Average Error: 4.0 → 1.6
Time: 31.0s
Precision: 64
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
↓
\[\frac{x}{\mathsf{fma}\left(y, e^{2 \cdot \mathsf{fma}\left(c - b, \frac{5}{6} - \left(\frac{\frac{2}{t}}{3} - a\right), \frac{\frac{\frac{\sqrt{a + t}}{\sqrt[3]{t}}}{\sqrt[3]{t}}}{\frac{\sqrt[3]{t}}{z}}\right)}, x\right)}\]
\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(\frac{\frac{2}{t}}{3} - a\right), \frac{\frac{\frac{\sqrt{a + t}}{\sqrt[3]{t}}}{\sqrt[3]{t}}}{\frac{\sqrt[3]{t}}{z}}\right)}, x\right)}
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r3121538 = x;
        double r3121539 = y;
        double r3121540 = 2.0;
        double r3121541 = z;
        double r3121542 = t;
        double r3121543 = a;
        double r3121544 = r3121542 + r3121543;
        double r3121545 = sqrt(r3121544);
        double r3121546 = r3121541 * r3121545;
        double r3121547 = r3121546 / r3121542;
        double r3121548 = b;
        double r3121549 = c;
        double r3121550 = r3121548 - r3121549;
        double r3121551 = 5.0;
        double r3121552 = 6.0;
        double r3121553 = r3121551 / r3121552;
        double r3121554 = r3121543 + r3121553;
        double r3121555 = 3.0;
        double r3121556 = r3121542 * r3121555;
        double r3121557 = r3121540 / r3121556;
        double r3121558 = r3121554 - r3121557;
        double r3121559 = r3121550 * r3121558;
        double r3121560 = r3121547 - r3121559;
        double r3121561 = r3121540 * r3121560;
        double r3121562 = exp(r3121561);
        double r3121563 = r3121539 * r3121562;
        double r3121564 = r3121538 + r3121563;
        double r3121565 = r3121538 / r3121564;
        return r3121565;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r3121566 = x;
        double r3121567 = y;
        double r3121568 = 2.0;
        double r3121569 = c;
        double r3121570 = b;
        double r3121571 = r3121569 - r3121570;
        double r3121572 = 5.0;
        double r3121573 = 6.0;
        double r3121574 = r3121572 / r3121573;
        double r3121575 = t;
        double r3121576 = r3121568 / r3121575;
        double r3121577 = 3.0;
        double r3121578 = r3121576 / r3121577;
        double r3121579 = a;
        double r3121580 = r3121578 - r3121579;
        double r3121581 = r3121574 - r3121580;
        double r3121582 = r3121579 + r3121575;
        double r3121583 = sqrt(r3121582);
        double r3121584 = cbrt(r3121575);
        double r3121585 = r3121583 / r3121584;
        double r3121586 = r3121585 / r3121584;
        double r3121587 = z;
        double r3121588 = r3121584 / r3121587;
        double r3121589 = r3121586 / r3121588;
        double r3121590 = fma(r3121571, r3121581, r3121589);
        double r3121591 = r3121568 * r3121590;
        double r3121592 = exp(r3121591);
        double r3121593 = fma(r3121567, r3121592, r3121566);
        double r3121594 = r3121566 / r3121593;
        return r3121594;
}

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. Simplified1.7

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

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

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

  5. Applied add-cube-cbrt1.7

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

  6. Applied times-frac1.7

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

  7. Applied associate-/r*1.6

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

  8. Simplified1.6

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

  9. Final simplification1.6

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

Reproduce

herbie shell --seed 2019172 +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)))))))))))