Average Error: 2.1 → 0.6
Time: 9.4s
Precision: 64
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}\]
↓
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)}\]
x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
↓
x \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)}
double f(double x, double y, double z, double t, double a, double b) {
        double r86475 = x;
        double r86476 = y;
        double r86477 = z;
        double r86478 = log(r86477);
        double r86479 = t;
        double r86480 = r86478 - r86479;
        double r86481 = r86476 * r86480;
        double r86482 = a;
        double r86483 = 1.0;
        double r86484 = r86483 - r86477;
        double r86485 = log(r86484);
        double r86486 = b;
        double r86487 = r86485 - r86486;
        double r86488 = r86482 * r86487;
        double r86489 = r86481 + r86488;
        double r86490 = exp(r86489);
        double r86491 = r86475 * r86490;
        return r86491;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r86492 = x;
        double r86493 = y;
        double r86494 = z;
        double r86495 = log(r86494);
        double r86496 = t;
        double r86497 = r86495 - r86496;
        double r86498 = r86493 * r86497;
        double r86499 = a;
        double r86500 = 1.0;
        double r86501 = log(r86500);
        double r86502 = 0.5;
        double r86503 = 2.0;
        double r86504 = pow(r86494, r86503);
        double r86505 = pow(r86500, r86503);
        double r86506 = r86504 / r86505;
        double r86507 = r86502 * r86506;
        double r86508 = r86500 * r86494;
        double r86509 = r86507 + r86508;
        double r86510 = r86501 - r86509;
        double r86511 = b;
        double r86512 = r86510 - r86511;
        double r86513 = r86499 * r86512;
        double r86514 = r86498 + r86513;
        double r86515 = exp(r86514);
        double r86516 = r86492 * r86515;
        return r86516;
}

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 2.1

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

  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. Final simplification0.6

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

Reproduce

herbie shell --seed 2020056 +o rules:numerics
(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))))))