Average Error: 1.9 → 0.4
Time: 19.0s
Precision: 64
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\]
↓
\[\frac{1}{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\frac{{\left(\frac{1}{\sqrt{a}}\right)}^{1}}{\sqrt{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}}} \cdot \frac{x}{\frac{\sqrt[3]{y}}{\frac{{\left(\frac{1}{\sqrt{a}}\right)}^{1}}{\sqrt{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}}}\]
\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}
↓
\frac{1}{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\frac{{\left(\frac{1}{\sqrt{a}}\right)}^{1}}{\sqrt{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 \frac{x}{\frac{\sqrt[3]{y}}{\frac{{\left(\frac{1}{\sqrt{a}}\right)}^{1}}{\sqrt{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 r434296 = x;
        double r434297 = y;
        double r434298 = z;
        double r434299 = log(r434298);
        double r434300 = r434297 * r434299;
        double r434301 = t;
        double r434302 = 1.0;
        double r434303 = r434301 - r434302;
        double r434304 = a;
        double r434305 = log(r434304);
        double r434306 = r434303 * r434305;
        double r434307 = r434300 + r434306;
        double r434308 = b;
        double r434309 = r434307 - r434308;
        double r434310 = exp(r434309);
        double r434311 = r434296 * r434310;
        double r434312 = r434311 / r434297;
        return r434312;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r434313 = 1.0;
        double r434314 = y;
        double r434315 = cbrt(r434314);
        double r434316 = r434315 * r434315;
        double r434317 = a;
        double r434318 = sqrt(r434317);
        double r434319 = r434313 / r434318;
        double r434320 = 1.0;
        double r434321 = pow(r434319, r434320);
        double r434322 = z;
        double r434323 = r434313 / r434322;
        double r434324 = log(r434323);
        double r434325 = r434313 / r434317;
        double r434326 = log(r434325);
        double r434327 = t;
        double r434328 = b;
        double r434329 = fma(r434326, r434327, r434328);
        double r434330 = fma(r434314, r434324, r434329);
        double r434331 = exp(r434330);
        double r434332 = sqrt(r434331);
        double r434333 = r434321 / r434332;
        double r434334 = r434316 / r434333;
        double r434335 = r434313 / r434334;
        double r434336 = x;
        double r434337 = r434315 / r434333;
        double r434338 = r434336 / r434337;
        double r434339 = r434335 * r434338;
        return r434339;
}

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

Target

Original1.9
Target11.7
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;t \lt -0.88458485041274715:\\ \;\;\;\;\frac{x \cdot \frac{{a}^{\left(t - 1\right)}}{y}}{\left(b + 1\right) - y \cdot \log z}\\ \mathbf{elif}\;t \lt 852031.22883740731:\\ \;\;\;\;\frac{\frac{x}{y} \cdot {a}^{\left(t - 1\right)}}{e^{b - \log z \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{{a}^{\left(t - 1\right)}}{y}}{\left(b + 1\right) - y \cdot \log z}\\ \end{array}\]

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

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

  3. Simplified1.2

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

  5. Applied associate-/l*1.3

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

  7. Applied add-sqr-sqrt1.3

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

  8. Applied add-sqr-sqrt1.3

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

  9. Applied *-un-lft-identity1.3

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

  10. Applied times-frac1.3

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

  11. Applied unpow-prod-down1.3

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

  12. Applied times-frac1.3

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

  13. Applied add-cube-cbrt1.4

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

  14. Applied times-frac1.4

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

  15. Applied *-un-lft-identity1.4

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

  16. Applied times-frac0.4

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

  17. Final simplification0.4

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

Reproduce

herbie shell --seed 2020046 +o rules:numerics
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:incompleteBetaWorker from math-functions-0.1.5.2, A"
  :precision binary64

  :herbie-target
  (if (< t -0.8845848504127471) (/ (* x (/ (pow a (- t 1)) y)) (- (+ b 1) (* y (log z)))) (if (< t 852031.2288374073) (/ (* (/ x y) (pow a (- t 1))) (exp (- b (* (log z) y)))) (/ (* x (/ (pow a (- t 1)) y)) (- (+ b 1) (* y (log z))))))

  (/ (* x (exp (- (+ (* y (log z)) (* (- t 1) (log a))) b))) y))