Average Error: 6.6 → 2.8
Time: 27.9s
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 r456011 = x;
        double r456012 = y;
        double r456013 = r456012 - r456011;
        double r456014 = z;
        double r456015 = r456013 * r456014;
        double r456016 = t;
        double r456017 = r456015 / r456016;
        double r456018 = r456011 + r456017;
        return r456018;
}
↓
double f(double x, double y, double z, double t) {
        double r456019 = z;
        double r456020 = -7.373144133272254e-181;
        bool r456021 = r456019 <= r456020;
        double r456022 = y;
        double r456023 = x;
        double r456024 = r456022 - r456023;
        double r456025 = t;
        double r456026 = r456024 / r456025;
        double r456027 = fma(r456026, r456019, r456023);
        double r456028 = r456025 / r456019;
        double r456029 = r456024 / r456028;
        double r456030 = r456029 + r456023;
        double r456031 = r456021 ? r456027 : r456030;
        return r456031;
}

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)))