Average Error: 3.8 → 3.2
Time: 37.5s
Precision: 64
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.3988457469696705 \cdot 10^{-242} \lor \neg \left(t \le 3.22201208794726605 \cdot 10^{-194}\right):\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{\left(\left|\sqrt[3]{t + a}\right| \cdot z\right) \cdot \sqrt{\sqrt[3]{t + a}}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{\left(\left(\left|\sqrt[3]{t + a}\right| \cdot z\right) \cdot \sqrt{\sqrt[3]{t + a}}\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right) - t \cdot \left(\left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right) - \left(a - \frac{5}{6}\right) \cdot 2\right)\right)}{t \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right)}}}\\ \end{array}\]
\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 -1.3988457469696705 \cdot 10^{-242} \lor \neg \left(t \le 3.22201208794726605 \cdot 10^{-194}\right):\\
\;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{\left(\left|\sqrt[3]{t + a}\right| \cdot z\right) \cdot \sqrt{\sqrt[3]{t + a}}}{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(\left(\left|\sqrt[3]{t + a}\right| \cdot z\right) \cdot \sqrt{\sqrt[3]{t + a}}\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right) - t \cdot \left(\left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right) - \left(a - \frac{5}{6}\right) \cdot 2\right)\right)}{t \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right)}}}\\

\end{array}
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r524208 = x;
        double r524209 = y;
        double r524210 = 2.0;
        double r524211 = z;
        double r524212 = t;
        double r524213 = a;
        double r524214 = r524212 + r524213;
        double r524215 = sqrt(r524214);
        double r524216 = r524211 * r524215;
        double r524217 = r524216 / r524212;
        double r524218 = b;
        double r524219 = c;
        double r524220 = r524218 - r524219;
        double r524221 = 5.0;
        double r524222 = 6.0;
        double r524223 = r524221 / r524222;
        double r524224 = r524213 + r524223;
        double r524225 = 3.0;
        double r524226 = r524212 * r524225;
        double r524227 = r524210 / r524226;
        double r524228 = r524224 - r524227;
        double r524229 = r524220 * r524228;
        double r524230 = r524217 - r524229;
        double r524231 = r524210 * r524230;
        double r524232 = exp(r524231);
        double r524233 = r524209 * r524232;
        double r524234 = r524208 + r524233;
        double r524235 = r524208 / r524234;
        return r524235;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r524236 = t;
        double r524237 = -1.3988457469696705e-242;
        bool r524238 = r524236 <= r524237;
        double r524239 = 3.222012087947266e-194;
        bool r524240 = r524236 <= r524239;
        double r524241 = !r524240;
        bool r524242 = r524238 || r524241;
        double r524243 = x;
        double r524244 = y;
        double r524245 = 2.0;
        double r524246 = a;
        double r524247 = r524236 + r524246;
        double r524248 = cbrt(r524247);
        double r524249 = fabs(r524248);
        double r524250 = z;
        double r524251 = r524249 * r524250;
        double r524252 = sqrt(r524248);
        double r524253 = r524251 * r524252;
        double r524254 = r524253 / r524236;
        double r524255 = b;
        double r524256 = c;
        double r524257 = r524255 - r524256;
        double r524258 = 5.0;
        double r524259 = 6.0;
        double r524260 = r524258 / r524259;
        double r524261 = r524246 + r524260;
        double r524262 = 3.0;
        double r524263 = r524236 * r524262;
        double r524264 = r524245 / r524263;
        double r524265 = r524261 - r524264;
        double r524266 = r524257 * r524265;
        double r524267 = r524254 - r524266;
        double r524268 = r524245 * r524267;
        double r524269 = exp(r524268);
        double r524270 = r524244 * r524269;
        double r524271 = r524243 + r524270;
        double r524272 = r524243 / r524271;
        double r524273 = r524246 - r524260;
        double r524274 = r524273 * r524263;
        double r524275 = r524253 * r524274;
        double r524276 = r524261 * r524274;
        double r524277 = r524273 * r524245;
        double r524278 = r524276 - r524277;
        double r524279 = r524257 * r524278;
        double r524280 = r524236 * r524279;
        double r524281 = r524275 - r524280;
        double r524282 = r524236 * r524274;
        double r524283 = r524281 / r524282;
        double r524284 = r524245 * r524283;
        double r524285 = exp(r524284);
        double r524286 = r524244 * r524285;
        double r524287 = r524243 + r524286;
        double r524288 = r524243 / r524287;
        double r524289 = r524242 ? r524272 : r524288;
        return r524289;
}

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

Original3.8
Target2.7
Herbie3.2
\[\begin{array}{l} \mathbf{if}\;t \lt -2.1183266448915811 \cdot 10^{-50}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\left(a \cdot c + 0.83333333333333337 \cdot c\right) - a \cdot b\right)}}\\ \mathbf{elif}\;t \lt 5.19658877065154709 \cdot 10^{-123}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{\left(z \cdot \sqrt{t + a}\right) \cdot \left(\left(3 \cdot t\right) \cdot \left(a - \frac{5}{6}\right)\right) - \left(\left(\frac{5}{6} + a\right) \cdot \left(3 \cdot t\right) - 2\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(\left(b - c\right) \cdot t\right)\right)}{\left(\left(t \cdot t\right) \cdot 3\right) \cdot \left(a - \frac{5}{6}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if t < -1.3988457469696705e-242 or 3.222012087947266e-194 < t

    1. Initial program 2.9

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

    3. Applied add-cube-cbrt2.9

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

    4. Applied sqrt-prod2.9

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

    5. Applied associate-*r*2.9

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

    6. Simplified2.9

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

    if -1.3988457469696705e-242 < t < 3.222012087947266e-194

    1. Initial program 9.2

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

    3. Applied add-cube-cbrt9.2

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

    4. Applied sqrt-prod9.2

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

    5. Applied associate-*r*9.2

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

    6. Simplified9.2

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

    8. Applied flip-+13.9

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

    9. Applied frac-sub13.9

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

    10. Applied associate-*r/13.9

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

    11. Applied frac-sub9.8

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

    13. Applied difference-of-squares9.8

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

    14. Applied associate-*l*4.9

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

  4. Final simplification3.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.3988457469696705 \cdot 10^{-242} \lor \neg \left(t \le 3.22201208794726605 \cdot 10^{-194}\right):\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{\left(\left|\sqrt[3]{t + a}\right| \cdot z\right) \cdot \sqrt{\sqrt[3]{t + a}}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{\left(\left(\left|\sqrt[3]{t + a}\right| \cdot z\right) \cdot \sqrt{\sqrt[3]{t + a}}\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right) - t \cdot \left(\left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right) - \left(a - \frac{5}{6}\right) \cdot 2\right)\right)}{t \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right)}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020027 
(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 -2.118326644891581e-50) (/ x (+ x (* y (exp (* 2 (- (+ (* a c) (* 0.8333333333333334 c)) (* a b))))))) (if (< t 5.196588770651547e-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)))))))))))