Average Error: 6.6 → 2.8
Time: 13.1s
Precision: 64
\[x + \frac{\left(y - x\right) \cdot z}{t}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -7.3731441332722536 \cdot 10^{-181}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - x}{t}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y - x}{\frac{t}{z}} + x\\ \end{array}\]
x + \frac{\left(y - x\right) \cdot z}{t}
↓
\begin{array}{l}
\mathbf{if}\;z \le -7.3731441332722536 \cdot 10^{-181}:\\
\;\;\;\;\mathsf{fma}\left(\frac{y - x}{t}, z, x\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{y - x}{\frac{t}{z}} + x\\

\end{array}
double f(double x, double y, double z, double t) {
        double r549084 = x;
        double r549085 = y;
        double r549086 = r549085 - r549084;
        double r549087 = z;
        double r549088 = r549086 * r549087;
        double r549089 = t;
        double r549090 = r549088 / r549089;
        double r549091 = r549084 + r549090;
        return r549091;
}
↓
double f(double x, double y, double z, double t) {
        double r549092 = z;
        double r549093 = -7.373144133272254e-181;
        bool r549094 = r549092 <= r549093;
        double r549095 = y;
        double r549096 = x;
        double r549097 = r549095 - r549096;
        double r549098 = t;
        double r549099 = r549097 / r549098;
        double r549100 = fma(r549099, r549092, r549096);
        double r549101 = r549098 / r549092;
        double r549102 = r549097 / r549101;
        double r549103 = r549102 + r549096;
        double r549104 = r549094 ? r549100 : r549103;
        return r549104;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original6.6
Target2.2
Herbie2.8
\[\begin{array}{l} \mathbf{if}\;x \lt -9.0255111955330046 \cdot 10^{-135}:\\ \;\;\;\;x - \frac{z}{t} \cdot \left(x - y\right)\\ \mathbf{elif}\;x \lt 4.2750321637007147 \cdot 10^{-250}:\\ \;\;\;\;x + \frac{y - x}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -7.373144133272254e-181

    1. Initial program 8.2

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

    2. Simplified4.3

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

    if -7.373144133272254e-181 < z

    1. Initial program 5.8

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

    2. Simplified8.0

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

    4. Applied fma-udef8.0

      \[\leadsto \color{blue}{\frac{y - x}{t} \cdot z + x}\]

    5. Simplified2.0

      \[\leadsto \color{blue}{\frac{y - x}{\frac{t}{z}}} + x\]
    6. Using strategy rm

    7. Applied pow12.0

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

  4. Final simplification2.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -7.3731441332722536 \cdot 10^{-181}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - x}{t}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y - x}{\frac{t}{z}} + x\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (x y z t)
  :name "Numeric.Histogram:binBounds from Chart-1.5.3"
  :precision binary64

  :herbie-target
  (if (< x -9.025511195533005e-135) (- x (* (/ z t) (- x y))) (if (< x 4.275032163700715e-250) (+ x (* (/ (- y x) t) z)) (+ x (/ (- y x) (/ t z)))))

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