Average Error: 0.9 → 0.7
Time: 15.6s
Precision: 64
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}\]
↓
\[x \cdot \left(\sqrt[3]{\sqrt{{\left(e^{y \cdot \left(\log z - t\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)}\right)}^{3}}} \cdot \sqrt[3]{\sqrt{\sqrt{{\left(e^{y \cdot \left(\log z - t\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)}\right)}^{3}}} \cdot \sqrt{\sqrt{{\left(e^{y \cdot \left(\log z - t\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)}\right)}^{3}}}}\right)\]
x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
↓
x \cdot \left(\sqrt[3]{\sqrt{{\left(e^{y \cdot \left(\log z - t\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)}\right)}^{3}}} \cdot \sqrt[3]{\sqrt{\sqrt{{\left(e^{y \cdot \left(\log z - t\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)}\right)}^{3}}} \cdot \sqrt{\sqrt{{\left(e^{y \cdot \left(\log z - t\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)}\right)}^{3}}}}\right)
double f(double x, double y, double z, double t, double a, double b) {
        double r130345 = x;
        double r130346 = y;
        double r130347 = z;
        double r130348 = log(r130347);
        double r130349 = t;
        double r130350 = r130348 - r130349;
        double r130351 = r130346 * r130350;
        double r130352 = a;
        double r130353 = 1.0;
        double r130354 = r130353 - r130347;
        double r130355 = log(r130354);
        double r130356 = b;
        double r130357 = r130355 - r130356;
        double r130358 = r130352 * r130357;
        double r130359 = r130351 + r130358;
        double r130360 = exp(r130359);
        double r130361 = r130345 * r130360;
        return r130361;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r130362 = x;
        double r130363 = y;
        double r130364 = z;
        double r130365 = log(r130364);
        double r130366 = t;
        double r130367 = r130365 - r130366;
        double r130368 = r130363 * r130367;
        double r130369 = a;
        double r130370 = 1.0;
        double r130371 = log(r130370);
        double r130372 = 0.5;
        double r130373 = 2.0;
        double r130374 = pow(r130364, r130373);
        double r130375 = pow(r130370, r130373);
        double r130376 = r130374 / r130375;
        double r130377 = r130372 * r130376;
        double r130378 = r130370 * r130364;
        double r130379 = r130377 + r130378;
        double r130380 = r130371 - r130379;
        double r130381 = b;
        double r130382 = r130380 - r130381;
        double r130383 = r130369 * r130382;
        double r130384 = r130368 + r130383;
        double r130385 = exp(r130384);
        double r130386 = 3.0;
        double r130387 = pow(r130385, r130386);
        double r130388 = sqrt(r130387);
        double r130389 = cbrt(r130388);
        double r130390 = sqrt(r130388);
        double r130391 = r130390 * r130390;
        double r130392 = cbrt(r130391);
        double r130393 = r130389 * r130392;
        double r130394 = r130362 * r130393;
        return r130394;
}

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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.9

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

  2. Taylor expanded around 0 0.6

    \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\color{blue}{\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right)} - b\right)}\]
  3. Using strategy rm

  4. Applied add-cbrt-cube0.7

    \[\leadsto x \cdot \color{blue}{\sqrt[3]{\left(e^{y \cdot \left(\log z - t\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)} \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)}\right) \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)}}}\]

  5. Simplified0.7

    \[\leadsto x \cdot \sqrt[3]{\color{blue}{{\left(e^{y \cdot \left(\log z - t\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)}\right)}^{3}}}\]
  6. Using strategy rm

  7. Applied add-sqr-sqrt0.7

    \[\leadsto x \cdot \sqrt[3]{\color{blue}{\sqrt{{\left(e^{y \cdot \left(\log z - t\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)}\right)}^{3}} \cdot \sqrt{{\left(e^{y \cdot \left(\log z - t\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)}\right)}^{3}}}}\]

  8. Applied cbrt-prod0.7

    \[\leadsto x \cdot \color{blue}{\left(\sqrt[3]{\sqrt{{\left(e^{y \cdot \left(\log z - t\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)}\right)}^{3}}} \cdot \sqrt[3]{\sqrt{{\left(e^{y \cdot \left(\log z - t\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)}\right)}^{3}}}\right)}\]
  9. Using strategy rm

  10. Applied add-sqr-sqrt0.7

    \[\leadsto x \cdot \left(\sqrt[3]{\sqrt{{\left(e^{y \cdot \left(\log z - t\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)}\right)}^{3}}} \cdot \sqrt[3]{\sqrt{\color{blue}{\sqrt{{\left(e^{y \cdot \left(\log z - t\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)}\right)}^{3}} \cdot \sqrt{{\left(e^{y \cdot \left(\log z - t\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)}\right)}^{3}}}}}\right)\]

  11. Applied sqrt-prod0.7

    \[\leadsto x \cdot \left(\sqrt[3]{\sqrt{{\left(e^{y \cdot \left(\log z - t\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)}\right)}^{3}}} \cdot \sqrt[3]{\color{blue}{\sqrt{\sqrt{{\left(e^{y \cdot \left(\log z - t\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)}\right)}^{3}}} \cdot \sqrt{\sqrt{{\left(e^{y \cdot \left(\log z - t\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)}\right)}^{3}}}}}\right)\]

  12. Final simplification0.7

    \[\leadsto x \cdot \left(\sqrt[3]{\sqrt{{\left(e^{y \cdot \left(\log z - t\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)}\right)}^{3}}} \cdot \sqrt[3]{\sqrt{\sqrt{{\left(e^{y \cdot \left(\log z - t\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)}\right)}^{3}}} \cdot \sqrt{\sqrt{{\left(e^{y \cdot \left(\log z - t\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)}\right)}^{3}}}}\right)\]

Reproduce

herbie shell --seed 2020021 
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, B"
  :precision binary64
  (* x (exp (+ (* y (- (log z) t)) (* a (- (log (- 1 z)) b))))))