Average Error: 3.9 → 2.6
Time: 9.2s
Precision: 64
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
↓
\[\frac{x}{x + y \cdot e^{2 \cdot \sqrt[3]{{\left(\mathsf{fma}\left(-\left(b - c\right), \left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}, \frac{z \cdot \sqrt{t + a}}{t}\right)\right)}^{3}}}}\]
\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 \sqrt[3]{{\left(\mathsf{fma}\left(-\left(b - c\right), \left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}, \frac{z \cdot \sqrt{t + a}}{t}\right)\right)}^{3}}}}
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r75494 = x;
        double r75495 = y;
        double r75496 = 2.0;
        double r75497 = z;
        double r75498 = t;
        double r75499 = a;
        double r75500 = r75498 + r75499;
        double r75501 = sqrt(r75500);
        double r75502 = r75497 * r75501;
        double r75503 = r75502 / r75498;
        double r75504 = b;
        double r75505 = c;
        double r75506 = r75504 - r75505;
        double r75507 = 5.0;
        double r75508 = 6.0;
        double r75509 = r75507 / r75508;
        double r75510 = r75499 + r75509;
        double r75511 = 3.0;
        double r75512 = r75498 * r75511;
        double r75513 = r75496 / r75512;
        double r75514 = r75510 - r75513;
        double r75515 = r75506 * r75514;
        double r75516 = r75503 - r75515;
        double r75517 = r75496 * r75516;
        double r75518 = exp(r75517);
        double r75519 = r75495 * r75518;
        double r75520 = r75494 + r75519;
        double r75521 = r75494 / r75520;
        return r75521;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r75522 = x;
        double r75523 = y;
        double r75524 = 2.0;
        double r75525 = b;
        double r75526 = c;
        double r75527 = r75525 - r75526;
        double r75528 = -r75527;
        double r75529 = a;
        double r75530 = 5.0;
        double r75531 = 6.0;
        double r75532 = r75530 / r75531;
        double r75533 = r75529 + r75532;
        double r75534 = t;
        double r75535 = 3.0;
        double r75536 = r75534 * r75535;
        double r75537 = r75524 / r75536;
        double r75538 = r75533 - r75537;
        double r75539 = z;
        double r75540 = r75534 + r75529;
        double r75541 = sqrt(r75540);
        double r75542 = r75539 * r75541;
        double r75543 = r75542 / r75534;
        double r75544 = fma(r75528, r75538, r75543);
        double r75545 = 3.0;
        double r75546 = pow(r75544, r75545);
        double r75547 = cbrt(r75546);
        double r75548 = r75524 * r75547;
        double r75549 = exp(r75548);
        double r75550 = r75523 * r75549;
        double r75551 = r75522 + r75550;
        double r75552 = r75522 / r75551;
        return r75552;
}

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 3.9

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

  3. Applied add-cbrt-cube3.9

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

  4. Applied add-cbrt-cube7.0

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

  5. Applied cbrt-unprod7.0

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

  6. Applied add-cbrt-cube7.0

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

  7. Applied cbrt-undiv7.1

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

  8. Simplified7.1

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

  10. Applied add-cube-cbrt7.1

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

  11. Applied times-frac5.9

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

  12. Applied fma-neg5.2

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

  13. Simplified1.9

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

  15. Applied add-cbrt-cube1.9

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

  16. Simplified2.6

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

  17. Final simplification2.6

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

Reproduce

herbie shell --seed 2019353 +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)))))))))))