Average Error: 2.0 → 1.4
Time: 18.1s
Precision: 64
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\]
↓
\[\frac{x}{\frac{y}{\frac{{\left(\frac{1}{a}\right)}^{1}}{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}}\]
\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}
↓
\frac{x}{\frac{y}{\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)}}}}
double f(double x, double y, double z, double t, double a, double b) {
        double r100202 = x;
        double r100203 = y;
        double r100204 = z;
        double r100205 = log(r100204);
        double r100206 = r100203 * r100205;
        double r100207 = t;
        double r100208 = 1.0;
        double r100209 = r100207 - r100208;
        double r100210 = a;
        double r100211 = log(r100210);
        double r100212 = r100209 * r100211;
        double r100213 = r100206 + r100212;
        double r100214 = b;
        double r100215 = r100213 - r100214;
        double r100216 = exp(r100215);
        double r100217 = r100202 * r100216;
        double r100218 = r100217 / r100203;
        return r100218;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r100219 = x;
        double r100220 = y;
        double r100221 = 1.0;
        double r100222 = a;
        double r100223 = r100221 / r100222;
        double r100224 = 1.0;
        double r100225 = pow(r100223, r100224);
        double r100226 = z;
        double r100227 = r100221 / r100226;
        double r100228 = log(r100227);
        double r100229 = log(r100223);
        double r100230 = t;
        double r100231 = b;
        double r100232 = fma(r100229, r100230, r100231);
        double r100233 = fma(r100220, r100228, r100232);
        double r100234 = exp(r100233);
        double r100235 = r100225 / r100234;
        double r100236 = r100220 / r100235;
        double r100237 = r100219 / r100236;
        return r100237;
}

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 associate-/l*1.4

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

  6. Final simplification1.4

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

Reproduce

herbie shell --seed 2020057 +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))