Average Error: 4.0 → 1.4
Time: 8.6s
Precision: 64
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
↓
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\mathsf{fma}\left(a + \left(\frac{5}{6} - \frac{2}{t \cdot 3}\right), -\left(b - c\right), \frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}}\right) + \left(a + \left(\frac{5}{6} - \frac{2}{t \cdot 3}\right)\right) \cdot \left(\left(-\left(b - c\right)\right) + \left(b - c\right)\right)\right)}}\]
\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 \left(\mathsf{fma}\left(a + \left(\frac{5}{6} - \frac{2}{t \cdot 3}\right), -\left(b - c\right), \frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}}\right) + \left(a + \left(\frac{5}{6} - \frac{2}{t \cdot 3}\right)\right) \cdot \left(\left(-\left(b - c\right)\right) + \left(b - c\right)\right)\right)}}
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r89641 = x;
        double r89642 = y;
        double r89643 = 2.0;
        double r89644 = z;
        double r89645 = t;
        double r89646 = a;
        double r89647 = r89645 + r89646;
        double r89648 = sqrt(r89647);
        double r89649 = r89644 * r89648;
        double r89650 = r89649 / r89645;
        double r89651 = b;
        double r89652 = c;
        double r89653 = r89651 - r89652;
        double r89654 = 5.0;
        double r89655 = 6.0;
        double r89656 = r89654 / r89655;
        double r89657 = r89646 + r89656;
        double r89658 = 3.0;
        double r89659 = r89645 * r89658;
        double r89660 = r89643 / r89659;
        double r89661 = r89657 - r89660;
        double r89662 = r89653 * r89661;
        double r89663 = r89650 - r89662;
        double r89664 = r89643 * r89663;
        double r89665 = exp(r89664);
        double r89666 = r89642 * r89665;
        double r89667 = r89641 + r89666;
        double r89668 = r89641 / r89667;
        return r89668;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r89669 = x;
        double r89670 = y;
        double r89671 = 2.0;
        double r89672 = a;
        double r89673 = 5.0;
        double r89674 = 6.0;
        double r89675 = r89673 / r89674;
        double r89676 = t;
        double r89677 = 3.0;
        double r89678 = r89676 * r89677;
        double r89679 = r89671 / r89678;
        double r89680 = r89675 - r89679;
        double r89681 = r89672 + r89680;
        double r89682 = b;
        double r89683 = c;
        double r89684 = r89682 - r89683;
        double r89685 = -r89684;
        double r89686 = z;
        double r89687 = cbrt(r89676);
        double r89688 = r89687 * r89687;
        double r89689 = r89686 / r89688;
        double r89690 = r89676 + r89672;
        double r89691 = sqrt(r89690);
        double r89692 = r89691 / r89687;
        double r89693 = r89689 * r89692;
        double r89694 = fma(r89681, r89685, r89693);
        double r89695 = r89685 + r89684;
        double r89696 = r89681 * r89695;
        double r89697 = r89694 + r89696;
        double r89698 = r89671 * r89697;
        double r89699 = exp(r89698);
        double r89700 = r89670 * r89699;
        double r89701 = r89669 + r89700;
        double r89702 = r89669 / r89701;
        return r89702;
}

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 log1p-expm1-u10.5

    \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{2}{t \cdot 3}\right)\right)}\right)\right)}}\]
  4. Using strategy rm

  5. Applied add-cube-cbrt10.5

    \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{\color{blue}{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}}} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{2}{t \cdot 3}\right)\right)\right)\right)}}\]

  6. Applied times-frac9.2

    \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\color{blue}{\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}}} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{2}{t \cdot 3}\right)\right)\right)\right)}}\]

  7. Applied prod-diff39.1

    \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(\mathsf{fma}\left(\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}}, \frac{\sqrt{t + a}}{\sqrt[3]{t}}, -\left(\left(a + \frac{5}{6}\right) - \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{2}{t \cdot 3}\right)\right)\right) \cdot \left(b - c\right)\right) + \mathsf{fma}\left(-\left(\left(a + \frac{5}{6}\right) - \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{2}{t \cdot 3}\right)\right)\right), b - c, \left(\left(a + \frac{5}{6}\right) - \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{2}{t \cdot 3}\right)\right)\right) \cdot \left(b - c\right)\right)\right)}}}\]

  8. Simplified39.1

    \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\color{blue}{\mathsf{fma}\left(a + \left(\frac{5}{6} - \frac{2}{t \cdot 3}\right), -\left(b - c\right), \frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}}\right)} + \mathsf{fma}\left(-\left(\left(a + \frac{5}{6}\right) - \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{2}{t \cdot 3}\right)\right)\right), b - c, \left(\left(a + \frac{5}{6}\right) - \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{2}{t \cdot 3}\right)\right)\right) \cdot \left(b - c\right)\right)\right)}}\]

  9. Simplified1.4

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

  10. Final simplification1.4

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

Reproduce

herbie shell --seed 2020047 +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)))))))))))