Average Error: 0.5 → 0.3
Time: 22.6s
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{\frac{1}{\sqrt{2} \cdot \pi}}{t} - \left(\frac{\left(\left(v \cdot v\right) \cdot \left(v \cdot v\right)\right) \cdot \frac{53}{8}}{t \cdot \left(\sqrt{2} \cdot \pi\right)} + \frac{\frac{5}{2} \cdot \left(v \cdot v\right)}{t \cdot \left(\sqrt{2} \cdot \pi\right)}\right)\]
\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{\frac{1}{\sqrt{2} \cdot \pi}}{t} - \left(\frac{\left(\left(v \cdot v\right) \cdot \left(v \cdot v\right)\right) \cdot \frac{53}{8}}{t \cdot \left(\sqrt{2} \cdot \pi\right)} + \frac{\frac{5}{2} \cdot \left(v \cdot v\right)}{t \cdot \left(\sqrt{2} \cdot \pi\right)}\right)
double f(double v, double t) {
        double r3088779 = 1.0;
        double r3088780 = 5.0;
        double r3088781 = v;
        double r3088782 = r3088781 * r3088781;
        double r3088783 = r3088780 * r3088782;
        double r3088784 = r3088779 - r3088783;
        double r3088785 = atan2(1.0, 0.0);
        double r3088786 = t;
        double r3088787 = r3088785 * r3088786;
        double r3088788 = 2.0;
        double r3088789 = 3.0;
        double r3088790 = r3088789 * r3088782;
        double r3088791 = r3088779 - r3088790;
        double r3088792 = r3088788 * r3088791;
        double r3088793 = sqrt(r3088792);
        double r3088794 = r3088787 * r3088793;
        double r3088795 = r3088779 - r3088782;
        double r3088796 = r3088794 * r3088795;
        double r3088797 = r3088784 / r3088796;
        return r3088797;
}
↓
double f(double v, double t) {
        double r3088798 = 1.0;
        double r3088799 = 2.0;
        double r3088800 = sqrt(r3088799);
        double r3088801 = atan2(1.0, 0.0);
        double r3088802 = r3088800 * r3088801;
        double r3088803 = r3088798 / r3088802;
        double r3088804 = t;
        double r3088805 = r3088803 / r3088804;
        double r3088806 = v;
        double r3088807 = r3088806 * r3088806;
        double r3088808 = r3088807 * r3088807;
        double r3088809 = 6.625;
        double r3088810 = r3088808 * r3088809;
        double r3088811 = r3088804 * r3088802;
        double r3088812 = r3088810 / r3088811;
        double r3088813 = 2.5;
        double r3088814 = r3088813 * r3088807;
        double r3088815 = r3088814 / r3088811;
        double r3088816 = r3088812 + r3088815;
        double r3088817 = r3088805 - r3088816;
        return r3088817;
}

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. Taylor expanded around 0 0.5

    \[\leadsto \color{blue}{\frac{1}{t \cdot \left(\sqrt{2} \cdot \pi\right)} - \left(\frac{53}{8} \cdot \frac{{v}^{4}}{t \cdot \left(\sqrt{2} \cdot \pi\right)} + \frac{5}{2} \cdot \frac{{v}^{2}}{t \cdot \left(\sqrt{2} \cdot \pi\right)}\right)}\]

  3. Simplified0.5

    \[\leadsto \color{blue}{\frac{\frac{1}{t}}{\sqrt{2} \cdot \pi} - \left(\frac{\frac{5}{2} \cdot \left(v \cdot v\right)}{\left(\sqrt{2} \cdot \pi\right) \cdot t} + \frac{\frac{53}{8} \cdot \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right)\right)}{\left(\sqrt{2} \cdot \pi\right) \cdot t}\right)}\]
  4. Using strategy rm

  5. Applied associate-/l/0.5

    \[\leadsto \color{blue}{\frac{1}{\left(\sqrt{2} \cdot \pi\right) \cdot t}} - \left(\frac{\frac{5}{2} \cdot \left(v \cdot v\right)}{\left(\sqrt{2} \cdot \pi\right) \cdot t} + \frac{\frac{53}{8} \cdot \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right)\right)}{\left(\sqrt{2} \cdot \pi\right) \cdot t}\right)\]
  6. Using strategy rm

  7. Applied associate-/r*0.3

    \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{2} \cdot \pi}}{t}} - \left(\frac{\frac{5}{2} \cdot \left(v \cdot v\right)}{\left(\sqrt{2} \cdot \pi\right) \cdot t} + \frac{\frac{53}{8} \cdot \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right)\right)}{\left(\sqrt{2} \cdot \pi\right) \cdot t}\right)\]

  8. Final simplification0.3

    \[\leadsto \frac{\frac{1}{\sqrt{2} \cdot \pi}}{t} - \left(\frac{\left(\left(v \cdot v\right) \cdot \left(v \cdot v\right)\right) \cdot \frac{53}{8}}{t \cdot \left(\sqrt{2} \cdot \pi\right)} + \frac{\frac{5}{2} \cdot \left(v \cdot v\right)}{t \cdot \left(\sqrt{2} \cdot \pi\right)}\right)\]

Reproduce

herbie shell --seed 2019152 
(FPCore (v t)
  :name "Falkner and Boettcher, Equation (20:1,3)"
  (/ (- 1 (* 5 (* v v))) (* (* (* PI t) (sqrt (* 2 (- 1 (* 3 (* v v)))))) (- 1 (* v v)))))