Average Error: 0.5 → 0.1
Time: 5.1s
Precision: binary64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\left(\sqrt[3]{1 - v \cdot \left(v \cdot 5\right)} \cdot \frac{\frac{\sqrt[3]{1 - v \cdot \left(v \cdot 5\right)}}{\sqrt{2 \cdot \left(1 - v \cdot \left(v \cdot 3\right)\right)} \cdot \pi}}{t}\right) \cdot \frac{\sqrt[3]{1 - v \cdot \left(v \cdot 5\right)}}{1 - v \cdot v}\]
\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)}
↓
\left(\sqrt[3]{1 - v \cdot \left(v \cdot 5\right)} \cdot \frac{\frac{\sqrt[3]{1 - v \cdot \left(v \cdot 5\right)}}{\sqrt{2 \cdot \left(1 - v \cdot \left(v \cdot 3\right)\right)} \cdot \pi}}{t}\right) \cdot \frac{\sqrt[3]{1 - v \cdot \left(v \cdot 5\right)}}{1 - v \cdot v}
double code(double v, double t) {
	return (((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) (v * ((double) (v * 5.0)))))))) * ((((double) cbrt(((double) (1.0 - ((double) (v * ((double) (v * 5.0)))))))) / ((double) (((double) sqrt(((double) (2.0 * ((double) (1.0 - ((double) (v * ((double) (v * 3.0)))))))))) * ((double) M_PI)))) / t))) * (((double) cbrt(((double) (1.0 - ((double) (v * ((double) (v * 5.0)))))))) / ((double) (1.0 - ((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

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

  4. Applied times-frac0.5

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

  5. Simplified0.5

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

  6. Simplified0.5

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

  8. Applied div-inv0.5

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

  9. Applied associate-*l*0.5

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

  10. Simplified0.5

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

  12. Applied associate-/r*0.1

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

  13. Simplified0.1

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

  14. Final simplification0.1

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

Reproduce

herbie shell --seed 2020196 
(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)))))