Average Error: 2.0 → 1.3
Time: 31.5s
Precision: 64
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\]
↓
\[\frac{\left(\sqrt{\frac{{a}^{\left(-1\right)}}{e^{\left(b - \log a \cdot t\right) - \log z \cdot y}}} \cdot x\right) \cdot \sqrt{\frac{{a}^{\left(-1\right)}}{e^{\left(-\left(\log z \cdot y + \log a \cdot t\right)\right) + b}}}}{y}\]
\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}
↓
\frac{\left(\sqrt{\frac{{a}^{\left(-1\right)}}{e^{\left(b - \log a \cdot t\right) - \log z \cdot y}}} \cdot x\right) \cdot \sqrt{\frac{{a}^{\left(-1\right)}}{e^{\left(-\left(\log z \cdot y + \log a \cdot t\right)\right) + b}}}}{y}
double f(double x, double y, double z, double t, double a, double b) {
        double r299626 = x;
        double r299627 = y;
        double r299628 = z;
        double r299629 = log(r299628);
        double r299630 = r299627 * r299629;
        double r299631 = t;
        double r299632 = 1.0;
        double r299633 = r299631 - r299632;
        double r299634 = a;
        double r299635 = log(r299634);
        double r299636 = r299633 * r299635;
        double r299637 = r299630 + r299636;
        double r299638 = b;
        double r299639 = r299637 - r299638;
        double r299640 = exp(r299639);
        double r299641 = r299626 * r299640;
        double r299642 = r299641 / r299627;
        return r299642;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r299643 = a;
        double r299644 = 1.0;
        double r299645 = -r299644;
        double r299646 = pow(r299643, r299645);
        double r299647 = b;
        double r299648 = log(r299643);
        double r299649 = t;
        double r299650 = r299648 * r299649;
        double r299651 = r299647 - r299650;
        double r299652 = z;
        double r299653 = log(r299652);
        double r299654 = y;
        double r299655 = r299653 * r299654;
        double r299656 = r299651 - r299655;
        double r299657 = exp(r299656);
        double r299658 = r299646 / r299657;
        double r299659 = sqrt(r299658);
        double r299660 = x;
        double r299661 = r299659 * r299660;
        double r299662 = r299655 + r299650;
        double r299663 = -r299662;
        double r299664 = r299663 + r299647;
        double r299665 = exp(r299664);
        double r299666 = r299646 / r299665;
        double r299667 = sqrt(r299666);
        double r299668 = r299661 * r299667;
        double r299669 = r299668 / r299654;
        return r299669;
}

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
Target11.0
Herbie1.3
\[\begin{array}{l} \mathbf{if}\;t \lt -0.8845848504127471478852839936735108494759:\\ \;\;\;\;\frac{x \cdot \frac{{a}^{\left(t - 1\right)}}{y}}{\left(b + 1\right) - y \cdot \log z}\\ \mathbf{elif}\;t \lt 852031.228837407310493290424346923828125:\\ \;\;\;\;\frac{\frac{x}{y} \cdot {a}^{\left(t - 1\right)}}{e^{b - \log z \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{{a}^{\left(t - 1\right)}}{y}}{\left(b + 1\right) - y \cdot \log z}\\ \end{array}\]

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. Taylor expanded around inf 2.0

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

  3. Simplified1.3

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

  5. Applied add-sqr-sqrt1.3

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

  6. Applied associate-*r*1.3

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

  7. Simplified1.3

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

  8. Final simplification1.3

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

Reproduce

herbie shell --seed 2019235 
(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 -0.88458485041274715) (/ (* x (/ (pow a (- t 1)) y)) (- (+ b 1) (* y (log z)))) (if (< t 852031.22883740731) (/ (* (/ 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))