Average Error: 4.0 → 2.3
Time: 5.9s
Precision: 64
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
↓
\[\frac{x}{x + y \cdot e^{2 \cdot \mathsf{fma}\left(\frac{z}{1}, \frac{\sqrt{t + a}}{t}, -\left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
\frac{x}{x + y \cdot e^{2 \cdot \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(\frac{z}{1}, \frac{\sqrt{t + a}}{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 r58551 = x;
        double r58552 = y;
        double r58553 = 2.0;
        double r58554 = z;
        double r58555 = t;
        double r58556 = a;
        double r58557 = r58555 + r58556;
        double r58558 = sqrt(r58557);
        double r58559 = r58554 * r58558;
        double r58560 = r58559 / r58555;
        double r58561 = b;
        double r58562 = c;
        double r58563 = r58561 - r58562;
        double r58564 = 5.0;
        double r58565 = 6.0;
        double r58566 = r58564 / r58565;
        double r58567 = r58556 + r58566;
        double r58568 = 3.0;
        double r58569 = r58555 * r58568;
        double r58570 = r58553 / r58569;
        double r58571 = r58567 - r58570;
        double r58572 = r58563 * r58571;
        double r58573 = r58560 - r58572;
        double r58574 = r58553 * r58573;
        double r58575 = exp(r58574);
        double r58576 = r58552 * r58575;
        double r58577 = r58551 + r58576;
        double r58578 = r58551 / r58577;
        return r58578;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r58579 = x;
        double r58580 = y;
        double r58581 = 2.0;
        double r58582 = z;
        double r58583 = 1.0;
        double r58584 = r58582 / r58583;
        double r58585 = t;
        double r58586 = a;
        double r58587 = r58585 + r58586;
        double r58588 = sqrt(r58587);
        double r58589 = r58588 / r58585;
        double r58590 = b;
        double r58591 = c;
        double r58592 = r58590 - r58591;
        double r58593 = 5.0;
        double r58594 = 6.0;
        double r58595 = r58593 / r58594;
        double r58596 = r58586 + r58595;
        double r58597 = 3.0;
        double r58598 = r58585 * r58597;
        double r58599 = r58581 / r58598;
        double r58600 = r58596 - r58599;
        double r58601 = r58592 * r58600;
        double r58602 = -r58601;
        double r58603 = fma(r58584, r58589, r58602);
        double r58604 = r58581 * r58603;
        double r58605 = exp(r58604);
        double r58606 = r58580 * r58605;
        double r58607 = r58579 + r58606;
        double r58608 = r58579 / r58607;
        return r58608;
}

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 4.0

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

  3. Applied *-un-lft-identity4.0

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

  4. Applied times-frac3.5

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

  5. Applied fma-neg2.3

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

  6. Final simplification2.3

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

Reproduce

herbie shell --seed 2020049 +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)))))))))))