Average Error: 26.9 → 16.0
Time: 6.4s
Precision: 64
\[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -1.5789142602105122 \cdot 10^{46} \lor \neg \left(y \le 3.00522161855546254 \cdot 10^{112}\right):\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b\right) \cdot \frac{1}{\left(x + t\right) + y}\\ \end{array}\]
\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}
↓
\begin{array}{l}
\mathbf{if}\;y \le -1.5789142602105122 \cdot 10^{46} \lor \neg \left(y \le 3.00522161855546254 \cdot 10^{112}\right):\\
\;\;\;\;\left(a + z\right) - b\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r951262 = x;
        double r951263 = y;
        double r951264 = r951262 + r951263;
        double r951265 = z;
        double r951266 = r951264 * r951265;
        double r951267 = t;
        double r951268 = r951267 + r951263;
        double r951269 = a;
        double r951270 = r951268 * r951269;
        double r951271 = r951266 + r951270;
        double r951272 = b;
        double r951273 = r951263 * r951272;
        double r951274 = r951271 - r951273;
        double r951275 = r951262 + r951267;
        double r951276 = r951275 + r951263;
        double r951277 = r951274 / r951276;
        return r951277;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r951278 = y;
        double r951279 = -1.5789142602105122e+46;
        bool r951280 = r951278 <= r951279;
        double r951281 = 3.0052216185554625e+112;
        bool r951282 = r951278 <= r951281;
        double r951283 = !r951282;
        bool r951284 = r951280 || r951283;
        double r951285 = a;
        double r951286 = z;
        double r951287 = r951285 + r951286;
        double r951288 = b;
        double r951289 = r951287 - r951288;
        double r951290 = x;
        double r951291 = r951290 + r951278;
        double r951292 = r951291 * r951286;
        double r951293 = t;
        double r951294 = r951293 + r951278;
        double r951295 = r951294 * r951285;
        double r951296 = r951292 + r951295;
        double r951297 = r951278 * r951288;
        double r951298 = r951296 - r951297;
        double r951299 = 1.0;
        double r951300 = r951290 + r951293;
        double r951301 = r951300 + r951278;
        double r951302 = r951299 / r951301;
        double r951303 = r951298 * r951302;
        double r951304 = r951284 ? r951289 : r951303;
        return r951304;
}

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

Original26.9
Target11.4
Herbie16.0
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \lt -3.5813117084150564 \cdot 10^{153}:\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{elif}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \lt 1.2285964308315609 \cdot 10^{82}:\\ \;\;\;\;\frac{1}{\frac{\left(x + t\right) + y}{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\left(z + a\right) - b\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -1.5789142602105122e+46 or 3.0052216185554625e+112 < y

    1. Initial program 43.7

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

    3. Applied clear-num43.8

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

    4. Simplified43.8

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

    5. Taylor expanded around 0 14.3

      \[\leadsto \color{blue}{\left(a + z\right) - b}\]

    if -1.5789142602105122e+46 < y < 3.0052216185554625e+112

    1. Initial program 16.9

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

    3. Applied div-inv17.0

      \[\leadsto \color{blue}{\left(\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b\right) \cdot \frac{1}{\left(x + t\right) + y}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification16.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.5789142602105122 \cdot 10^{46} \lor \neg \left(y \le 3.00522161855546254 \cdot 10^{112}\right):\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b\right) \cdot \frac{1}{\left(x + t\right) + y}\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 +o rules:numerics
(FPCore (x y z t a b)
  :name "AI.Clustering.Hierarchical.Internal:ward from clustering-0.2.1"
  :precision binary64

  :herbie-target
  (if (< (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)) -3.5813117084150564e+153) (- (+ z a) b) (if (< (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)) 1.2285964308315609e+82) (/ 1 (/ (+ (+ x t) y) (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)))) (- (+ z a) b)))

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