Average Error: 3.7 → 3.3
Time: 36.9s
Precision: 64
\[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.978001093890994 \cdot 10^{+55}:\\ \;\;\;\;\frac{x}{x + e^{\left(\frac{\left(\left(\sqrt{t + a} \cdot z\right)\right)}{t} - \left(\left(\frac{5.0}{6.0} + a\right) - \frac{2.0}{3.0 \cdot t}\right) \cdot \left(b - c\right)\right) \cdot 2.0} \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2.0 \cdot \left(\log \left(\sqrt{e^{\sqrt{t + a} \cdot \frac{z}{t} - \left(\frac{5.0}{6.0} + \left(a - \frac{\frac{2.0}{t}}{3.0}\right)\right) \cdot \left(b - c\right)}}\right) + \log \left(\sqrt{e^{\sqrt{t + a} \cdot \frac{z}{t} - \left(\frac{5.0}{6.0} + \left(a - \frac{\frac{2.0}{t}}{3.0}\right)\right) \cdot \left(b - c\right)}}\right)\right)}}\\ \end{array}\]
\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)}}
↓
\begin{array}{l}
\mathbf{if}\;b \le -2.978001093890994 \cdot 10^{+55}:\\
\;\;\;\;\frac{x}{x + e^{\left(\frac{\left(\left(\sqrt{t + a} \cdot z\right)\right)}{t} - \left(\left(\frac{5.0}{6.0} + a\right) - \frac{2.0}{3.0 \cdot t}\right) \cdot \left(b - c\right)\right) \cdot 2.0} \cdot y}\\

\mathbf{else}:\\
\;\;\;\;\frac{x}{x + y \cdot e^{2.0 \cdot \left(\log \left(\sqrt{e^{\sqrt{t + a} \cdot \frac{z}{t} - \left(\frac{5.0}{6.0} + \left(a - \frac{\frac{2.0}{t}}{3.0}\right)\right) \cdot \left(b - c\right)}}\right) + \log \left(\sqrt{e^{\sqrt{t + a} \cdot \frac{z}{t} - \left(\frac{5.0}{6.0} + \left(a - \frac{\frac{2.0}{t}}{3.0}\right)\right) \cdot \left(b - c\right)}}\right)\right)}}\\

\end{array}
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r23456928 = x;
        double r23456929 = y;
        double r23456930 = 2.0;
        double r23456931 = z;
        double r23456932 = t;
        double r23456933 = a;
        double r23456934 = r23456932 + r23456933;
        double r23456935 = sqrt(r23456934);
        double r23456936 = r23456931 * r23456935;
        double r23456937 = r23456936 / r23456932;
        double r23456938 = b;
        double r23456939 = c;
        double r23456940 = r23456938 - r23456939;
        double r23456941 = 5.0;
        double r23456942 = 6.0;
        double r23456943 = r23456941 / r23456942;
        double r23456944 = r23456933 + r23456943;
        double r23456945 = 3.0;
        double r23456946 = r23456932 * r23456945;
        double r23456947 = r23456930 / r23456946;
        double r23456948 = r23456944 - r23456947;
        double r23456949 = r23456940 * r23456948;
        double r23456950 = r23456937 - r23456949;
        double r23456951 = r23456930 * r23456950;
        double r23456952 = exp(r23456951);
        double r23456953 = r23456929 * r23456952;
        double r23456954 = r23456928 + r23456953;
        double r23456955 = r23456928 / r23456954;
        return r23456955;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r23456956 = b;
        double r23456957 = -2.978001093890994e+55;
        bool r23456958 = r23456956 <= r23456957;
        double r23456959 = x;
        double r23456960 = t;
        double r23456961 = a;
        double r23456962 = r23456960 + r23456961;
        double r23456963 = sqrt(r23456962);
        double r23456964 = z;
        double r23456965 = r23456963 * r23456964;
        double r23456966 = /* ERROR: no posit support in C */;
        double r23456967 = /* ERROR: no posit support in C */;
        double r23456968 = r23456967 / r23456960;
        double r23456969 = 5.0;
        double r23456970 = 6.0;
        double r23456971 = r23456969 / r23456970;
        double r23456972 = r23456971 + r23456961;
        double r23456973 = 2.0;
        double r23456974 = 3.0;
        double r23456975 = r23456974 * r23456960;
        double r23456976 = r23456973 / r23456975;
        double r23456977 = r23456972 - r23456976;
        double r23456978 = c;
        double r23456979 = r23456956 - r23456978;
        double r23456980 = r23456977 * r23456979;
        double r23456981 = r23456968 - r23456980;
        double r23456982 = r23456981 * r23456973;
        double r23456983 = exp(r23456982);
        double r23456984 = y;
        double r23456985 = r23456983 * r23456984;
        double r23456986 = r23456959 + r23456985;
        double r23456987 = r23456959 / r23456986;
        double r23456988 = r23456964 / r23456960;
        double r23456989 = r23456963 * r23456988;
        double r23456990 = r23456973 / r23456960;
        double r23456991 = r23456990 / r23456974;
        double r23456992 = r23456961 - r23456991;
        double r23456993 = r23456971 + r23456992;
        double r23456994 = r23456993 * r23456979;
        double r23456995 = r23456989 - r23456994;
        double r23456996 = exp(r23456995);
        double r23456997 = sqrt(r23456996);
        double r23456998 = log(r23456997);
        double r23456999 = r23456998 + r23456998;
        double r23457000 = r23456973 * r23456999;
        double r23457001 = exp(r23457000);
        double r23457002 = r23456984 * r23457001;
        double r23457003 = r23456959 + r23457002;
        double r23457004 = r23456959 / r23457003;
        double r23457005 = r23456958 ? r23456987 : r23457004;
        return r23457005;
}

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

Original3.7
Target3.0
Herbie3.3
\[\begin{array}{l} \mathbf{if}\;t \lt -2.118326644891581 \cdot 10^{-50}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2.0 \cdot \left(\left(a \cdot c + 0.8333333333333334 \cdot c\right) - a \cdot b\right)}}\\ \mathbf{elif}\;t \lt 5.196588770651547 \cdot 10^{-123}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2.0 \cdot \frac{\left(z \cdot \sqrt{t + a}\right) \cdot \left(\left(3.0 \cdot t\right) \cdot \left(a - \frac{5.0}{6.0}\right)\right) - \left(\left(\frac{5.0}{6.0} + a\right) \cdot \left(3.0 \cdot t\right) - 2.0\right) \cdot \left(\left(a - \frac{5.0}{6.0}\right) \cdot \left(\left(b - c\right) \cdot t\right)\right)}{\left(\left(t \cdot t\right) \cdot 3.0\right) \cdot \left(a - \frac{5.0}{6.0}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if b < -2.978001093890994e+55

    1. Initial program 5.4

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

    3. Applied insert-posit165.7

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

    if -2.978001093890994e+55 < b

    1. Initial program 3.2

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

    3. Applied add-log-exp8.1

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

    4. Applied add-log-exp15.6

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

    5. Applied diff-log15.6

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

    6. Simplified2.6

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

    8. Applied add-sqr-sqrt2.6

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

    9. Applied log-prod2.6

      \[\leadsto \frac{x}{x + y \cdot e^{2.0 \cdot \color{blue}{\left(\log \left(\sqrt{e^{\frac{z}{t} \cdot \sqrt{t + a} - \left(\frac{5.0}{6.0} + \left(a - \frac{\frac{2.0}{t}}{3.0}\right)\right) \cdot \left(b - c\right)}}\right) + \log \left(\sqrt{e^{\frac{z}{t} \cdot \sqrt{t + a} - \left(\frac{5.0}{6.0} + \left(a - \frac{\frac{2.0}{t}}{3.0}\right)\right) \cdot \left(b - c\right)}}\right)\right)}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification3.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.978001093890994 \cdot 10^{+55}:\\ \;\;\;\;\frac{x}{x + e^{\left(\frac{\left(\left(\sqrt{t + a} \cdot z\right)\right)}{t} - \left(\left(\frac{5.0}{6.0} + a\right) - \frac{2.0}{3.0 \cdot t}\right) \cdot \left(b - c\right)\right) \cdot 2.0} \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2.0 \cdot \left(\log \left(\sqrt{e^{\sqrt{t + a} \cdot \frac{z}{t} - \left(\frac{5.0}{6.0} + \left(a - \frac{\frac{2.0}{t}}{3.0}\right)\right) \cdot \left(b - c\right)}}\right) + \log \left(\sqrt{e^{\sqrt{t + a} \cdot \frac{z}{t} - \left(\frac{5.0}{6.0} + \left(a - \frac{\frac{2.0}{t}}{3.0}\right)\right) \cdot \left(b - c\right)}}\right)\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 
(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)))))))))))