Average Error: 2.0 → 2.1
Time: 1.7m
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 r453028 = x;
        double r453029 = y;
        double r453030 = z;
        double r453031 = log(r453030);
        double r453032 = r453029 * r453031;
        double r453033 = t;
        double r453034 = 1.0;
        double r453035 = r453033 - r453034;
        double r453036 = a;
        double r453037 = log(r453036);
        double r453038 = r453035 * r453037;
        double r453039 = r453032 + r453038;
        double r453040 = b;
        double r453041 = r453039 - r453040;
        double r453042 = exp(r453041);
        double r453043 = r453028 * r453042;
        double r453044 = r453043 / r453029;
        return r453044;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r453045 = x;
        double r453046 = z;
        double r453047 = log(r453046);
        double r453048 = y;
        double r453049 = r453047 * r453048;
        double r453050 = b;
        double r453051 = a;
        double r453052 = log(r453051);
        double r453053 = t;
        double r453054 = 1.0;
        double r453055 = r453053 - r453054;
        double r453056 = r453052 * r453055;
        double r453057 = r453050 - r453056;
        double r453058 = r453049 - r453057;
        double r453059 = exp(r453058);
        double r453060 = r453059 / r453048;
        double r453061 = r453045 * r453060;
        return r453061;
}

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))