Average Error: 23.8 → 15.7
Time: 13.3s
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 r480230 = x;
        double r480231 = y;
        double r480232 = r480230 * r480231;
        double r480233 = z;
        double r480234 = t;
        double r480235 = a;
        double r480236 = r480234 - r480235;
        double r480237 = r480233 * r480236;
        double r480238 = r480232 + r480237;
        double r480239 = b;
        double r480240 = r480239 - r480231;
        double r480241 = r480233 * r480240;
        double r480242 = r480231 + r480241;
        double r480243 = r480238 / r480242;
        return r480243;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r480244 = x;
        double r480245 = y;
        double r480246 = r480244 * r480245;
        double r480247 = z;
        double r480248 = t;
        double r480249 = a;
        double r480250 = r480248 - r480249;
        double r480251 = r480247 * r480250;
        double r480252 = r480246 + r480251;
        double r480253 = b;
        double r480254 = r480253 - r480245;
        double r480255 = r480247 * r480254;
        double r480256 = r480245 + r480255;
        double r480257 = r480252 / r480256;
        double r480258 = -inf.0;
        bool r480259 = r480257 <= r480258;
        double r480260 = -1.045310505849654e-301;
        bool r480261 = r480257 <= r480260;
        double r480262 = 0.0;
        bool r480263 = r480257 <= r480262;
        double r480264 = 1.9961735121647787e+271;
        bool r480265 = r480257 <= r480264;
        double r480266 = !r480265;
        bool r480267 = r480263 || r480266;
        double r480268 = r480248 / r480253;
        double r480269 = r480249 / r480253;
        double r480270 = r480268 - r480269;
        double r480271 = 1.0;
        double r480272 = r480271 / r480256;
        double r480273 = r480252 * r480272;
        double r480274 = r480267 ? r480270 : r480273;
        double r480275 = r480261 ? r480257 : r480274;
        double r480276 = r480259 ? r480244 : r480275;
        return r480276;
}

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)))))