Average Error: 0.4 → 0.3
Time: 46.1s
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{\frac{\frac{\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\sqrt{\pi}}}{\sqrt{\pi}}}{t}}{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(6, \left(-v\right) \cdot v, 2\right)}, \left(-v\right) \cdot v, \sqrt{\mathsf{fma}\left(6, \left(-v\right) \cdot v, 2\right)}\right)}\]
\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{\frac{\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\sqrt{\pi}}}{\sqrt{\pi}}}{t}}{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(6, \left(-v\right) \cdot v, 2\right)}, \left(-v\right) \cdot v, \sqrt{\mathsf{fma}\left(6, \left(-v\right) \cdot v, 2\right)}\right)}
double f(double v, double t) {
        double r9039340 = 1.0;
        double r9039341 = 5.0;
        double r9039342 = v;
        double r9039343 = r9039342 * r9039342;
        double r9039344 = r9039341 * r9039343;
        double r9039345 = r9039340 - r9039344;
        double r9039346 = atan2(1.0, 0.0);
        double r9039347 = t;
        double r9039348 = r9039346 * r9039347;
        double r9039349 = 2.0;
        double r9039350 = 3.0;
        double r9039351 = r9039350 * r9039343;
        double r9039352 = r9039340 - r9039351;
        double r9039353 = r9039349 * r9039352;
        double r9039354 = sqrt(r9039353);
        double r9039355 = r9039348 * r9039354;
        double r9039356 = r9039340 - r9039343;
        double r9039357 = r9039355 * r9039356;
        double r9039358 = r9039345 / r9039357;
        return r9039358;
}
↓
double f(double v, double t) {
        double r9039359 = v;
        double r9039360 = r9039359 * r9039359;
        double r9039361 = -5.0;
        double r9039362 = 1.0;
        double r9039363 = fma(r9039360, r9039361, r9039362);
        double r9039364 = atan2(1.0, 0.0);
        double r9039365 = sqrt(r9039364);
        double r9039366 = r9039363 / r9039365;
        double r9039367 = r9039366 / r9039365;
        double r9039368 = t;
        double r9039369 = r9039367 / r9039368;
        double r9039370 = 6.0;
        double r9039371 = -r9039359;
        double r9039372 = r9039371 * r9039359;
        double r9039373 = 2.0;
        double r9039374 = fma(r9039370, r9039372, r9039373);
        double r9039375 = sqrt(r9039374);
        double r9039376 = fma(r9039375, r9039372, r9039375);
        double r9039377 = r9039369 / r9039376;
        return r9039377;
}

Error

Bits error versus v

Bits error versus t

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

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

  4. Applied add-sqr-sqrt1.0

    \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\color{blue}{\sqrt{\pi} \cdot \sqrt{\pi}}}}{t}}{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(6, \left(-v\right) \cdot v, 2\right)}, \left(-v\right) \cdot v, \sqrt{\mathsf{fma}\left(6, \left(-v\right) \cdot v, 2\right)}\right)}\]

  5. Applied associate-/r*0.3

    \[\leadsto \frac{\frac{\color{blue}{\frac{\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\sqrt{\pi}}}{\sqrt{\pi}}}}{t}}{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(6, \left(-v\right) \cdot v, 2\right)}, \left(-v\right) \cdot v, \sqrt{\mathsf{fma}\left(6, \left(-v\right) \cdot v, 2\right)}\right)}\]

  6. Final simplification0.3

    \[\leadsto \frac{\frac{\frac{\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\sqrt{\pi}}}{\sqrt{\pi}}}{t}}{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(6, \left(-v\right) \cdot v, 2\right)}, \left(-v\right) \cdot v, \sqrt{\mathsf{fma}\left(6, \left(-v\right) \cdot v, 2\right)}\right)}\]

Reproduce

herbie shell --seed 2019158 +o rules:numerics
(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)))))