Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2

Average Error: 3.5 → 1.2

Time: 2.4m

Precision: 64

Math

\[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]

↓

\[\frac{x}{\mathsf{fma}\left(y, \left(e^{2.0 \cdot \left(\left(\sqrt[3]{\mathsf{fma}\left(\left(\frac{\sqrt[3]{z}}{\sqrt[3]{t}} \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{t}}\right), \left(\frac{\sqrt{t + a}}{\frac{\sqrt[3]{t}}{\sqrt[3]{z}}}\right), \left(\left(\left(a - \frac{2.0}{3.0 \cdot t}\right) + \frac{5.0}{6.0}\right) \cdot \left(-\left(b - c\right)\right)\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\left(\frac{\sqrt[3]{z}}{\sqrt[3]{t}} \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{t}}\right), \left(\frac{\sqrt{t + a}}{\frac{\sqrt[3]{t}}{\sqrt[3]{z}}}\right), \left(\left(\left(a - \frac{2.0}{3.0 \cdot t}\right) + \frac{5.0}{6.0}\right) \cdot \left(-\left(b - c\right)\right)\right)\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(\left(\frac{\sqrt[3]{z}}{\sqrt[3]{t}} \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{t}}\right), \left(\frac{\sqrt{t + a}}{\frac{\sqrt[3]{t}}{\sqrt[3]{z}}}\right), \left(\left(\left(a - \frac{2.0}{3.0 \cdot t}\right) + \frac{5.0}{6.0}\right) \cdot \left(-\left(b - c\right)\right)\right)\right)}\right)}\right), x\right)}\]

C

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r12589218 = x;
        double r12589219 = y;
        double r12589220 = 2.0;
        double r12589221 = z;
        double r12589222 = t;
        double r12589223 = a;
        double r12589224 = r12589222 + r12589223;
        double r12589225 = sqrt(r12589224);
        double r12589226 = r12589221 * r12589225;
        double r12589227 = r12589226 / r12589222;
        double r12589228 = b;
        double r12589229 = c;
        double r12589230 = r12589228 - r12589229;
        double r12589231 = 5.0;
        double r12589232 = 6.0;
        double r12589233 = r12589231 / r12589232;
        double r12589234 = r12589223 + r12589233;
        double r12589235 = 3.0;
        double r12589236 = r12589222 * r12589235;
        double r12589237 = r12589220 / r12589236;
        double r12589238 = r12589234 - r12589237;
        double r12589239 = r12589230 * r12589238;
        double r12589240 = r12589227 - r12589239;
        double r12589241 = r12589220 * r12589240;
        double r12589242 = exp(r12589241);
        double r12589243 = r12589219 * r12589242;
        double r12589244 = r12589218 + r12589243;
        double r12589245 = r12589218 / r12589244;
        return r12589245;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r12589246 = x;
        double r12589247 = y;
        double r12589248 = 2.0;
        double r12589249 = z;
        double r12589250 = cbrt(r12589249);
        double r12589251 = t;
        double r12589252 = cbrt(r12589251);
        double r12589253 = r12589250 / r12589252;
        double r12589254 = r12589253 * r12589253;
        double r12589255 = a;
        double r12589256 = r12589251 + r12589255;
        double r12589257 = sqrt(r12589256);
        double r12589258 = r12589252 / r12589250;
        double r12589259 = r12589257 / r12589258;
        double r12589260 = 3.0;
        double r12589261 = r12589260 * r12589251;
        double r12589262 = r12589248 / r12589261;
        double r12589263 = r12589255 - r12589262;
        double r12589264 = 5.0;
        double r12589265 = 6.0;
        double r12589266 = r12589264 / r12589265;
        double r12589267 = r12589263 + r12589266;
        double r12589268 = b;
        double r12589269 = c;
        double r12589270 = r12589268 - r12589269;
        double r12589271 = -r12589270;
        double r12589272 = r12589267 * r12589271;
        double r12589273 = fma(r12589254, r12589259, r12589272);
        double r12589274 = cbrt(r12589273);
        double r12589275 = r12589274 * r12589274;
        double r12589276 = r12589275 * r12589274;
        double r12589277 = r12589248 * r12589276;
        double r12589278 = exp(r12589277);
        double r12589279 = fma(r12589247, r12589278, r12589246);
        double r12589280 = r12589246 / r12589279;
        return r12589280;
}

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

Bits error versus c

Derivation

1. Initial program 3.5

   \[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]

2. Simplified 2.6

   \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, \left(e^{2.0 \cdot \left(\frac{\sqrt{a + t}}{\frac{t}{z}} - \left(\left(a - \frac{2.0}{t \cdot 3.0}\right) + \frac{5.0}{6.0}\right) \cdot \left(b - c\right)\right)}\right), x\right)}}\]
3. Using strategy rm

4. Applied add-cube-cbrt 2.6

   \[\leadsto \frac{x}{\mathsf{fma}\left(y, \left(e^{2.0 \cdot \left(\frac{\sqrt{a + t}}{\frac{t}{\color{blue}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}}} - \left(\left(a - \frac{2.0}{t \cdot 3.0}\right) + \frac{5.0}{6.0}\right) \cdot \left(b - c\right)\right)}\right), x\right)}\]

5. Applied add-cube-cbrt 2.6

   \[\leadsto \frac{x}{\mathsf{fma}\left(y, \left(e^{2.0 \cdot \left(\frac{\sqrt{a + t}}{\frac{\color{blue}{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}}}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}} - \left(\left(a - \frac{2.0}{t \cdot 3.0}\right) + \frac{5.0}{6.0}\right) \cdot \left(b - c\right)\right)}\right), x\right)}\]

6. Applied times-frac 2.6

   \[\leadsto \frac{x}{\mathsf{fma}\left(y, \left(e^{2.0 \cdot \left(\frac{\sqrt{a + t}}{\color{blue}{\frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{\sqrt[3]{t}}{\sqrt[3]{z}}}} - \left(\left(a - \frac{2.0}{t \cdot 3.0}\right) + \frac{5.0}{6.0}\right) \cdot \left(b - c\right)\right)}\right), x\right)}\]

7. Applied *-un-lft-identity 2.6

   \[\leadsto \frac{x}{\mathsf{fma}\left(y, \left(e^{2.0 \cdot \left(\frac{\color{blue}{1 \cdot \sqrt{a + t}}}{\frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{\sqrt[3]{t}}{\sqrt[3]{z}}} - \left(\left(a - \frac{2.0}{t \cdot 3.0}\right) + \frac{5.0}{6.0}\right) \cdot \left(b - c\right)\right)}\right), x\right)}\]

8. Applied times-frac 2.3

   \[\leadsto \frac{x}{\mathsf{fma}\left(y, \left(e^{2.0 \cdot \left(\color{blue}{\frac{1}{\frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}} \cdot \frac{\sqrt{a + t}}{\frac{\sqrt[3]{t}}{\sqrt[3]{z}}}} - \left(\left(a - \frac{2.0}{t \cdot 3.0}\right) + \frac{5.0}{6.0}\right) \cdot \left(b - c\right)\right)}\right), x\right)}\]

9. Applied fma-neg 1.2

   \[\leadsto \frac{x}{\mathsf{fma}\left(y, \left(e^{2.0 \cdot \color{blue}{\mathsf{fma}\left(\left(\frac{1}{\frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}}\right), \left(\frac{\sqrt{a + t}}{\frac{\sqrt[3]{t}}{\sqrt[3]{z}}}\right), \left(-\left(\left(a - \frac{2.0}{t \cdot 3.0}\right) + \frac{5.0}{6.0}\right) \cdot \left(b - c\right)\right)\right)}}\right), x\right)}\]

10. Simplified 1.2

    \[\leadsto \frac{x}{\mathsf{fma}\left(y, \left(e^{2.0 \cdot \mathsf{fma}\left(\color{blue}{\left(\frac{\sqrt[3]{z}}{\sqrt[3]{t}} \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{t}}\right)}, \left(\frac{\sqrt{a + t}}{\frac{\sqrt[3]{t}}{\sqrt[3]{z}}}\right), \left(-\left(\left(a - \frac{2.0}{t \cdot 3.0}\right) + \frac{5.0}{6.0}\right) \cdot \left(b - c\right)\right)\right)}\right), x\right)}\]
11. Using strategy rm

12. Applied add-cube-cbrt 1.2

    \[\leadsto \frac{x}{\mathsf{fma}\left(y, \left(e^{2.0 \cdot \color{blue}{\left(\left(\sqrt[3]{\mathsf{fma}\left(\left(\frac{\sqrt[3]{z}}{\sqrt[3]{t}} \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{t}}\right), \left(\frac{\sqrt{a + t}}{\frac{\sqrt[3]{t}}{\sqrt[3]{z}}}\right), \left(-\left(\left(a - \frac{2.0}{t \cdot 3.0}\right) + \frac{5.0}{6.0}\right) \cdot \left(b - c\right)\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\left(\frac{\sqrt[3]{z}}{\sqrt[3]{t}} \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{t}}\right), \left(\frac{\sqrt{a + t}}{\frac{\sqrt[3]{t}}{\sqrt[3]{z}}}\right), \left(-\left(\left(a - \frac{2.0}{t \cdot 3.0}\right) + \frac{5.0}{6.0}\right) \cdot \left(b - c\right)\right)\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(\left(\frac{\sqrt[3]{z}}{\sqrt[3]{t}} \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{t}}\right), \left(\frac{\sqrt{a + t}}{\frac{\sqrt[3]{t}}{\sqrt[3]{z}}}\right), \left(-\left(\left(a - \frac{2.0}{t \cdot 3.0}\right) + \frac{5.0}{6.0}\right) \cdot \left(b - c\right)\right)\right)}\right)}}\right), x\right)}\]

13. Final simplification 1.2

    \[\leadsto \frac{x}{\mathsf{fma}\left(y, \left(e^{2.0 \cdot \left(\left(\sqrt[3]{\mathsf{fma}\left(\left(\frac{\sqrt[3]{z}}{\sqrt[3]{t}} \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{t}}\right), \left(\frac{\sqrt{t + a}}{\frac{\sqrt[3]{t}}{\sqrt[3]{z}}}\right), \left(\left(\left(a - \frac{2.0}{3.0 \cdot t}\right) + \frac{5.0}{6.0}\right) \cdot \left(-\left(b - c\right)\right)\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\left(\frac{\sqrt[3]{z}}{\sqrt[3]{t}} \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{t}}\right), \left(\frac{\sqrt{t + a}}{\frac{\sqrt[3]{t}}{\sqrt[3]{z}}}\right), \left(\left(\left(a - \frac{2.0}{3.0 \cdot t}\right) + \frac{5.0}{6.0}\right) \cdot \left(-\left(b - c\right)\right)\right)\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(\left(\frac{\sqrt[3]{z}}{\sqrt[3]{t}} \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{t}}\right), \left(\frac{\sqrt{t + a}}{\frac{\sqrt[3]{t}}{\sqrt[3]{z}}}\right), \left(\left(\left(a - \frac{2.0}{3.0 \cdot t}\right) + \frac{5.0}{6.0}\right) \cdot \left(-\left(b - c\right)\right)\right)\right)}\right)}\right), x\right)}\]

Reproduce

herbie shell --seed 2019120 +o rules:numerics
(FPCore (x y z t a b c)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2"
  (/ x (+ x (* y (exp (* 2.0 (- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0)))))))))))