Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, I

Average Error: 3.7 → 2.6

Time: 33.1s

Precision: 64

Math

\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]

↓

\[\begin{array}{l} \mathbf{if}\;t \le \frac{-3044132996692187}{1.852673427797059126777135760139006525652 \cdot 10^{78}} \lor \neg \left(t \le \frac{8271322624711839}{1.774508604237321510130185077851573570199 \cdot 10^{131}}\right):\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{\left(z \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}}\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right) + \left(-\sqrt[3]{t} \cdot \left(\sqrt[3]{t} \cdot \left(\left(b - c\right) \cdot \left(\left(a \cdot a - \frac{5}{6} \cdot \frac{5}{6}\right) \cdot \left(t \cdot 3\right) - \left(a - \frac{5}{6}\right) \cdot 2\right)\right)\right)\right)}{\sqrt[3]{t} \cdot \left(\sqrt[3]{t} \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right)\right)}}}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r326246 = x;
        double r326247 = y;
        double r326248 = 2.0;
        double r326249 = z;
        double r326250 = t;
        double r326251 = a;
        double r326252 = r326250 + r326251;
        double r326253 = sqrt(r326252);
        double r326254 = r326249 * r326253;
        double r326255 = r326254 / r326250;
        double r326256 = b;
        double r326257 = c;
        double r326258 = r326256 - r326257;
        double r326259 = 5.0;
        double r326260 = 6.0;
        double r326261 = r326259 / r326260;
        double r326262 = r326251 + r326261;
        double r326263 = 3.0;
        double r326264 = r326250 * r326263;
        double r326265 = r326248 / r326264;
        double r326266 = r326262 - r326265;
        double r326267 = r326258 * r326266;
        double r326268 = r326255 - r326267;
        double r326269 = r326248 * r326268;
        double r326270 = exp(r326269);
        double r326271 = r326247 * r326270;
        double r326272 = r326246 + r326271;
        double r326273 = r326246 / r326272;
        return r326273;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r326274 = t;
        double r326275 = -3044132996692187.0;
        double r326276 = 1.8526734277970591e+78;
        double r326277 = r326275 / r326276;
        bool r326278 = r326274 <= r326277;
        double r326279 = 8271322624711839.0;
        double r326280 = 1.7745086042373215e+131;
        double r326281 = r326279 / r326280;
        bool r326282 = r326274 <= r326281;
        double r326283 = !r326282;
        bool r326284 = r326278 || r326283;
        double r326285 = x;
        double r326286 = y;
        double r326287 = 2.0;
        double r326288 = z;
        double r326289 = cbrt(r326274);
        double r326290 = r326289 * r326289;
        double r326291 = r326288 / r326290;
        double r326292 = a;
        double r326293 = r326274 + r326292;
        double r326294 = sqrt(r326293);
        double r326295 = r326294 / r326289;
        double r326296 = r326291 * r326295;
        double r326297 = b;
        double r326298 = c;
        double r326299 = r326297 - r326298;
        double r326300 = 5.0;
        double r326301 = 6.0;
        double r326302 = r326300 / r326301;
        double r326303 = r326292 + r326302;
        double r326304 = 3.0;
        double r326305 = r326274 * r326304;
        double r326306 = r326287 / r326305;
        double r326307 = r326303 - r326306;
        double r326308 = r326299 * r326307;
        double r326309 = r326296 - r326308;
        double r326310 = r326287 * r326309;
        double r326311 = exp(r326310);
        double r326312 = r326286 * r326311;
        double r326313 = r326285 + r326312;
        double r326314 = r326285 / r326313;
        double r326315 = r326288 * r326295;
        double r326316 = r326292 - r326302;
        double r326317 = r326316 * r326305;
        double r326318 = r326315 * r326317;
        double r326319 = r326292 * r326292;
        double r326320 = r326302 * r326302;
        double r326321 = r326319 - r326320;
        double r326322 = r326321 * r326305;
        double r326323 = r326316 * r326287;
        double r326324 = r326322 - r326323;
        double r326325 = r326299 * r326324;
        double r326326 = r326289 * r326325;
        double r326327 = r326289 * r326326;
        double r326328 = -r326327;
        double r326329 = r326318 + r326328;
        double r326330 = r326289 * r326317;
        double r326331 = r326289 * r326330;
        double r326332 = r326329 / r326331;
        double r326333 = r326287 * r326332;
        double r326334 = exp(r326333);
        double r326335 = r326286 * r326334;
        double r326336 = r326285 + r326335;
        double r326337 = r326285 / r326336;
        double r326338 = r326284 ? r326314 : r326337;
        return r326338;
}

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

Target

Original	3.7
Target	2.8
Herbie	2.6

\[\begin{array}{l} \mathbf{if}\;t \lt -2.118326644891581057561884576920117070548 \cdot 10^{-50}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\left(a \cdot c + 0.8333333333333333703407674875052180141211 \cdot c\right) - a \cdot b\right)}}\\ \mathbf{elif}\;t \lt 5.196588770651547088010424937268931048836 \cdot 10^{-123}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{\left(z \cdot \sqrt{t + a}\right) \cdot \left(\left(3 \cdot t\right) \cdot \left(a - \frac{5}{6}\right)\right) - \left(\left(\frac{5}{6} + a\right) \cdot \left(3 \cdot t\right) - 2\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(\left(b - c\right) \cdot t\right)\right)}{\left(\left(t \cdot t\right) \cdot 3\right) \cdot \left(a - \frac{5}{6}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if t < -1.6431028539724054e-63 or 4.661190486741443e-116 < t

   1. Initial program 2.3

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 2.3

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{\color{blue}{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}}} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]

   4. Applied times-frac 0.5

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\color{blue}{\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}}} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]

   if -1.6431028539724054e-63 < t < 4.661190486741443e-116

   1. Initial program 6.6

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 6.6

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{\color{blue}{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}}} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]

   4. Applied times-frac 6.8

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\color{blue}{\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}}} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
   5. Using strategy rm

   6. Applied add-log-exp 33.0

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z}{\sqrt[3]{t} \cdot \color{blue}{\log \left(e^{\sqrt[3]{t}}\right)}} \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
   7. Using strategy rm

   8. Applied flip-+ 34.8

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z}{\sqrt[3]{t} \cdot \log \left(e^{\sqrt[3]{t}}\right)} \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}} - \left(b - c\right) \cdot \left(\color{blue}{\frac{a \cdot a - \frac{5}{6} \cdot \frac{5}{6}}{a - \frac{5}{6}}} - \frac{2}{t \cdot 3}\right)\right)}}\]

   9. Applied frac-sub 34.8

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z}{\sqrt[3]{t} \cdot \log \left(e^{\sqrt[3]{t}}\right)} \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}} - \left(b - c\right) \cdot \color{blue}{\frac{\left(a \cdot a - \frac{5}{6} \cdot \frac{5}{6}\right) \cdot \left(t \cdot 3\right) - \left(a - \frac{5}{6}\right) \cdot 2}{\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)}}\right)}}\]

   10. Applied associate-*r/ 34.8

       \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z}{\sqrt[3]{t} \cdot \log \left(e^{\sqrt[3]{t}}\right)} \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}} - \color{blue}{\frac{\left(b - c\right) \cdot \left(\left(a \cdot a - \frac{5}{6} \cdot \frac{5}{6}\right) \cdot \left(t \cdot 3\right) - \left(a - \frac{5}{6}\right) \cdot 2\right)}{\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)}}\right)}}\]

   11. Applied associate-*l/ 34.9

       \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\color{blue}{\frac{z \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}}}{\sqrt[3]{t} \cdot \log \left(e^{\sqrt[3]{t}}\right)}} - \frac{\left(b - c\right) \cdot \left(\left(a \cdot a - \frac{5}{6} \cdot \frac{5}{6}\right) \cdot \left(t \cdot 3\right) - \left(a - \frac{5}{6}\right) \cdot 2\right)}{\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)}\right)}}\]

   12. Applied frac-sub 46.1

       \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\frac{\left(z \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}}\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right) - \left(\sqrt[3]{t} \cdot \log \left(e^{\sqrt[3]{t}}\right)\right) \cdot \left(\left(b - c\right) \cdot \left(\left(a \cdot a - \frac{5}{6} \cdot \frac{5}{6}\right) \cdot \left(t \cdot 3\right) - \left(a - \frac{5}{6}\right) \cdot 2\right)\right)}{\left(\sqrt[3]{t} \cdot \log \left(e^{\sqrt[3]{t}}\right)\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right)}}}}\]

   13. Simplified 32.1

       \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \frac{\color{blue}{\left(z \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}}\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right) + \left(-\sqrt[3]{t} \cdot \left(\sqrt[3]{t} \cdot \left(\left(b - c\right) \cdot \left(\left(a \cdot a - \frac{5}{6} \cdot \frac{5}{6}\right) \cdot \left(t \cdot 3\right) - \left(a - \frac{5}{6}\right) \cdot 2\right)\right)\right)\right)}}{\left(\sqrt[3]{t} \cdot \log \left(e^{\sqrt[3]{t}}\right)\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right)}}}\]

   14. Simplified 6.7

       \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \frac{\left(z \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}}\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right) + \left(-\sqrt[3]{t} \cdot \left(\sqrt[3]{t} \cdot \left(\left(b - c\right) \cdot \left(\left(a \cdot a - \frac{5}{6} \cdot \frac{5}{6}\right) \cdot \left(t \cdot 3\right) - \left(a - \frac{5}{6}\right) \cdot 2\right)\right)\right)\right)}{\color{blue}{\sqrt[3]{t} \cdot \left(\sqrt[3]{t} \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right)\right)}}}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 2.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le \frac{-3044132996692187}{1.852673427797059126777135760139006525652 \cdot 10^{78}} \lor \neg \left(t \le \frac{8271322624711839}{1.774508604237321510130185077851573570199 \cdot 10^{131}}\right):\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{\left(z \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}}\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right) + \left(-\sqrt[3]{t} \cdot \left(\sqrt[3]{t} \cdot \left(\left(b - c\right) \cdot \left(\left(a \cdot a - \frac{5}{6} \cdot \frac{5}{6}\right) \cdot \left(t \cdot 3\right) - \left(a - \frac{5}{6}\right) \cdot 2\right)\right)\right)\right)}{\sqrt[3]{t} \cdot \left(\sqrt[3]{t} \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right)\right)}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 
(FPCore (x y z t a b c)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, I"
  :precision binary64

  :herbie-target
  (if (< t -2.1183266448915811e-50) (/ x (+ x (* y (exp (* 2 (- (+ (* a c) (* 0.83333333333333337 c)) (* a b))))))) (if (< t 5.19658877065154709e-123) (/ x (+ x (* y (exp (* 2 (/ (- (* (* z (sqrt (+ t a))) (* (* 3 t) (- a (/ 5 6)))) (* (- (* (+ (/ 5 6) a) (* 3 t)) 2) (* (- a (/ 5 6)) (* (- b c) t)))) (* (* (* t t) 3) (- a (/ 5 6))))))))) (/ x (+ x (* y (exp (* 2 (- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5 6)) (/ 2 (* t 3))))))))))))

  (/ x (+ x (* y (exp (* 2 (- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5 6)) (/ 2 (* t 3)))))))))))