Average Error: 3.6 → 1.4
Time: 35.8s
Precision: 64
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
↓
\[\frac{x}{\mathsf{fma}\left(y, e^{2 \cdot \mathsf{fma}\left(c - b, \frac{5}{6} - \left(\frac{\frac{2}{t}}{3} - a\right), \frac{\sqrt{a + t}}{\frac{t}{\sqrt[3]{z}}} \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right)}, x\right)}\]
\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}{\mathsf{fma}\left(y, e^{2 \cdot \mathsf{fma}\left(c - b, \frac{5}{6} - \left(\frac{\frac{2}{t}}{3} - a\right), \frac{\sqrt{a + t}}{\frac{t}{\sqrt[3]{z}}} \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right)}, x\right)}
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r3518307 = x;
        double r3518308 = y;
        double r3518309 = 2.0;
        double r3518310 = z;
        double r3518311 = t;
        double r3518312 = a;
        double r3518313 = r3518311 + r3518312;
        double r3518314 = sqrt(r3518313);
        double r3518315 = r3518310 * r3518314;
        double r3518316 = r3518315 / r3518311;
        double r3518317 = b;
        double r3518318 = c;
        double r3518319 = r3518317 - r3518318;
        double r3518320 = 5.0;
        double r3518321 = 6.0;
        double r3518322 = r3518320 / r3518321;
        double r3518323 = r3518312 + r3518322;
        double r3518324 = 3.0;
        double r3518325 = r3518311 * r3518324;
        double r3518326 = r3518309 / r3518325;
        double r3518327 = r3518323 - r3518326;
        double r3518328 = r3518319 * r3518327;
        double r3518329 = r3518316 - r3518328;
        double r3518330 = r3518309 * r3518329;
        double r3518331 = exp(r3518330);
        double r3518332 = r3518308 * r3518331;
        double r3518333 = r3518307 + r3518332;
        double r3518334 = r3518307 / r3518333;
        return r3518334;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r3518335 = x;
        double r3518336 = y;
        double r3518337 = 2.0;
        double r3518338 = c;
        double r3518339 = b;
        double r3518340 = r3518338 - r3518339;
        double r3518341 = 5.0;
        double r3518342 = 6.0;
        double r3518343 = r3518341 / r3518342;
        double r3518344 = t;
        double r3518345 = r3518337 / r3518344;
        double r3518346 = 3.0;
        double r3518347 = r3518345 / r3518346;
        double r3518348 = a;
        double r3518349 = r3518347 - r3518348;
        double r3518350 = r3518343 - r3518349;
        double r3518351 = r3518348 + r3518344;
        double r3518352 = sqrt(r3518351);
        double r3518353 = z;
        double r3518354 = cbrt(r3518353);
        double r3518355 = r3518344 / r3518354;
        double r3518356 = r3518352 / r3518355;
        double r3518357 = r3518354 * r3518354;
        double r3518358 = r3518356 * r3518357;
        double r3518359 = fma(r3518340, r3518350, r3518358);
        double r3518360 = r3518337 * r3518359;
        double r3518361 = exp(r3518360);
        double r3518362 = fma(r3518336, r3518361, r3518335);
        double r3518363 = r3518335 / r3518362;
        return r3518363;
}

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 3.6

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

  2. Simplified1.5

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

  4. Applied add-cube-cbrt1.5

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

  5. Applied *-un-lft-identity1.5

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

  6. Applied times-frac1.5

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

  7. Applied *-un-lft-identity1.5

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

  8. Applied sqrt-prod1.5

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

  9. Applied times-frac1.4

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

  10. Simplified1.4

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

  11. Final simplification1.4

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

Reproduce

herbie shell --seed 2019170 +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)))))))))))