Average Error: 7.0 → 1.7
Time: 24.9s
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 r403629 = x;
        double r403630 = y;
        double r403631 = r403630 - r403629;
        double r403632 = z;
        double r403633 = r403631 * r403632;
        double r403634 = t;
        double r403635 = r403633 / r403634;
        double r403636 = r403629 + r403635;
        return r403636;
}
↓
double f(double x, double y, double z, double t) {
        double r403637 = t;
        double r403638 = -7.148917450038407e-67;
        bool r403639 = r403637 <= r403638;
        double r403640 = 3.2712300343252506e+39;
        bool r403641 = r403637 <= r403640;
        double r403642 = !r403641;
        bool r403643 = r403639 || r403642;
        double r403644 = y;
        double r403645 = x;
        double r403646 = r403644 - r403645;
        double r403647 = r403646 / r403637;
        double r403648 = z;
        double r403649 = fma(r403647, r403648, r403645);
        double r403650 = r403646 * r403648;
        double r403651 = r403650 / r403637;
        double r403652 = r403651 + r403645;
        double r403653 = r403643 ? r403649 : r403652;
        return r403653;
}

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)))