Average Error: 2.1 → 1.4
Time: 34.4s
Precision: 64
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\]
↓
\[\frac{x \cdot \frac{{a}^{\left(-1\right)}}{e^{\left(\sqrt[3]{b - \mathsf{fma}\left(y, \log z, t \cdot \log a\right)} \cdot \sqrt[3]{b - \mathsf{fma}\left(y, \log z, t \cdot \log a\right)}\right) \cdot \sqrt[3]{b - \mathsf{fma}\left(y, \log z, t \cdot \log a\right)}}}}{y}\]
\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^{\left(\sqrt[3]{b - \mathsf{fma}\left(y, \log z, t \cdot \log a\right)} \cdot \sqrt[3]{b - \mathsf{fma}\left(y, \log z, t \cdot \log a\right)}\right) \cdot \sqrt[3]{b - \mathsf{fma}\left(y, \log z, t \cdot \log a\right)}}}}{y}
double f(double x, double y, double z, double t, double a, double b) {
        double r54054 = x;
        double r54055 = y;
        double r54056 = z;
        double r54057 = log(r54056);
        double r54058 = r54055 * r54057;
        double r54059 = t;
        double r54060 = 1.0;
        double r54061 = r54059 - r54060;
        double r54062 = a;
        double r54063 = log(r54062);
        double r54064 = r54061 * r54063;
        double r54065 = r54058 + r54064;
        double r54066 = b;
        double r54067 = r54065 - r54066;
        double r54068 = exp(r54067);
        double r54069 = r54054 * r54068;
        double r54070 = r54069 / r54055;
        return r54070;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r54071 = x;
        double r54072 = a;
        double r54073 = 1.0;
        double r54074 = -r54073;
        double r54075 = pow(r54072, r54074);
        double r54076 = b;
        double r54077 = y;
        double r54078 = z;
        double r54079 = log(r54078);
        double r54080 = t;
        double r54081 = log(r54072);
        double r54082 = r54080 * r54081;
        double r54083 = fma(r54077, r54079, r54082);
        double r54084 = r54076 - r54083;
        double r54085 = cbrt(r54084);
        double r54086 = r54085 * r54085;
        double r54087 = r54086 * r54085;
        double r54088 = exp(r54087);
        double r54089 = r54075 / r54088;
        double r54090 = r54071 * r54089;
        double r54091 = r54090 / r54077;
        return r54091;
}

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 2.1

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

  2. Taylor expanded around inf 2.1

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

  3. Simplified1.4

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

  5. Applied add-cube-cbrt1.4

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

  6. Simplified1.4

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

  7. Simplified1.4

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

  8. Final simplification1.4

    \[\leadsto \frac{x \cdot \frac{{a}^{\left(-1\right)}}{e^{\left(\sqrt[3]{b - \mathsf{fma}\left(y, \log z, t \cdot \log a\right)} \cdot \sqrt[3]{b - \mathsf{fma}\left(y, \log z, t \cdot \log a\right)}\right) \cdot \sqrt[3]{b - \mathsf{fma}\left(y, \log z, t \cdot \log a\right)}}}}{y}\]

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:incompleteBetaWorker from math-functions-0.1.5.2"
  (/ (* x (exp (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))) y))