Average Error: 0.5 → 0.5
Time: 6.2s
Precision: binary64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{\sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)}}{1 - v \cdot v} \cdot \frac{\frac{\sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)}}{\pi \cdot t} \cdot \sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)}}{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}}\]
\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}
↓
\frac{\sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)}}{1 - v \cdot v} \cdot \frac{\frac{\sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)}}{\pi \cdot t} \cdot \sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)}}{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}}
double code(double v, double t) {
	return ((double) (((double) (1.0 - ((double) (5.0 * ((double) (v * v)))))) / ((double) (((double) (((double) (((double) M_PI) * t)) * ((double) sqrt(((double) (2.0 * ((double) (1.0 - ((double) (3.0 * ((double) (v * v)))))))))))) * ((double) (1.0 - ((double) (v * v))))))));
}
↓
double code(double v, double t) {
	return ((double) (((double) (((double) cbrt(((double) (1.0 - ((double) (5.0 * ((double) (v * v)))))))) / ((double) (1.0 - ((double) (v * v)))))) * ((double) (((double) (((double) (((double) cbrt(((double) (1.0 - ((double) (5.0 * ((double) (v * v)))))))) / ((double) (((double) M_PI) * t)))) * ((double) cbrt(((double) (1.0 - ((double) (5.0 * ((double) (v * v)))))))))) / ((double) sqrt(((double) (2.0 * ((double) (1.0 - ((double) (3.0 * ((double) (v * v))))))))))))));
}

Error

Bits error versus v

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.5

    \[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
  2. Using strategy rm

  3. Applied add-cube-cbrt0.5

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

  4. Applied associate-*r*0.5

    \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\left(\left(\left(\pi \cdot t\right) \cdot \left(\sqrt[3]{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}} \cdot \sqrt[3]{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}}\right)\right) \cdot \sqrt[3]{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}}\right)} \cdot \left(1 - v \cdot v\right)}\]
  5. Using strategy rm

  6. Applied add-cube-cbrt0.5

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

  7. Applied times-frac0.5

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

  8. Simplified0.5

    \[\leadsto \color{blue}{\left(\frac{\sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)}}{\pi \cdot t} \cdot \frac{\sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)}}{{\left(\sqrt[3]{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}}\right)}^{3}}\right)} \cdot \frac{\sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)}}{1 - v \cdot v}\]
  9. Using strategy rm

  10. Applied pow10.5

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

  11. Applied pow10.5

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

  12. Applied pow-prod-down0.5

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

  13. Simplified0.5

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

  14. Final simplification0.5

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

Reproduce

herbie shell --seed 2020162 
(FPCore (v t)
  :name "Falkner and Boettcher, Equation (20:1,3)"
  :precision binary64
  (/ (- 1.0 (* 5.0 (* v v))) (* (* (* PI t) (sqrt (* 2.0 (- 1.0 (* 3.0 (* v v)))))) (- 1.0 (* v v)))))