Average Error: 2.0 → 1.3
Time: 13.2s
Precision: 64
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\]
↓
\[\frac{\left(x \cdot \sqrt{\frac{{\left(\frac{1}{a}\right)}^{1}}{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}\right) \cdot \sqrt{\frac{{\left(\frac{1}{a}\right)}^{1}}{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}}{y}\]
\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}
↓
\frac{\left(x \cdot \sqrt{\frac{{\left(\frac{1}{a}\right)}^{1}}{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}\right) \cdot \sqrt{\frac{{\left(\frac{1}{a}\right)}^{1}}{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}}{y}
double f(double x, double y, double z, double t, double a, double b) {
        double r85204 = x;
        double r85205 = y;
        double r85206 = z;
        double r85207 = log(r85206);
        double r85208 = r85205 * r85207;
        double r85209 = t;
        double r85210 = 1.0;
        double r85211 = r85209 - r85210;
        double r85212 = a;
        double r85213 = log(r85212);
        double r85214 = r85211 * r85213;
        double r85215 = r85208 + r85214;
        double r85216 = b;
        double r85217 = r85215 - r85216;
        double r85218 = exp(r85217);
        double r85219 = r85204 * r85218;
        double r85220 = r85219 / r85205;
        return r85220;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r85221 = x;
        double r85222 = 1.0;
        double r85223 = a;
        double r85224 = r85222 / r85223;
        double r85225 = 1.0;
        double r85226 = pow(r85224, r85225);
        double r85227 = y;
        double r85228 = z;
        double r85229 = r85222 / r85228;
        double r85230 = log(r85229);
        double r85231 = log(r85224);
        double r85232 = t;
        double r85233 = b;
        double r85234 = fma(r85231, r85232, r85233);
        double r85235 = fma(r85227, r85230, r85234);
        double r85236 = exp(r85235);
        double r85237 = r85226 / r85236;
        double r85238 = sqrt(r85237);
        double r85239 = r85221 * r85238;
        double r85240 = r85239 * r85238;
        double r85241 = r85240 / r85227;
        return r85241;
}

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

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

  5. Applied add-sqr-sqrt1.3

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

  6. Applied associate-*r*1.3

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

  7. Final simplification1.3

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

Reproduce

herbie shell --seed 2020081 +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))