Average Error: 1.9 → 0.3
Time: 18.2s
Precision: 64
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}\]
↓
\[{e}^{\left(\frac{\mathsf{fma}\left(y, \log z - t, a \cdot \left(\left(\log 1 - \mathsf{fma}\left(\frac{1}{2}, \frac{{z}^{2}}{{1}^{2}}, 1 \cdot z\right)\right) - b\right)\right)}{2}\right)} \cdot \left(x \cdot {e}^{\left(\frac{\mathsf{fma}\left(\log z - t, y, a \cdot \left(\log 1 - \mathsf{fma}\left(\frac{{z}^{2}}{{1}^{2}}, \frac{1}{2}, \mathsf{fma}\left(1, z, b\right)\right)\right)\right)}{2}\right)}\right)\]
x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
↓
{e}^{\left(\frac{\mathsf{fma}\left(y, \log z - t, a \cdot \left(\left(\log 1 - \mathsf{fma}\left(\frac{1}{2}, \frac{{z}^{2}}{{1}^{2}}, 1 \cdot z\right)\right) - b\right)\right)}{2}\right)} \cdot \left(x \cdot {e}^{\left(\frac{\mathsf{fma}\left(\log z - t, y, a \cdot \left(\log 1 - \mathsf{fma}\left(\frac{{z}^{2}}{{1}^{2}}, \frac{1}{2}, \mathsf{fma}\left(1, z, b\right)\right)\right)\right)}{2}\right)}\right)
double f(double x, double y, double z, double t, double a, double b) {
        double r114218 = x;
        double r114219 = y;
        double r114220 = z;
        double r114221 = log(r114220);
        double r114222 = t;
        double r114223 = r114221 - r114222;
        double r114224 = r114219 * r114223;
        double r114225 = a;
        double r114226 = 1.0;
        double r114227 = r114226 - r114220;
        double r114228 = log(r114227);
        double r114229 = b;
        double r114230 = r114228 - r114229;
        double r114231 = r114225 * r114230;
        double r114232 = r114224 + r114231;
        double r114233 = exp(r114232);
        double r114234 = r114218 * r114233;
        return r114234;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r114235 = exp(1.0);
        double r114236 = y;
        double r114237 = z;
        double r114238 = log(r114237);
        double r114239 = t;
        double r114240 = r114238 - r114239;
        double r114241 = a;
        double r114242 = 1.0;
        double r114243 = log(r114242);
        double r114244 = 0.5;
        double r114245 = 2.0;
        double r114246 = pow(r114237, r114245);
        double r114247 = pow(r114242, r114245);
        double r114248 = r114246 / r114247;
        double r114249 = r114242 * r114237;
        double r114250 = fma(r114244, r114248, r114249);
        double r114251 = r114243 - r114250;
        double r114252 = b;
        double r114253 = r114251 - r114252;
        double r114254 = r114241 * r114253;
        double r114255 = fma(r114236, r114240, r114254);
        double r114256 = r114255 / r114245;
        double r114257 = pow(r114235, r114256);
        double r114258 = x;
        double r114259 = fma(r114242, r114237, r114252);
        double r114260 = fma(r114248, r114244, r114259);
        double r114261 = r114243 - r114260;
        double r114262 = r114241 * r114261;
        double r114263 = fma(r114240, r114236, r114262);
        double r114264 = r114263 / r114245;
        double r114265 = pow(r114235, r114264);
        double r114266 = r114258 * r114265;
        double r114267 = r114257 * r114266;
        return r114267;
}

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

Derivation

  1. Initial program 1.9

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

  2. Simplified1.8

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

  3. Taylor expanded around 0 0.3

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

  4. Simplified0.3

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

  6. Applied *-un-lft-identity0.3

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

  7. Applied exp-prod0.3

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

  8. Simplified0.3

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

  10. Applied sqr-pow0.3

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

  11. Applied associate-*l*0.3

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

  12. Simplified0.3

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

  13. Final simplification0.3

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

Reproduce

herbie shell --seed 2019351 +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))))))