Average Error: 4.1 → 1.6
Time: 38.4s
Precision: 64
\[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]
↓
\[\frac{x}{\mathsf{fma}\left(y, e^{\mathsf{fma}\left(c - b, \frac{5.0}{6.0} - \left(\frac{\frac{2.0}{t}}{3.0} - a\right), \left(\left(\frac{\sqrt[3]{\sqrt{a + t}}}{\sqrt[3]{t}} \cdot \frac{\sqrt[3]{\sqrt{a + t}}}{\sqrt[3]{t}}\right) \cdot z\right) \cdot \frac{\sqrt[3]{\sqrt{a + t}}}{\sqrt[3]{t}}\right) \cdot 2.0}, x\right)}\]
\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}
↓
\frac{x}{\mathsf{fma}\left(y, e^{\mathsf{fma}\left(c - b, \frac{5.0}{6.0} - \left(\frac{\frac{2.0}{t}}{3.0} - a\right), \left(\left(\frac{\sqrt[3]{\sqrt{a + t}}}{\sqrt[3]{t}} \cdot \frac{\sqrt[3]{\sqrt{a + t}}}{\sqrt[3]{t}}\right) \cdot z\right) \cdot \frac{\sqrt[3]{\sqrt{a + t}}}{\sqrt[3]{t}}\right) \cdot 2.0}, x\right)}
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r2433278 = x;
        double r2433279 = y;
        double r2433280 = 2.0;
        double r2433281 = z;
        double r2433282 = t;
        double r2433283 = a;
        double r2433284 = r2433282 + r2433283;
        double r2433285 = sqrt(r2433284);
        double r2433286 = r2433281 * r2433285;
        double r2433287 = r2433286 / r2433282;
        double r2433288 = b;
        double r2433289 = c;
        double r2433290 = r2433288 - r2433289;
        double r2433291 = 5.0;
        double r2433292 = 6.0;
        double r2433293 = r2433291 / r2433292;
        double r2433294 = r2433283 + r2433293;
        double r2433295 = 3.0;
        double r2433296 = r2433282 * r2433295;
        double r2433297 = r2433280 / r2433296;
        double r2433298 = r2433294 - r2433297;
        double r2433299 = r2433290 * r2433298;
        double r2433300 = r2433287 - r2433299;
        double r2433301 = r2433280 * r2433300;
        double r2433302 = exp(r2433301);
        double r2433303 = r2433279 * r2433302;
        double r2433304 = r2433278 + r2433303;
        double r2433305 = r2433278 / r2433304;
        return r2433305;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r2433306 = x;
        double r2433307 = y;
        double r2433308 = c;
        double r2433309 = b;
        double r2433310 = r2433308 - r2433309;
        double r2433311 = 5.0;
        double r2433312 = 6.0;
        double r2433313 = r2433311 / r2433312;
        double r2433314 = 2.0;
        double r2433315 = t;
        double r2433316 = r2433314 / r2433315;
        double r2433317 = 3.0;
        double r2433318 = r2433316 / r2433317;
        double r2433319 = a;
        double r2433320 = r2433318 - r2433319;
        double r2433321 = r2433313 - r2433320;
        double r2433322 = r2433319 + r2433315;
        double r2433323 = sqrt(r2433322);
        double r2433324 = cbrt(r2433323);
        double r2433325 = cbrt(r2433315);
        double r2433326 = r2433324 / r2433325;
        double r2433327 = r2433326 * r2433326;
        double r2433328 = z;
        double r2433329 = r2433327 * r2433328;
        double r2433330 = r2433329 * r2433326;
        double r2433331 = fma(r2433310, r2433321, r2433330);
        double r2433332 = r2433331 * r2433314;
        double r2433333 = exp(r2433332);
        double r2433334 = fma(r2433307, r2433333, r2433306);
        double r2433335 = r2433306 / r2433334;
        return r2433335;
}

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 4.1

    \[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]

  2. Simplified2.2

    \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, e^{2.0 \cdot \mathsf{fma}\left(c - b, \frac{5.0}{6.0} - \left(\frac{\frac{2.0}{t}}{3.0} - a\right), z \cdot \frac{\sqrt{a + t}}{t}\right)}, x\right)}}\]
  3. Using strategy rm

  4. Applied add-cube-cbrt2.2

    \[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{2.0 \cdot \mathsf{fma}\left(c - b, \frac{5.0}{6.0} - \left(\frac{\frac{2.0}{t}}{3.0} - a\right), z \cdot \frac{\sqrt{a + t}}{\color{blue}{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}}}\right)}, x\right)}\]

  5. Applied add-cube-cbrt2.2

    \[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{2.0 \cdot \mathsf{fma}\left(c - b, \frac{5.0}{6.0} - \left(\frac{\frac{2.0}{t}}{3.0} - a\right), z \cdot \frac{\color{blue}{\left(\sqrt[3]{\sqrt{a + t}} \cdot \sqrt[3]{\sqrt{a + t}}\right) \cdot \sqrt[3]{\sqrt{a + t}}}}{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}}\right)}, x\right)}\]

  6. Applied times-frac2.2

    \[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{2.0 \cdot \mathsf{fma}\left(c - b, \frac{5.0}{6.0} - \left(\frac{\frac{2.0}{t}}{3.0} - a\right), z \cdot \color{blue}{\left(\frac{\sqrt[3]{\sqrt{a + t}} \cdot \sqrt[3]{\sqrt{a + t}}}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt[3]{\sqrt{a + t}}}{\sqrt[3]{t}}\right)}\right)}, x\right)}\]

  7. Applied associate-*r*1.6

    \[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{2.0 \cdot \mathsf{fma}\left(c - b, \frac{5.0}{6.0} - \left(\frac{\frac{2.0}{t}}{3.0} - a\right), \color{blue}{\left(z \cdot \frac{\sqrt[3]{\sqrt{a + t}} \cdot \sqrt[3]{\sqrt{a + t}}}{\sqrt[3]{t} \cdot \sqrt[3]{t}}\right) \cdot \frac{\sqrt[3]{\sqrt{a + t}}}{\sqrt[3]{t}}}\right)}, x\right)}\]

  8. Simplified1.6

    \[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{2.0 \cdot \mathsf{fma}\left(c - b, \frac{5.0}{6.0} - \left(\frac{\frac{2.0}{t}}{3.0} - a\right), \color{blue}{\left(\left(\frac{\sqrt[3]{\sqrt{a + t}}}{\sqrt[3]{t}} \cdot \frac{\sqrt[3]{\sqrt{a + t}}}{\sqrt[3]{t}}\right) \cdot z\right)} \cdot \frac{\sqrt[3]{\sqrt{a + t}}}{\sqrt[3]{t}}\right)}, x\right)}\]

  9. Final simplification1.6

    \[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{\mathsf{fma}\left(c - b, \frac{5.0}{6.0} - \left(\frac{\frac{2.0}{t}}{3.0} - a\right), \left(\left(\frac{\sqrt[3]{\sqrt{a + t}}}{\sqrt[3]{t}} \cdot \frac{\sqrt[3]{\sqrt{a + t}}}{\sqrt[3]{t}}\right) \cdot z\right) \cdot \frac{\sqrt[3]{\sqrt{a + t}}}{\sqrt[3]{t}}\right) \cdot 2.0}, x\right)}\]

Reproduce

herbie shell --seed 2019142 +o rules:numerics
(FPCore (x y z t a b c)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2"
  (/ x (+ x (* y (exp (* 2.0 (- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0)))))))))))