Average Error: 1.9 → 1.2
Time: 20.6s
Precision: 64
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\]
↓
\[\frac{x \cdot \frac{{\left(\frac{1}{a}\right)}^{1}}{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}{y}\]
\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}
↓
\frac{x \cdot \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}
double f(double x, double y, double z, double t, double a, double b) {
        double r553196 = x;
        double r553197 = y;
        double r553198 = z;
        double r553199 = log(r553198);
        double r553200 = r553197 * r553199;
        double r553201 = t;
        double r553202 = 1.0;
        double r553203 = r553201 - r553202;
        double r553204 = a;
        double r553205 = log(r553204);
        double r553206 = r553203 * r553205;
        double r553207 = r553200 + r553206;
        double r553208 = b;
        double r553209 = r553207 - r553208;
        double r553210 = exp(r553209);
        double r553211 = r553196 * r553210;
        double r553212 = r553211 / r553197;
        return r553212;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r553213 = x;
        double r553214 = 1.0;
        double r553215 = a;
        double r553216 = r553214 / r553215;
        double r553217 = 1.0;
        double r553218 = pow(r553216, r553217);
        double r553219 = y;
        double r553220 = z;
        double r553221 = r553214 / r553220;
        double r553222 = log(r553221);
        double r553223 = log(r553216);
        double r553224 = t;
        double r553225 = b;
        double r553226 = fma(r553223, r553224, r553225);
        double r553227 = fma(r553219, r553222, r553226);
        double r553228 = exp(r553227);
        double r553229 = r553218 / r553228;
        double r553230 = r553213 * r553229;
        double r553231 = r553230 / r553219;
        return r553231;
}

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
Target11.4
Herbie1.2
\[\begin{array}{l} \mathbf{if}\;t \lt -0.88458485041274715:\\ \;\;\;\;\frac{x \cdot \frac{{a}^{\left(t - 1\right)}}{y}}{\left(b + 1\right) - y \cdot \log z}\\ \mathbf{elif}\;t \lt 852031.22883740731:\\ \;\;\;\;\frac{\frac{x}{y} \cdot {a}^{\left(t - 1\right)}}{e^{b - \log z \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{{a}^{\left(t - 1\right)}}{y}}{\left(b + 1\right) - y \cdot \log z}\\ \end{array}\]

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. Final simplification1.2

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

Reproduce

herbie shell --seed 2020083 +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))