Average Error: 3.9 → 1.6
Time: 26.6s
Precision: 64
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
↓
\[\frac{x}{\mathsf{fma}\left(y, e^{2 \cdot \mathsf{fma}\left(c - b, \frac{5}{6} + \left(a - \frac{\frac{2}{t}}{3}\right), \left(\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt{t + a}\right) \cdot \frac{\sqrt[3]{z}}{t}\right)}, x\right)}\]
\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(a - \frac{\frac{2}{t}}{3}\right), \left(\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt{t + a}\right) \cdot \frac{\sqrt[3]{z}}{t}\right)}, x\right)}
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r2604622 = x;
        double r2604623 = y;
        double r2604624 = 2.0;
        double r2604625 = z;
        double r2604626 = t;
        double r2604627 = a;
        double r2604628 = r2604626 + r2604627;
        double r2604629 = sqrt(r2604628);
        double r2604630 = r2604625 * r2604629;
        double r2604631 = r2604630 / r2604626;
        double r2604632 = b;
        double r2604633 = c;
        double r2604634 = r2604632 - r2604633;
        double r2604635 = 5.0;
        double r2604636 = 6.0;
        double r2604637 = r2604635 / r2604636;
        double r2604638 = r2604627 + r2604637;
        double r2604639 = 3.0;
        double r2604640 = r2604626 * r2604639;
        double r2604641 = r2604624 / r2604640;
        double r2604642 = r2604638 - r2604641;
        double r2604643 = r2604634 * r2604642;
        double r2604644 = r2604631 - r2604643;
        double r2604645 = r2604624 * r2604644;
        double r2604646 = exp(r2604645);
        double r2604647 = r2604623 * r2604646;
        double r2604648 = r2604622 + r2604647;
        double r2604649 = r2604622 / r2604648;
        return r2604649;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r2604650 = x;
        double r2604651 = y;
        double r2604652 = 2.0;
        double r2604653 = c;
        double r2604654 = b;
        double r2604655 = r2604653 - r2604654;
        double r2604656 = 5.0;
        double r2604657 = 6.0;
        double r2604658 = r2604656 / r2604657;
        double r2604659 = a;
        double r2604660 = t;
        double r2604661 = r2604652 / r2604660;
        double r2604662 = 3.0;
        double r2604663 = r2604661 / r2604662;
        double r2604664 = r2604659 - r2604663;
        double r2604665 = r2604658 + r2604664;
        double r2604666 = z;
        double r2604667 = cbrt(r2604666);
        double r2604668 = r2604667 * r2604667;
        double r2604669 = r2604660 + r2604659;
        double r2604670 = sqrt(r2604669);
        double r2604671 = r2604668 * r2604670;
        double r2604672 = r2604667 / r2604660;
        double r2604673 = r2604671 * r2604672;
        double r2604674 = fma(r2604655, r2604665, r2604673);
        double r2604675 = r2604652 * r2604674;
        double r2604676 = exp(r2604675);
        double r2604677 = fma(r2604651, r2604676, r2604650);
        double r2604678 = r2604650 / r2604677;
        return r2604678;
}

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. Simplified1.7

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

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

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

  5. Applied add-cube-cbrt1.7

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

  6. Applied times-frac1.7

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

  7. Applied associate-*r*1.6

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

  8. Simplified1.6

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

  9. Final simplification1.6

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

Reproduce

herbie shell --seed 2019171 +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)))))))))))