Average Error: 0.4 → 0.1
Time: 1.3m
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{\frac{\frac{\frac{1 - \left(5 \cdot v\right) \cdot v}{1 - v \cdot v}}{\pi}}{\sqrt{8 + -216 \cdot \left(\left(\left(v \cdot v\right) \cdot \left(v \cdot v\right)\right) \cdot \left(v \cdot v\right)\right)}}}{t} \cdot \sqrt{\left(4 - 2 \cdot \left(\left(v \cdot v\right) \cdot -6\right)\right) + \left(\left(v \cdot v\right) \cdot -6\right) \cdot \left(\left(v \cdot v\right) \cdot -6\right)}\]
\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{\frac{\frac{1 - \left(5 \cdot v\right) \cdot v}{1 - v \cdot v}}{\pi}}{\sqrt{8 + -216 \cdot \left(\left(\left(v \cdot v\right) \cdot \left(v \cdot v\right)\right) \cdot \left(v \cdot v\right)\right)}}}{t} \cdot \sqrt{\left(4 - 2 \cdot \left(\left(v \cdot v\right) \cdot -6\right)\right) + \left(\left(v \cdot v\right) \cdot -6\right) \cdot \left(\left(v \cdot v\right) \cdot -6\right)}
double f(double v, double t) {
        double r9279176 = 1.0;
        double r9279177 = 5.0;
        double r9279178 = v;
        double r9279179 = r9279178 * r9279178;
        double r9279180 = r9279177 * r9279179;
        double r9279181 = r9279176 - r9279180;
        double r9279182 = atan2(1.0, 0.0);
        double r9279183 = t;
        double r9279184 = r9279182 * r9279183;
        double r9279185 = 2.0;
        double r9279186 = 3.0;
        double r9279187 = r9279186 * r9279179;
        double r9279188 = r9279176 - r9279187;
        double r9279189 = r9279185 * r9279188;
        double r9279190 = sqrt(r9279189);
        double r9279191 = r9279184 * r9279190;
        double r9279192 = r9279176 - r9279179;
        double r9279193 = r9279191 * r9279192;
        double r9279194 = r9279181 / r9279193;
        return r9279194;
}
↓
double f(double v, double t) {
        double r9279195 = 1.0;
        double r9279196 = 5.0;
        double r9279197 = v;
        double r9279198 = r9279196 * r9279197;
        double r9279199 = r9279198 * r9279197;
        double r9279200 = r9279195 - r9279199;
        double r9279201 = r9279197 * r9279197;
        double r9279202 = r9279195 - r9279201;
        double r9279203 = r9279200 / r9279202;
        double r9279204 = atan2(1.0, 0.0);
        double r9279205 = r9279203 / r9279204;
        double r9279206 = 8.0;
        double r9279207 = -216.0;
        double r9279208 = r9279201 * r9279201;
        double r9279209 = r9279208 * r9279201;
        double r9279210 = r9279207 * r9279209;
        double r9279211 = r9279206 + r9279210;
        double r9279212 = sqrt(r9279211);
        double r9279213 = r9279205 / r9279212;
        double r9279214 = t;
        double r9279215 = r9279213 / r9279214;
        double r9279216 = 4.0;
        double r9279217 = 2.0;
        double r9279218 = -6.0;
        double r9279219 = r9279201 * r9279218;
        double r9279220 = r9279217 * r9279219;
        double r9279221 = r9279216 - r9279220;
        double r9279222 = r9279219 * r9279219;
        double r9279223 = r9279221 + r9279222;
        double r9279224 = sqrt(r9279223);
        double r9279225 = r9279215 * r9279224;
        return r9279225;
}

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. Simplified0.3

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

  4. Applied flip3-+0.3

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

  5. Applied sqrt-div0.3

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

  6. Applied associate-*r/0.3

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

  7. Applied associate-/r/0.3

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

  8. Simplified0.1

    \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{1 - v \cdot \left(5 \cdot v\right)}{1 - v \cdot v}}{\pi}}{\sqrt{8 + -216 \cdot \left(\left(\left(v \cdot v\right) \cdot \left(v \cdot v\right)\right) \cdot \left(v \cdot v\right)\right)}}}{t}} \cdot \sqrt{\left(-6 \cdot \left(v \cdot v\right)\right) \cdot \left(-6 \cdot \left(v \cdot v\right)\right) + \left(2 \cdot 2 - \left(-6 \cdot \left(v \cdot v\right)\right) \cdot 2\right)}\]

  9. Final simplification0.1

    \[\leadsto \frac{\frac{\frac{\frac{1 - \left(5 \cdot v\right) \cdot v}{1 - v \cdot v}}{\pi}}{\sqrt{8 + -216 \cdot \left(\left(\left(v \cdot v\right) \cdot \left(v \cdot v\right)\right) \cdot \left(v \cdot v\right)\right)}}}{t} \cdot \sqrt{\left(4 - 2 \cdot \left(\left(v \cdot v\right) \cdot -6\right)\right) + \left(\left(v \cdot v\right) \cdot -6\right) \cdot \left(\left(v \cdot v\right) \cdot -6\right)}\]

Reproduce

herbie shell --seed 2019151 
(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)))))