Average Error: 3.9 → 5.6
Time: 39.9s
Precision: 64
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -699293331120.752197265625:\\ \;\;\;\;\frac{x}{x + e^{\left(\frac{\sqrt{t + a} \cdot z}{t} - \left(\left(\frac{5}{6} + a\right) - \log \left(e^{\frac{2}{t \cdot 3}}\right)\right) \cdot \left(b - c\right)\right) \cdot 2} \cdot y}\\ \mathbf{elif}\;t \le 9.289679889769380475433857569100837281439 \cdot 10^{-4}:\\ \;\;\;\;\frac{x}{y \cdot e^{\frac{\left(\left(b + c\right) \cdot \sqrt{t + a}\right) \cdot z - \left(t \cdot \left(\left(\frac{5}{6} - \frac{\frac{2}{t}}{3}\right) + a\right)\right) \cdot \left(\left(b + c\right) \cdot \left(b - c\right)\right)}{t \cdot \left(b + c\right)} \cdot 2} + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + e^{\left(\frac{\sqrt{t + a} \cdot z}{t} - \left(\left(\frac{5}{6} + a\right) - \log \left(e^{\frac{2}{t \cdot 3}}\right)\right) \cdot \left(b - c\right)\right) \cdot 2} \cdot y}\\ \end{array}\]
\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 -699293331120.752197265625:\\
\;\;\;\;\frac{x}{x + e^{\left(\frac{\sqrt{t + a} \cdot z}{t} - \left(\left(\frac{5}{6} + a\right) - \log \left(e^{\frac{2}{t \cdot 3}}\right)\right) \cdot \left(b - c\right)\right) \cdot 2} \cdot y}\\

\mathbf{elif}\;t \le 9.289679889769380475433857569100837281439 \cdot 10^{-4}:\\
\;\;\;\;\frac{x}{y \cdot e^{\frac{\left(\left(b + c\right) \cdot \sqrt{t + a}\right) \cdot z - \left(t \cdot \left(\left(\frac{5}{6} - \frac{\frac{2}{t}}{3}\right) + a\right)\right) \cdot \left(\left(b + c\right) \cdot \left(b - c\right)\right)}{t \cdot \left(b + c\right)} \cdot 2} + x}\\

\mathbf{else}:\\
\;\;\;\;\frac{x}{x + e^{\left(\frac{\sqrt{t + a} \cdot z}{t} - \left(\left(\frac{5}{6} + a\right) - \log \left(e^{\frac{2}{t \cdot 3}}\right)\right) \cdot \left(b - c\right)\right) \cdot 2} \cdot y}\\

\end{array}
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r25146231 = x;
        double r25146232 = y;
        double r25146233 = 2.0;
        double r25146234 = z;
        double r25146235 = t;
        double r25146236 = a;
        double r25146237 = r25146235 + r25146236;
        double r25146238 = sqrt(r25146237);
        double r25146239 = r25146234 * r25146238;
        double r25146240 = r25146239 / r25146235;
        double r25146241 = b;
        double r25146242 = c;
        double r25146243 = r25146241 - r25146242;
        double r25146244 = 5.0;
        double r25146245 = 6.0;
        double r25146246 = r25146244 / r25146245;
        double r25146247 = r25146236 + r25146246;
        double r25146248 = 3.0;
        double r25146249 = r25146235 * r25146248;
        double r25146250 = r25146233 / r25146249;
        double r25146251 = r25146247 - r25146250;
        double r25146252 = r25146243 * r25146251;
        double r25146253 = r25146240 - r25146252;
        double r25146254 = r25146233 * r25146253;
        double r25146255 = exp(r25146254);
        double r25146256 = r25146232 * r25146255;
        double r25146257 = r25146231 + r25146256;
        double r25146258 = r25146231 / r25146257;
        return r25146258;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r25146259 = t;
        double r25146260 = -699293331120.7522;
        bool r25146261 = r25146259 <= r25146260;
        double r25146262 = x;
        double r25146263 = a;
        double r25146264 = r25146259 + r25146263;
        double r25146265 = sqrt(r25146264);
        double r25146266 = z;
        double r25146267 = r25146265 * r25146266;
        double r25146268 = r25146267 / r25146259;
        double r25146269 = 5.0;
        double r25146270 = 6.0;
        double r25146271 = r25146269 / r25146270;
        double r25146272 = r25146271 + r25146263;
        double r25146273 = 2.0;
        double r25146274 = 3.0;
        double r25146275 = r25146259 * r25146274;
        double r25146276 = r25146273 / r25146275;
        double r25146277 = exp(r25146276);
        double r25146278 = log(r25146277);
        double r25146279 = r25146272 - r25146278;
        double r25146280 = b;
        double r25146281 = c;
        double r25146282 = r25146280 - r25146281;
        double r25146283 = r25146279 * r25146282;
        double r25146284 = r25146268 - r25146283;
        double r25146285 = r25146284 * r25146273;
        double r25146286 = exp(r25146285);
        double r25146287 = y;
        double r25146288 = r25146286 * r25146287;
        double r25146289 = r25146262 + r25146288;
        double r25146290 = r25146262 / r25146289;
        double r25146291 = 0.000928967988976938;
        bool r25146292 = r25146259 <= r25146291;
        double r25146293 = r25146280 + r25146281;
        double r25146294 = r25146293 * r25146265;
        double r25146295 = r25146294 * r25146266;
        double r25146296 = r25146273 / r25146259;
        double r25146297 = r25146296 / r25146274;
        double r25146298 = r25146271 - r25146297;
        double r25146299 = r25146298 + r25146263;
        double r25146300 = r25146259 * r25146299;
        double r25146301 = r25146293 * r25146282;
        double r25146302 = r25146300 * r25146301;
        double r25146303 = r25146295 - r25146302;
        double r25146304 = r25146259 * r25146293;
        double r25146305 = r25146303 / r25146304;
        double r25146306 = r25146305 * r25146273;
        double r25146307 = exp(r25146306);
        double r25146308 = r25146287 * r25146307;
        double r25146309 = r25146308 + r25146262;
        double r25146310 = r25146262 / r25146309;
        double r25146311 = r25146292 ? r25146310 : r25146290;
        double r25146312 = r25146261 ? r25146290 : r25146311;
        return r25146312;
}

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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original3.9
Target2.9
Herbie5.6
\[\begin{array}{l} \mathbf{if}\;t \lt -2.118326644891581057561884576920117070548 \cdot 10^{-50}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\left(a \cdot c + 0.8333333333333333703407674875052180141211 \cdot c\right) - a \cdot b\right)}}\\ \mathbf{elif}\;t \lt 5.196588770651547088010424937268931048836 \cdot 10^{-123}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{\left(z \cdot \sqrt{t + a}\right) \cdot \left(\left(3 \cdot t\right) \cdot \left(a - \frac{5}{6}\right)\right) - \left(\left(\frac{5}{6} + a\right) \cdot \left(3 \cdot t\right) - 2\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(\left(b - c\right) \cdot t\right)\right)}{\left(\left(t \cdot t\right) \cdot 3\right) \cdot \left(a - \frac{5}{6}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if t < -699293331120.7522 or 0.000928967988976938 < t

    1. Initial program 2.8

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

    3. Applied add-log-exp2.8

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

    if -699293331120.7522 < t < 0.000928967988976938

    1. Initial program 5.1

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

    3. Applied flip--9.0

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

    4. Applied associate-*l/9.7

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

    5. Applied frac-sub11.4

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

    6. Simplified8.4

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

  4. Final simplification5.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -699293331120.752197265625:\\ \;\;\;\;\frac{x}{x + e^{\left(\frac{\sqrt{t + a} \cdot z}{t} - \left(\left(\frac{5}{6} + a\right) - \log \left(e^{\frac{2}{t \cdot 3}}\right)\right) \cdot \left(b - c\right)\right) \cdot 2} \cdot y}\\ \mathbf{elif}\;t \le 9.289679889769380475433857569100837281439 \cdot 10^{-4}:\\ \;\;\;\;\frac{x}{y \cdot e^{\frac{\left(\left(b + c\right) \cdot \sqrt{t + a}\right) \cdot z - \left(t \cdot \left(\left(\frac{5}{6} - \frac{\frac{2}{t}}{3}\right) + a\right)\right) \cdot \left(\left(b + c\right) \cdot \left(b - c\right)\right)}{t \cdot \left(b + c\right)} \cdot 2} + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + e^{\left(\frac{\sqrt{t + a} \cdot z}{t} - \left(\left(\frac{5}{6} + a\right) - \log \left(e^{\frac{2}{t \cdot 3}}\right)\right) \cdot \left(b - c\right)\right) \cdot 2} \cdot y}\\ \end{array}\]

Reproduce

herbie shell --seed 2019169 
(FPCore (x y z t a b c)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, I"

  :herbie-target
  (if (< t -2.118326644891581e-50) (/ x (+ x (* y (exp (* 2.0 (- (+ (* a c) (* 0.8333333333333334 c)) (* a b))))))) (if (< t 5.196588770651547e-123) (/ x (+ x (* y (exp (* 2.0 (/ (- (* (* z (sqrt (+ t a))) (* (* 3.0 t) (- a (/ 5.0 6.0)))) (* (- (* (+ (/ 5.0 6.0) a) (* 3.0 t)) 2.0) (* (- a (/ 5.0 6.0)) (* (- b c) t)))) (* (* (* t t) 3.0) (- a (/ 5.0 6.0))))))))) (/ x (+ x (* y (exp (* 2.0 (- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0))))))))))))

  (/ x (+ x (* y (exp (* 2.0 (- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0)))))))))))