Average Error: 27.0 → 7.8
Time: 22.7s
Precision: 64
\[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \le -8.416712259254574092270707749732520074762 \cdot 10^{242}:\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{elif}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \le 7.307468158284447910195168497396941423128 \cdot 10^{286}:\\ \;\;\;\;\frac{1}{\frac{x + \left(t + y\right)}{\mathsf{fma}\left(z, x + y, \left(t + y\right) \cdot a\right) - b \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;\left(a + z\right) - b\\ \end{array}\]
\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}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \le -8.416712259254574092270707749732520074762 \cdot 10^{242}:\\
\;\;\;\;\left(a + z\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} \le 7.307468158284447910195168497396941423128 \cdot 10^{286}:\\
\;\;\;\;\frac{1}{\frac{x + \left(t + y\right)}{\mathsf{fma}\left(z, x + y, \left(t + y\right) \cdot a\right) - b \cdot y}}\\

\mathbf{else}:\\
\;\;\;\;\left(a + z\right) - b\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r54878266 = x;
        double r54878267 = y;
        double r54878268 = r54878266 + r54878267;
        double r54878269 = z;
        double r54878270 = r54878268 * r54878269;
        double r54878271 = t;
        double r54878272 = r54878271 + r54878267;
        double r54878273 = a;
        double r54878274 = r54878272 * r54878273;
        double r54878275 = r54878270 + r54878274;
        double r54878276 = b;
        double r54878277 = r54878267 * r54878276;
        double r54878278 = r54878275 - r54878277;
        double r54878279 = r54878266 + r54878271;
        double r54878280 = r54878279 + r54878267;
        double r54878281 = r54878278 / r54878280;
        return r54878281;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r54878282 = x;
        double r54878283 = y;
        double r54878284 = r54878282 + r54878283;
        double r54878285 = z;
        double r54878286 = r54878284 * r54878285;
        double r54878287 = t;
        double r54878288 = r54878287 + r54878283;
        double r54878289 = a;
        double r54878290 = r54878288 * r54878289;
        double r54878291 = r54878286 + r54878290;
        double r54878292 = b;
        double r54878293 = r54878283 * r54878292;
        double r54878294 = r54878291 - r54878293;
        double r54878295 = r54878282 + r54878287;
        double r54878296 = r54878295 + r54878283;
        double r54878297 = r54878294 / r54878296;
        double r54878298 = -8.416712259254574e+242;
        bool r54878299 = r54878297 <= r54878298;
        double r54878300 = r54878289 + r54878285;
        double r54878301 = r54878300 - r54878292;
        double r54878302 = 7.307468158284448e+286;
        bool r54878303 = r54878297 <= r54878302;
        double r54878304 = 1.0;
        double r54878305 = r54878282 + r54878288;
        double r54878306 = fma(r54878285, r54878284, r54878290);
        double r54878307 = r54878292 * r54878283;
        double r54878308 = r54878306 - r54878307;
        double r54878309 = r54878305 / r54878308;
        double r54878310 = r54878304 / r54878309;
        double r54878311 = r54878303 ? r54878310 : r54878301;
        double r54878312 = r54878299 ? r54878301 : r54878311;
        return r54878312;
}

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

Target

Original27.0
Target11.3
Herbie7.8
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \lt -3.581311708415056427521064305370896655752 \cdot 10^{153}:\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{elif}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \lt 1.228596430831560895857110658734089400289 \cdot 10^{82}:\\ \;\;\;\;\frac{1}{\frac{\left(x + t\right) + y}{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\left(z + a\right) - b\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)) < -8.416712259254574e+242 or 7.307468158284448e+286 < (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y))

    1. Initial program 61.5

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

    3. Applied clear-num61.5

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

    4. Simplified61.5

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

    5. Taylor expanded around 0 17.4

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

    if -8.416712259254574e+242 < (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)) < 7.307468158284448e+286

    1. Initial program 0.3

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

    3. Applied clear-num0.5

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

    4. Simplified0.5

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

  4. Final simplification7.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \le -8.416712259254574092270707749732520074762 \cdot 10^{242}:\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{elif}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \le 7.307468158284447910195168497396941423128 \cdot 10^{286}:\\ \;\;\;\;\frac{1}{\frac{x + \left(t + y\right)}{\mathsf{fma}\left(z, x + y, \left(t + y\right) \cdot a\right) - b \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;\left(a + z\right) - b\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (x y z t a b)
  :name "AI.Clustering.Hierarchical.Internal:ward from clustering-0.2.1"

  :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) (/ 1.0 (/ (+ (+ 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)))