Average Error: 7.0 → 1.7
Time: 26.4s
Precision: 64
\[x + \frac{\left(y - x\right) \cdot z}{t}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -7.148917450038407011215425202135074047193 \cdot 10^{-67} \lor \neg \left(t \le 3271230034325250553802375920643693608960\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y - x}{t}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(y - x\right) \cdot z}{t} + x\\ \end{array}\]
x + \frac{\left(y - x\right) \cdot z}{t}
↓
\begin{array}{l}
\mathbf{if}\;t \le -7.148917450038407011215425202135074047193 \cdot 10^{-67} \lor \neg \left(t \le 3271230034325250553802375920643693608960\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{y - x}{t}, z, x\right)\\

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

\end{array}
double f(double x, double y, double z, double t) {
        double r371208 = x;
        double r371209 = y;
        double r371210 = r371209 - r371208;
        double r371211 = z;
        double r371212 = r371210 * r371211;
        double r371213 = t;
        double r371214 = r371212 / r371213;
        double r371215 = r371208 + r371214;
        return r371215;
}
↓
double f(double x, double y, double z, double t) {
        double r371216 = t;
        double r371217 = -7.148917450038407e-67;
        bool r371218 = r371216 <= r371217;
        double r371219 = 3.2712300343252506e+39;
        bool r371220 = r371216 <= r371219;
        double r371221 = !r371220;
        bool r371222 = r371218 || r371221;
        double r371223 = y;
        double r371224 = x;
        double r371225 = r371223 - r371224;
        double r371226 = r371225 / r371216;
        double r371227 = z;
        double r371228 = fma(r371226, r371227, r371224);
        double r371229 = r371225 * r371227;
        double r371230 = r371229 / r371216;
        double r371231 = r371230 + r371224;
        double r371232 = r371222 ? r371228 : r371231;
        return r371232;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original7.0
Target2.0
Herbie1.7
\[\begin{array}{l} \mathbf{if}\;x \lt -9.025511195533004570453352523209034680317 \cdot 10^{-135}:\\ \;\;\;\;x - \frac{z}{t} \cdot \left(x - y\right)\\ \mathbf{elif}\;x \lt 4.275032163700714748507147332551979944314 \cdot 10^{-250}:\\ \;\;\;\;x + \frac{y - x}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if t < -7.148917450038407e-67 or 3.2712300343252506e+39 < t

    1. Initial program 9.9

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

    2. Simplified1.4

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

    if -7.148917450038407e-67 < t < 3.2712300343252506e+39

    1. Initial program 2.2

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

    2. Simplified14.7

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - x}{t}, z, x\right)}\]
    3. Using strategy rm

    4. Applied *-un-lft-identity14.7

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

    5. Applied add-cube-cbrt15.3

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left(\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right) \cdot \sqrt[3]{y - x}}}{1 \cdot t}, z, x\right)\]

    6. Applied times-frac15.3

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}}{1} \cdot \frac{\sqrt[3]{y - x}}{t}}, z, x\right)\]

    7. Simplified15.3

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right)} \cdot \frac{\sqrt[3]{y - x}}{t}, z, x\right)\]
    8. Using strategy rm

    9. Applied fma-udef15.3

      \[\leadsto \color{blue}{\left(\left(\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right) \cdot \frac{\sqrt[3]{y - x}}{t}\right) \cdot z + x}\]

    10. Simplified2.2

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

  4. Final simplification1.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -7.148917450038407011215425202135074047193 \cdot 10^{-67} \lor \neg \left(t \le 3271230034325250553802375920643693608960\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y - x}{t}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(y - x\right) \cdot z}{t} + x\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 +o rules:numerics
(FPCore (x y z t)
  :name "Numeric.Histogram:binBounds from Chart-1.5.3"
  :precision binary64

  :herbie-target
  (if (< x -9.0255111955330046e-135) (- x (* (/ z t) (- x y))) (if (< x 4.2750321637007147e-250) (+ x (* (/ (- y x) t) z)) (+ x (/ (- y x) (/ t z)))))

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