Average Error: 2.0 → 2.1
Time: 1.8m
Precision: 64
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\]
↓
\[x \cdot \frac{e^{\log z \cdot y - \left(b - \log a \cdot \left(t - 1\right)\right)}}{y}\]
\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}
↓
x \cdot \frac{e^{\log z \cdot y - \left(b - \log a \cdot \left(t - 1\right)\right)}}{y}
double f(double x, double y, double z, double t, double a, double b) {
        double r448569 = x;
        double r448570 = y;
        double r448571 = z;
        double r448572 = log(r448571);
        double r448573 = r448570 * r448572;
        double r448574 = t;
        double r448575 = 1.0;
        double r448576 = r448574 - r448575;
        double r448577 = a;
        double r448578 = log(r448577);
        double r448579 = r448576 * r448578;
        double r448580 = r448573 + r448579;
        double r448581 = b;
        double r448582 = r448580 - r448581;
        double r448583 = exp(r448582);
        double r448584 = r448569 * r448583;
        double r448585 = r448584 / r448570;
        return r448585;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r448586 = x;
        double r448587 = z;
        double r448588 = log(r448587);
        double r448589 = y;
        double r448590 = r448588 * r448589;
        double r448591 = b;
        double r448592 = a;
        double r448593 = log(r448592);
        double r448594 = t;
        double r448595 = 1.0;
        double r448596 = r448594 - r448595;
        double r448597 = r448593 * r448596;
        double r448598 = r448591 - r448597;
        double r448599 = r448590 - r448598;
        double r448600 = exp(r448599);
        double r448601 = r448600 / r448589;
        double r448602 = r448586 * r448601;
        return r448602;
}

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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original2.0
Target11.2
Herbie2.1
\[\begin{array}{l} \mathbf{if}\;t \lt -0.8845848504127471478852839936735108494759:\\ \;\;\;\;\frac{x \cdot \frac{{a}^{\left(t - 1\right)}}{y}}{\left(b + 1\right) - y \cdot \log z}\\ \mathbf{elif}\;t \lt 852031.228837407310493290424346923828125:\\ \;\;\;\;\frac{\frac{x}{y} \cdot {a}^{\left(t - 1\right)}}{e^{b - \log z \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{{a}^{\left(t - 1\right)}}{y}}{\left(b + 1\right) - y \cdot \log z}\\ \end{array}\]

Derivation

  1. Initial program 2.0

    \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\]
  2. Using strategy rm

  3. Applied *-un-lft-identity2.0

    \[\leadsto \frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{\color{blue}{1 \cdot y}}\]

  4. Applied times-frac2.1

    \[\leadsto \color{blue}{\frac{x}{1} \cdot \frac{e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}}\]

  5. Simplified2.1

    \[\leadsto \color{blue}{x} \cdot \frac{e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\]

  6. Simplified18.2

    \[\leadsto x \cdot \color{blue}{\frac{\frac{{z}^{y}}{\frac{e^{b}}{{a}^{\left(t - 1\right)}}}}{y}}\]
  7. Using strategy rm

  8. Applied add-exp-log18.9

    \[\leadsto x \cdot \frac{\frac{{z}^{y}}{\frac{e^{b}}{{\color{blue}{\left(e^{\log a}\right)}}^{\left(t - 1\right)}}}}{y}\]

  9. Applied pow-exp18.9

    \[\leadsto x \cdot \frac{\frac{{z}^{y}}{\frac{e^{b}}{\color{blue}{e^{\log a \cdot \left(t - 1\right)}}}}}{y}\]

  10. Applied div-exp13.4

    \[\leadsto x \cdot \frac{\frac{{z}^{y}}{\color{blue}{e^{b - \log a \cdot \left(t - 1\right)}}}}{y}\]

  11. Applied add-exp-log13.4

    \[\leadsto x \cdot \frac{\frac{{\color{blue}{\left(e^{\log z}\right)}}^{y}}{e^{b - \log a \cdot \left(t - 1\right)}}}{y}\]

  12. Applied pow-exp13.4

    \[\leadsto x \cdot \frac{\frac{\color{blue}{e^{\log z \cdot y}}}{e^{b - \log a \cdot \left(t - 1\right)}}}{y}\]

  13. Applied div-exp2.1

    \[\leadsto x \cdot \frac{\color{blue}{e^{\log z \cdot y - \left(b - \log a \cdot \left(t - 1\right)\right)}}}{y}\]

  14. Final simplification2.1

    \[\leadsto x \cdot \frac{e^{\log z \cdot y - \left(b - \log a \cdot \left(t - 1\right)\right)}}{y}\]

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:incompleteBetaWorker from math-functions-0.1.5.2, A"
  :precision binary64

  :herbie-target
  (if (< t -0.8845848504127471) (/ (* x (/ (pow a (- t 1)) y)) (- (+ b 1) (* y (log z)))) (if (< t 852031.2288374073) (/ (* (/ x y) (pow a (- t 1))) (exp (- b (* (log z) y)))) (/ (* x (/ (pow a (- t 1)) y)) (- (+ b 1) (* y (log z))))))

  (/ (* x (exp (- (+ (* y (log z)) (* (- t 1) (log a))) b))) y))