Average Error: 2.0 → 1.3
Time: 35.2s
Precision: 64
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\]
↓
\[\frac{x \cdot \frac{{a}^{\left(-1\right)}}{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}}{y}\]
\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}
↓
\frac{x \cdot \frac{{a}^{\left(-1\right)}}{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}}{y}
double f(double x, double y, double z, double t, double a, double b) {
        double r70661 = x;
        double r70662 = y;
        double r70663 = z;
        double r70664 = log(r70663);
        double r70665 = r70662 * r70664;
        double r70666 = t;
        double r70667 = 1.0;
        double r70668 = r70666 - r70667;
        double r70669 = a;
        double r70670 = log(r70669);
        double r70671 = r70668 * r70670;
        double r70672 = r70665 + r70671;
        double r70673 = b;
        double r70674 = r70672 - r70673;
        double r70675 = exp(r70674);
        double r70676 = r70661 * r70675;
        double r70677 = r70676 / r70662;
        return r70677;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r70678 = x;
        double r70679 = a;
        double r70680 = 1.0;
        double r70681 = -r70680;
        double r70682 = pow(r70679, r70681);
        double r70683 = y;
        double r70684 = z;
        double r70685 = log(r70684);
        double r70686 = -r70685;
        double r70687 = log(r70679);
        double r70688 = -r70687;
        double r70689 = t;
        double r70690 = b;
        double r70691 = fma(r70688, r70689, r70690);
        double r70692 = fma(r70683, r70686, r70691);
        double r70693 = exp(r70692);
        double r70694 = r70682 / r70693;
        double r70695 = r70678 * r70694;
        double r70696 = r70695 / r70683;
        return r70696;
}

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

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

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

  4. Final simplification1.3

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

Reproduce

herbie shell --seed 2019305 +o rules:numerics
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:incompleteBetaWorker from math-functions-0.1.5.2"
  :precision binary64
  (/ (* x (exp (- (+ (* y (log z)) (* (- t 1) (log a))) b))) y))