Average Error: 3.8 → 2.3
Time: 28.8s
Precision: 64
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
↓
\[\frac{x}{\mathsf{fma}\left(y, e^{2 \cdot \left(\mathsf{fma}\left(z, \frac{\sqrt{t + a}}{t}, -\left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right) \cdot \left(b - c\right)\right) + \left(\left(-\left(b - c\right)\right) + \left(b - c\right)\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}, x\right)}\]
\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 \left(\mathsf{fma}\left(z, \frac{\sqrt{t + a}}{t}, -\left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right) \cdot \left(b - c\right)\right) + \left(\left(-\left(b - c\right)\right) + \left(b - c\right)\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}, x\right)}
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r79689 = x;
        double r79690 = y;
        double r79691 = 2.0;
        double r79692 = z;
        double r79693 = t;
        double r79694 = a;
        double r79695 = r79693 + r79694;
        double r79696 = sqrt(r79695);
        double r79697 = r79692 * r79696;
        double r79698 = r79697 / r79693;
        double r79699 = b;
        double r79700 = c;
        double r79701 = r79699 - r79700;
        double r79702 = 5.0;
        double r79703 = 6.0;
        double r79704 = r79702 / r79703;
        double r79705 = r79694 + r79704;
        double r79706 = 3.0;
        double r79707 = r79693 * r79706;
        double r79708 = r79691 / r79707;
        double r79709 = r79705 - r79708;
        double r79710 = r79701 * r79709;
        double r79711 = r79698 - r79710;
        double r79712 = r79691 * r79711;
        double r79713 = exp(r79712);
        double r79714 = r79690 * r79713;
        double r79715 = r79689 + r79714;
        double r79716 = r79689 / r79715;
        return r79716;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r79717 = x;
        double r79718 = y;
        double r79719 = 2.0;
        double r79720 = z;
        double r79721 = t;
        double r79722 = a;
        double r79723 = r79721 + r79722;
        double r79724 = sqrt(r79723);
        double r79725 = r79724 / r79721;
        double r79726 = 5.0;
        double r79727 = 6.0;
        double r79728 = r79726 / r79727;
        double r79729 = r79722 + r79728;
        double r79730 = 3.0;
        double r79731 = r79721 * r79730;
        double r79732 = r79719 / r79731;
        double r79733 = r79729 - r79732;
        double r79734 = b;
        double r79735 = c;
        double r79736 = r79734 - r79735;
        double r79737 = r79733 * r79736;
        double r79738 = -r79737;
        double r79739 = fma(r79720, r79725, r79738);
        double r79740 = -r79736;
        double r79741 = r79740 + r79736;
        double r79742 = r79741 * r79733;
        double r79743 = r79739 + r79742;
        double r79744 = r79719 * r79743;
        double r79745 = exp(r79744);
        double r79746 = fma(r79718, r79745, r79717);
        double r79747 = r79717 / r79746;
        return r79747;
}

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 3.8

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

  2. Simplified3.8

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

  4. Applied *-un-lft-identity3.8

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

  5. Applied times-frac3.5

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

  6. Applied prod-diff22.4

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

  7. Simplified22.4

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

  8. Simplified2.3

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

  9. Final simplification2.3

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

Reproduce

herbie shell --seed 2019323 +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)))))))))))