Average Error: 1.9 → 1.0
Time: 41.8s
Precision: 64
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\]
↓
\[\frac{x}{\frac{\sqrt[3]{y}}{\sqrt{e^{-\mathsf{fma}\left(1, \log a, \mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(t, -\log a, b\right)\right)\right)}}}} \cdot \frac{1}{\frac{\left(\sqrt[3]{\sqrt[3]{\sqrt[3]{y}} \cdot \left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right)} \cdot \left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right)\right) \cdot \sqrt[3]{y}}{\sqrt{e^{-\mathsf{fma}\left(1, \log a, \mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(t, -\log a, b\right)\right)\right)}}}}\]
\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}
↓
\frac{x}{\frac{\sqrt[3]{y}}{\sqrt{e^{-\mathsf{fma}\left(1, \log a, \mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(t, -\log a, b\right)\right)\right)}}}} \cdot \frac{1}{\frac{\left(\sqrt[3]{\sqrt[3]{\sqrt[3]{y}} \cdot \left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right)} \cdot \left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right)\right) \cdot \sqrt[3]{y}}{\sqrt{e^{-\mathsf{fma}\left(1, \log a, \mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(t, -\log a, b\right)\right)\right)}}}}
double f(double x, double y, double z, double t, double a, double b) {
        double r2961233 = x;
        double r2961234 = y;
        double r2961235 = z;
        double r2961236 = log(r2961235);
        double r2961237 = r2961234 * r2961236;
        double r2961238 = t;
        double r2961239 = 1.0;
        double r2961240 = r2961238 - r2961239;
        double r2961241 = a;
        double r2961242 = log(r2961241);
        double r2961243 = r2961240 * r2961242;
        double r2961244 = r2961237 + r2961243;
        double r2961245 = b;
        double r2961246 = r2961244 - r2961245;
        double r2961247 = exp(r2961246);
        double r2961248 = r2961233 * r2961247;
        double r2961249 = r2961248 / r2961234;
        return r2961249;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r2961250 = x;
        double r2961251 = y;
        double r2961252 = cbrt(r2961251);
        double r2961253 = 1.0;
        double r2961254 = a;
        double r2961255 = log(r2961254);
        double r2961256 = z;
        double r2961257 = log(r2961256);
        double r2961258 = -r2961257;
        double r2961259 = t;
        double r2961260 = -r2961255;
        double r2961261 = b;
        double r2961262 = fma(r2961259, r2961260, r2961261);
        double r2961263 = fma(r2961251, r2961258, r2961262);
        double r2961264 = fma(r2961253, r2961255, r2961263);
        double r2961265 = -r2961264;
        double r2961266 = exp(r2961265);
        double r2961267 = sqrt(r2961266);
        double r2961268 = r2961252 / r2961267;
        double r2961269 = r2961250 / r2961268;
        double r2961270 = 1.0;
        double r2961271 = cbrt(r2961252);
        double r2961272 = r2961271 * r2961271;
        double r2961273 = r2961271 * r2961272;
        double r2961274 = cbrt(r2961273);
        double r2961275 = r2961274 * r2961272;
        double r2961276 = r2961275 * r2961252;
        double r2961277 = r2961276 / r2961267;
        double r2961278 = r2961270 / r2961277;
        double r2961279 = r2961269 * r2961278;
        return r2961279;
}

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 1.9

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

  2. Taylor expanded around inf 1.9

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

  3. Simplified2.0

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

  5. Applied add-sqr-sqrt2.0

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

  6. Applied add-cube-cbrt2.0

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

  7. Applied times-frac2.0

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

  8. Applied *-un-lft-identity2.0

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

  9. Applied times-frac1.0

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

  11. Applied add-cube-cbrt1.0

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

  13. Applied add-cube-cbrt1.0

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

  14. Final simplification1.0

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

Reproduce

herbie shell --seed 2019192 +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))