Average Error: 0.6 → 0.7
Time: 4.9s
Precision: binary64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\frac{\pi}{2} - \sin^{-1} \left(\frac{\sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)} \cdot \sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)}}{\sqrt[3]{v \cdot v - 1} \cdot \sqrt[3]{v \cdot v - 1}} \cdot \frac{\sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)}}{\sqrt[3]{v \cdot v - 1}}\right)\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\frac{\pi}{2} - \sin^{-1} \left(\frac{\sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)} \cdot \sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)}}{\sqrt[3]{v \cdot v - 1} \cdot \sqrt[3]{v \cdot v - 1}} \cdot \frac{\sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)}}{\sqrt[3]{v \cdot v - 1}}\right)
(FPCore (v)
 :precision binary64
 (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))
↓
(FPCore (v)
 :precision binary64
 (-
  (/ PI 2.0)
  (asin
   (*
    (/
     (* (cbrt (- 1.0 (* 5.0 (* v v)))) (cbrt (- 1.0 (* 5.0 (* v v)))))
     (* (cbrt (- (* v v) 1.0)) (cbrt (- (* v v) 1.0))))
    (/ (cbrt (- 1.0 (* 5.0 (* v v)))) (cbrt (- (* v v) 1.0)))))))
double code(double v) {
	return acos((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0));
}
↓
double code(double v) {
	return (((double) M_PI) / 2.0) - asin(((cbrt(1.0 - (5.0 * (v * v))) * cbrt(1.0 - (5.0 * (v * v)))) / (cbrt((v * v) - 1.0) * cbrt((v * v) - 1.0))) * (cbrt(1.0 - (5.0 * (v * v))) / cbrt((v * v) - 1.0)));
}

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 acos-asin_binary640.6

    \[\leadsto \color{blue}{\frac{\pi}{2} - \sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)}\]
  4. Using strategy rm

  5. Applied add-cube-cbrt_binary640.6

    \[\leadsto \frac{\pi}{2} - \sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\left(\sqrt[3]{v \cdot v - 1} \cdot \sqrt[3]{v \cdot v - 1}\right) \cdot \sqrt[3]{v \cdot v - 1}}}\right)\]

  6. Applied add-cube-cbrt_binary640.7

    \[\leadsto \frac{\pi}{2} - \sin^{-1} \left(\frac{\color{blue}{\left(\sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)} \cdot \sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)}\right) \cdot \sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)}}}{\left(\sqrt[3]{v \cdot v - 1} \cdot \sqrt[3]{v \cdot v - 1}\right) \cdot \sqrt[3]{v \cdot v - 1}}\right)\]

  7. Applied times-frac_binary640.7

    \[\leadsto \frac{\pi}{2} - \sin^{-1} \color{blue}{\left(\frac{\sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)} \cdot \sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)}}{\sqrt[3]{v \cdot v - 1} \cdot \sqrt[3]{v \cdot v - 1}} \cdot \frac{\sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)}}{\sqrt[3]{v \cdot v - 1}}\right)}\]

  8. Final simplification0.7

    \[\leadsto \frac{\pi}{2} - \sin^{-1} \left(\frac{\sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)} \cdot \sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)}}{\sqrt[3]{v \cdot v - 1} \cdot \sqrt[3]{v \cdot v - 1}} \cdot \frac{\sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)}}{\sqrt[3]{v \cdot v - 1}}\right)\]

Reproduce

herbie shell --seed 2020232 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))