Average Error: 0.4 → 0.4
Time: 8.8s
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{1}{\frac{\frac{\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 \cdot 1 - \left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)}}{\sqrt{1 + 3 \cdot \left(v \cdot v\right)}} \cdot \left(1 - v \cdot v\right)}{1 - 5 \cdot \left(v \cdot v\right)}}\]
\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{1}{\frac{\frac{\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 \cdot 1 - \left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)}}{\sqrt{1 + 3 \cdot \left(v \cdot v\right)}} \cdot \left(1 - v \cdot v\right)}{1 - 5 \cdot \left(v \cdot v\right)}}
double f(double v, double t) {
        double r219070 = 1.0;
        double r219071 = 5.0;
        double r219072 = v;
        double r219073 = r219072 * r219072;
        double r219074 = r219071 * r219073;
        double r219075 = r219070 - r219074;
        double r219076 = atan2(1.0, 0.0);
        double r219077 = t;
        double r219078 = r219076 * r219077;
        double r219079 = 2.0;
        double r219080 = 3.0;
        double r219081 = r219080 * r219073;
        double r219082 = r219070 - r219081;
        double r219083 = r219079 * r219082;
        double r219084 = sqrt(r219083);
        double r219085 = r219078 * r219084;
        double r219086 = r219070 - r219073;
        double r219087 = r219085 * r219086;
        double r219088 = r219075 / r219087;
        return r219088;
}
↓
double f(double v, double t) {
        double r219089 = 1.0;
        double r219090 = atan2(1.0, 0.0);
        double r219091 = t;
        double r219092 = r219090 * r219091;
        double r219093 = 2.0;
        double r219094 = 1.0;
        double r219095 = r219094 * r219094;
        double r219096 = 3.0;
        double r219097 = v;
        double r219098 = r219097 * r219097;
        double r219099 = r219096 * r219098;
        double r219100 = r219099 * r219099;
        double r219101 = r219095 - r219100;
        double r219102 = r219093 * r219101;
        double r219103 = sqrt(r219102);
        double r219104 = r219092 * r219103;
        double r219105 = r219094 + r219099;
        double r219106 = sqrt(r219105);
        double r219107 = r219104 / r219106;
        double r219108 = r219094 - r219098;
        double r219109 = r219107 * r219108;
        double r219110 = 5.0;
        double r219111 = r219110 * r219098;
        double r219112 = r219094 - r219111;
        double r219113 = r219109 / r219112;
        double r219114 = r219089 / r219113;
        return r219114;
}

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 clear-num0.4

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

  5. Applied flip--0.4

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

  6. Applied associate-*r/0.4

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

  7. Applied sqrt-div0.4

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

  8. Applied associate-*r/0.4

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

  9. Final simplification0.4

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

Reproduce

herbie shell --seed 2020083 +o rules:numerics
(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)))))