Average Error: 23.8 → 15.7
Time: 13.1s
Precision: 64
\[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} = -\infty:\\ \;\;\;\;x\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \le -1.0453105058496541 \cdot 10^{-301}:\\ \;\;\;\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \le 0.0 \lor \neg \left(\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \le 1.9961735121647787 \cdot 10^{271}\right):\\ \;\;\;\;\frac{t}{b} - \frac{a}{b}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y + z \cdot \left(t - a\right)\right) \cdot \frac{1}{y + z \cdot \left(b - y\right)}\\ \end{array}\]
\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} = -\infty:\\
\;\;\;\;x\\

\mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \le -1.0453105058496541 \cdot 10^{-301}:\\
\;\;\;\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\

\mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \le 0.0 \lor \neg \left(\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \le 1.9961735121647787 \cdot 10^{271}\right):\\
\;\;\;\;\frac{t}{b} - \frac{a}{b}\\

\mathbf{else}:\\
\;\;\;\;\left(x \cdot y + z \cdot \left(t - a\right)\right) \cdot \frac{1}{y + z \cdot \left(b - y\right)}\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r489773 = x;
        double r489774 = y;
        double r489775 = r489773 * r489774;
        double r489776 = z;
        double r489777 = t;
        double r489778 = a;
        double r489779 = r489777 - r489778;
        double r489780 = r489776 * r489779;
        double r489781 = r489775 + r489780;
        double r489782 = b;
        double r489783 = r489782 - r489774;
        double r489784 = r489776 * r489783;
        double r489785 = r489774 + r489784;
        double r489786 = r489781 / r489785;
        return r489786;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r489787 = x;
        double r489788 = y;
        double r489789 = r489787 * r489788;
        double r489790 = z;
        double r489791 = t;
        double r489792 = a;
        double r489793 = r489791 - r489792;
        double r489794 = r489790 * r489793;
        double r489795 = r489789 + r489794;
        double r489796 = b;
        double r489797 = r489796 - r489788;
        double r489798 = r489790 * r489797;
        double r489799 = r489788 + r489798;
        double r489800 = r489795 / r489799;
        double r489801 = -inf.0;
        bool r489802 = r489800 <= r489801;
        double r489803 = -1.045310505849654e-301;
        bool r489804 = r489800 <= r489803;
        double r489805 = 0.0;
        bool r489806 = r489800 <= r489805;
        double r489807 = 1.9961735121647787e+271;
        bool r489808 = r489800 <= r489807;
        double r489809 = !r489808;
        bool r489810 = r489806 || r489809;
        double r489811 = r489791 / r489796;
        double r489812 = r489792 / r489796;
        double r489813 = r489811 - r489812;
        double r489814 = 1.0;
        double r489815 = r489814 / r489799;
        double r489816 = r489795 * r489815;
        double r489817 = r489810 ? r489813 : r489816;
        double r489818 = r489804 ? r489800 : r489817;
        double r489819 = r489802 ? r489787 : r489818;
        return r489819;
}

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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original23.8
Target18.2
Herbie15.7
\[\frac{z \cdot t + y \cdot x}{y + z \cdot \left(b - y\right)} - \frac{a}{\left(b - y\right) + \frac{y}{z}}\]

Derivation

  1. Split input into 4 regimes

  2. if (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))) < -inf.0

    1. Initial program 64.0

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\]
    2. Using strategy rm

    3. Applied clear-num64.0

      \[\leadsto \color{blue}{\frac{1}{\frac{y + z \cdot \left(b - y\right)}{x \cdot y + z \cdot \left(t - a\right)}}}\]
    4. Using strategy rm

    5. Applied div-inv64.0

      \[\leadsto \frac{1}{\color{blue}{\left(y + z \cdot \left(b - y\right)\right) \cdot \frac{1}{x \cdot y + z \cdot \left(t - a\right)}}}\]

    6. Applied associate-/r*64.0

      \[\leadsto \color{blue}{\frac{\frac{1}{y + z \cdot \left(b - y\right)}}{\frac{1}{x \cdot y + z \cdot \left(t - a\right)}}}\]

    7. Simplified64.0

      \[\leadsto \frac{\color{blue}{\frac{1}{z \cdot \left(b - y\right) + y}}}{\frac{1}{x \cdot y + z \cdot \left(t - a\right)}}\]

    8. Taylor expanded around 0 37.3

      \[\leadsto \color{blue}{x}\]

    if -inf.0 < (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))) < -1.045310505849654e-301

    1. Initial program 0.3

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\]

    if -1.045310505849654e-301 < (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))) < 0.0 or 1.9961735121647787e+271 < (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y))))

    1. Initial program 56.8

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\]
    2. Using strategy rm

    3. Applied clear-num56.8

      \[\leadsto \color{blue}{\frac{1}{\frac{y + z \cdot \left(b - y\right)}{x \cdot y + z \cdot \left(t - a\right)}}}\]
    4. Using strategy rm

    5. Applied div-inv56.8

      \[\leadsto \frac{1}{\color{blue}{\left(y + z \cdot \left(b - y\right)\right) \cdot \frac{1}{x \cdot y + z \cdot \left(t - a\right)}}}\]

    6. Applied associate-/r*56.8

      \[\leadsto \color{blue}{\frac{\frac{1}{y + z \cdot \left(b - y\right)}}{\frac{1}{x \cdot y + z \cdot \left(t - a\right)}}}\]

    7. Simplified56.8

      \[\leadsto \frac{\color{blue}{\frac{1}{z \cdot \left(b - y\right) + y}}}{\frac{1}{x \cdot y + z \cdot \left(t - a\right)}}\]

    8. Taylor expanded around inf 38.3

      \[\leadsto \color{blue}{\frac{t}{b} - \frac{a}{b}}\]

    if 0.0 < (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))) < 1.9961735121647787e+271

    1. Initial program 0.3

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\]
    2. Using strategy rm

    3. Applied div-inv0.4

      \[\leadsto \color{blue}{\left(x \cdot y + z \cdot \left(t - a\right)\right) \cdot \frac{1}{y + z \cdot \left(b - y\right)}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification15.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} = -\infty:\\ \;\;\;\;x\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \le -1.0453105058496541 \cdot 10^{-301}:\\ \;\;\;\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \le 0.0 \lor \neg \left(\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \le 1.9961735121647787 \cdot 10^{271}\right):\\ \;\;\;\;\frac{t}{b} - \frac{a}{b}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y + z \cdot \left(t - a\right)\right) \cdot \frac{1}{y + z \cdot \left(b - y\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 
(FPCore (x y z t a b)
  :name "Development.Shake.Progress:decay from shake-0.15.5"
  :precision binary64

  :herbie-target
  (- (/ (+ (* z t) (* y x)) (+ y (* z (- b y)))) (/ a (+ (- b y) (/ y z))))

  (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))))