Average Error: 2.0 → 0.5
Time: 17.9s
Precision: 64
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}\]
↓
\[\left(x \cdot \sqrt{e^{\sqrt[3]{{\left(\left(\log z - t\right) \cdot y + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)\right)}^{3}}}}\right) \cdot \sqrt{e^{y \cdot \left(\log z - t\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)}}\]
x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
↓
\left(x \cdot \sqrt{e^{\sqrt[3]{{\left(\left(\log z - t\right) \cdot y + 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) \cdot \sqrt{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)}}
double f(double x, double y, double z, double t, double a, double b) {
        double r191502 = x;
        double r191503 = y;
        double r191504 = z;
        double r191505 = log(r191504);
        double r191506 = t;
        double r191507 = r191505 - r191506;
        double r191508 = r191503 * r191507;
        double r191509 = a;
        double r191510 = 1.0;
        double r191511 = r191510 - r191504;
        double r191512 = log(r191511);
        double r191513 = b;
        double r191514 = r191512 - r191513;
        double r191515 = r191509 * r191514;
        double r191516 = r191508 + r191515;
        double r191517 = exp(r191516);
        double r191518 = r191502 * r191517;
        return r191518;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r191519 = x;
        double r191520 = z;
        double r191521 = log(r191520);
        double r191522 = t;
        double r191523 = r191521 - r191522;
        double r191524 = y;
        double r191525 = r191523 * r191524;
        double r191526 = a;
        double r191527 = 1.0;
        double r191528 = log(r191527);
        double r191529 = 0.5;
        double r191530 = 2.0;
        double r191531 = pow(r191520, r191530);
        double r191532 = pow(r191527, r191530);
        double r191533 = r191531 / r191532;
        double r191534 = r191529 * r191533;
        double r191535 = r191527 * r191520;
        double r191536 = r191534 + r191535;
        double r191537 = r191528 - r191536;
        double r191538 = b;
        double r191539 = r191537 - r191538;
        double r191540 = r191526 * r191539;
        double r191541 = r191525 + r191540;
        double r191542 = 3.0;
        double r191543 = pow(r191541, r191542);
        double r191544 = cbrt(r191543);
        double r191545 = exp(r191544);
        double r191546 = sqrt(r191545);
        double r191547 = r191519 * r191546;
        double r191548 = r191524 * r191523;
        double r191549 = r191548 + r191540;
        double r191550 = exp(r191549);
        double r191551 = sqrt(r191550);
        double r191552 = r191547 * r191551;
        return r191552;
}

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 2.0

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

  2. Taylor expanded around 0 0.5

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

  4. Applied add-sqr-sqrt0.5

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

  5. Applied associate-*r*0.5

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

  7. Applied add-cbrt-cube0.5

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

  8. Simplified0.5

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

  9. Final simplification0.5

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

Reproduce

herbie shell --seed 2020062 
(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))))))