{1} + \left|v\right|}}{\frac{\sqrt{v \cdot v} - \sqrt{1}}{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}}\right)\right)}\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
e^{\log \left(\cos^{-1} \left(\frac{\frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{\sqrt{1} + \left|v\right|}}{\frac{\sqrt{v \cdot v} - \sqrt{1}}{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}}\right)\right)}
double code(double v) {
	return ((double) acos(((double) (((double) (1.0 - ((double) (5.0 * ((double) (v * v)))))) / ((double) (((double) (v * v)) - 1.0))))));
}
↓
double code(double v) {
	return ((double) exp(((double) log(((double) acos(((double) (((double) (((double) sqrt(((double) (1.0 - ((double) (5.0 * ((double) (v * v)))))))) / ((double) (((double) sqrt(1.0)) + ((double) fabs(v)))))) / ((double) (((double) (((double) sqrt(((double) (v * v)))) - ((double) sqrt(1.0)))) / ((double) sqrt(((double) (1.0 - ((double) (5.0 * ((double) (v * v))))))))))))))))));
}

Error

Bits error versus v

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.6

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

  3. Applied clear-num0.6

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

  5. Applied add-exp-log0.6

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

  7. Applied add-sqr-sqrt0.7

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

  8. Applied add-sqr-sqrt0.7

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

  9. Applied add-sqr-sqrt0.7

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

  10. Applied difference-of-squares1.0

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

  11. Applied times-frac1.0

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

  12. Applied associate-/r*1.1

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

  13. Simplified1.1

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

  14. Final simplification1.1

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

Reproduce

herbie shell --seed 2020114 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))