Average Error: 3.5 → 2.3
Time: 14.2s
Precision: 64
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
↓
\[\frac{x}{x + y \cdot e^{2 \cdot \mathsf{fma}\left(z \cdot \sqrt{t + a}, \frac{1}{t}, -\left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}
↓
\frac{x}{x + y \cdot e^{2 \cdot \mathsf{fma}\left(z \cdot \sqrt{t + a}, \frac{1}{t}, -\left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r248 = x;
        double r249 = y;
        double r250 = 2.0;
        double r251 = z;
        double r252 = t;
        double r253 = a;
        double r254 = r252 + r253;
        double r255 = sqrt(r254);
        double r256 = r251 * r255;
        double r257 = r256 / r252;
        double r258 = b;
        double r259 = c;
        double r260 = r258 - r259;
        double r261 = 5.0;
        double r262 = 6.0;
        double r263 = r261 / r262;
        double r264 = r253 + r263;
        double r265 = 3.0;
        double r266 = r252 * r265;
        double r267 = r250 / r266;
        double r268 = r264 - r267;
        double r269 = r260 * r268;
        double r270 = r257 - r269;
        double r271 = r250 * r270;
        double r272 = exp(r271);
        double r273 = r249 * r272;
        double r274 = r248 + r273;
        double r275 = r248 / r274;
        return r275;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r276 = x;
        double r277 = y;
        double r278 = 2.0;
        double r279 = z;
        double r280 = t;
        double r281 = a;
        double r282 = r280 + r281;
        double r283 = sqrt(r282);
        double r284 = r279 * r283;
        double r285 = 1.0;
        double r286 = r285 / r280;
        double r287 = b;
        double r288 = c;
        double r289 = r287 - r288;
        double r290 = 5.0;
        double r291 = 6.0;
        double r292 = r290 / r291;
        double r293 = r281 + r292;
        double r294 = 3.0;
        double r295 = r280 * r294;
        double r296 = r278 / r295;
        double r297 = r293 - r296;
        double r298 = r289 * r297;
        double r299 = -r298;
        double r300 = fma(r284, r286, r299);
        double r301 = r278 * r300;
        double r302 = exp(r301);
        double r303 = r277 * r302;
        double r304 = r276 + r303;
        double r305 = r276 / r304;
        return r305;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 3.5

    \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
  2. Using strategy rm

  3. Applied div-inv3.5

    \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\color{blue}{\left(z \cdot \sqrt{t + a}\right) \cdot \frac{1}{t}} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]

  4. Applied fma-neg2.3

    \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\mathsf{fma}\left(z \cdot \sqrt{t + a}, \frac{1}{t}, -\left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}}\]

  5. Final simplification2.3

    \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \mathsf{fma}\left(z \cdot \sqrt{t + a}, \frac{1}{t}, -\left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]

Reproduce

herbie shell --seed 2020025 +o rules:numerics
(FPCore (x y z t a b c)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2"
  :precision binary64
  (/ x (+ x (* y (exp (* 2 (- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5 6)) (/ 2 (* t 3)))))))))))