Average Error: 0.4 → 0.3
Time: 14.2s
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(t \cdot \left(\sqrt{2} \cdot \pi\right)\right) \cdot \sqrt{{1}^{3} - {\left(3 \cdot \left(v \cdot v\right)\right)}^{3}}}}{{1}^{3} - {v}^{6}} \cdot \left(\sqrt{1 \cdot 1 + \left(\left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right) + 1 \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)} \cdot \left(1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)\right)\right)\]
\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 - 5 \cdot \left(v \cdot v\right)}{\left(t \cdot \left(\sqrt{2} \cdot \pi\right)\right) \cdot \sqrt{{1}^{3} - {\left(3 \cdot \left(v \cdot v\right)\right)}^{3}}}}{{1}^{3} - {v}^{6}} \cdot \left(\sqrt{1 \cdot 1 + \left(\left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right) + 1 \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)} \cdot \left(1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)\right)\right)
double f(double v, double t) {
        double r334042 = 1.0;
        double r334043 = 5.0;
        double r334044 = v;
        double r334045 = r334044 * r334044;
        double r334046 = r334043 * r334045;
        double r334047 = r334042 - r334046;
        double r334048 = atan2(1.0, 0.0);
        double r334049 = t;
        double r334050 = r334048 * r334049;
        double r334051 = 2.0;
        double r334052 = 3.0;
        double r334053 = r334052 * r334045;
        double r334054 = r334042 - r334053;
        double r334055 = r334051 * r334054;
        double r334056 = sqrt(r334055);
        double r334057 = r334050 * r334056;
        double r334058 = r334042 - r334045;
        double r334059 = r334057 * r334058;
        double r334060 = r334047 / r334059;
        return r334060;
}
↓
double f(double v, double t) {
        double r334061 = 1.0;
        double r334062 = 5.0;
        double r334063 = v;
        double r334064 = r334063 * r334063;
        double r334065 = r334062 * r334064;
        double r334066 = r334061 - r334065;
        double r334067 = t;
        double r334068 = 2.0;
        double r334069 = sqrt(r334068);
        double r334070 = atan2(1.0, 0.0);
        double r334071 = r334069 * r334070;
        double r334072 = r334067 * r334071;
        double r334073 = 3.0;
        double r334074 = pow(r334061, r334073);
        double r334075 = 3.0;
        double r334076 = r334075 * r334064;
        double r334077 = pow(r334076, r334073);
        double r334078 = r334074 - r334077;
        double r334079 = sqrt(r334078);
        double r334080 = r334072 * r334079;
        double r334081 = r334066 / r334080;
        double r334082 = 6.0;
        double r334083 = pow(r334063, r334082);
        double r334084 = r334074 - r334083;
        double r334085 = r334081 / r334084;
        double r334086 = r334061 * r334061;
        double r334087 = r334076 * r334076;
        double r334088 = r334061 * r334076;
        double r334089 = r334087 + r334088;
        double r334090 = r334086 + r334089;
        double r334091 = sqrt(r334090);
        double r334092 = r334064 * r334064;
        double r334093 = r334061 * r334064;
        double r334094 = r334092 + r334093;
        double r334095 = r334086 + r334094;
        double r334096 = r334091 * r334095;
        double r334097 = r334085 * r334096;
        return r334097;
}

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 0.4

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

  3. Applied sqrt-prod0.4

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

  4. Applied associate-*r*0.4

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

  5. Simplified0.3

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

  7. Applied flip3--0.3

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

  8. Applied flip3--0.3

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

  9. Applied sqrt-div0.3

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

  10. Applied associate-*r/0.3

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

  11. Applied frac-times0.3

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

  12. Applied associate-/r/0.3

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

  13. Simplified0.3

    \[\leadsto \color{blue}{\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(t \cdot \left(\sqrt{2} \cdot \pi\right)\right) \cdot \sqrt{{1}^{3} - {\left(3 \cdot \left(v \cdot v\right)\right)}^{3}}}}{{1}^{3} - {v}^{6}}} \cdot \left(\sqrt{1 \cdot 1 + \left(\left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right) + 1 \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)} \cdot \left(1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)\right)\right)\]

  14. Final simplification0.3

    \[\leadsto \frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(t \cdot \left(\sqrt{2} \cdot \pi\right)\right) \cdot \sqrt{{1}^{3} - {\left(3 \cdot \left(v \cdot v\right)\right)}^{3}}}}{{1}^{3} - {v}^{6}} \cdot \left(\sqrt{1 \cdot 1 + \left(\left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right) + 1 \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)} \cdot \left(1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)\right)\right)\]

Reproduce

herbie shell --seed 2019350 
(FPCore (v t)
  :name "Falkner and Boettcher, Equation (20:1,3)"
  :precision binary64
  (/ (- 1 (* 5 (* v v))) (* (* (* PI t) (sqrt (* 2 (- 1 (* 3 (* v v)))))) (- 1 (* v v)))))