Average Error: 6.7 → 1.9
Time: 14.7s
Precision: 64
\[x + \frac{\left(y - x\right) \cdot z}{t}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -3.165968678679720807368432635267843482511 \cdot 10^{-173}:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z}{t}, x\right)\\ \mathbf{elif}\;z \le 2.133640105926503877026549421733062970497 \cdot 10^{-14}:\\ \;\;\;\;\frac{\left(y - x\right) \cdot z}{t} + x\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z}{t} \cdot \left(y - x\right)\\ \end{array}\]
x + \frac{\left(y - x\right) \cdot z}{t}
↓
\begin{array}{l}
\mathbf{if}\;z \le -3.165968678679720807368432635267843482511 \cdot 10^{-173}:\\
\;\;\;\;\mathsf{fma}\left(y - x, \frac{z}{t}, x\right)\\

\mathbf{elif}\;z \le 2.133640105926503877026549421733062970497 \cdot 10^{-14}:\\
\;\;\;\;\frac{\left(y - x\right) \cdot z}{t} + x\\

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

\end{array}
double f(double x, double y, double z, double t) {
        double r19480312 = x;
        double r19480313 = y;
        double r19480314 = r19480313 - r19480312;
        double r19480315 = z;
        double r19480316 = r19480314 * r19480315;
        double r19480317 = t;
        double r19480318 = r19480316 / r19480317;
        double r19480319 = r19480312 + r19480318;
        return r19480319;
}
↓
double f(double x, double y, double z, double t) {
        double r19480320 = z;
        double r19480321 = -3.1659686786797208e-173;
        bool r19480322 = r19480320 <= r19480321;
        double r19480323 = y;
        double r19480324 = x;
        double r19480325 = r19480323 - r19480324;
        double r19480326 = t;
        double r19480327 = r19480320 / r19480326;
        double r19480328 = fma(r19480325, r19480327, r19480324);
        double r19480329 = 2.133640105926504e-14;
        bool r19480330 = r19480320 <= r19480329;
        double r19480331 = r19480325 * r19480320;
        double r19480332 = r19480331 / r19480326;
        double r19480333 = r19480332 + r19480324;
        double r19480334 = r19480327 * r19480325;
        double r19480335 = r19480324 + r19480334;
        double r19480336 = r19480330 ? r19480333 : r19480335;
        double r19480337 = r19480322 ? r19480328 : r19480336;
        return r19480337;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original6.7
Target2.1
Herbie1.9
\[\begin{array}{l} \mathbf{if}\;x \lt -9.025511195533004570453352523209034680317 \cdot 10^{-135}:\\ \;\;\;\;x - \frac{z}{t} \cdot \left(x - y\right)\\ \mathbf{elif}\;x \lt 4.275032163700714748507147332551979944314 \cdot 10^{-250}:\\ \;\;\;\;x + \frac{y - x}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -3.1659686786797208e-173

    1. Initial program 8.2

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

    2. Simplified2.0

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

    if -3.1659686786797208e-173 < z < 2.133640105926504e-14

    1. Initial program 1.1

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

    2. Simplified1.3

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

    4. Applied fma-udef1.3

      \[\leadsto \color{blue}{\left(y - x\right) \cdot \frac{z}{t} + x}\]
    5. Using strategy rm

    6. Applied associate-*r/1.1

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

    if 2.133640105926504e-14 < z

    1. Initial program 15.5

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

    2. Simplified3.3

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

    4. Applied fma-udef3.4

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

  4. Final simplification1.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -3.165968678679720807368432635267843482511 \cdot 10^{-173}:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z}{t}, x\right)\\ \mathbf{elif}\;z \le 2.133640105926503877026549421733062970497 \cdot 10^{-14}:\\ \;\;\;\;\frac{\left(y - x\right) \cdot z}{t} + x\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z}{t} \cdot \left(y - x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (x y z t)
  :name "Numeric.Histogram:binBounds from Chart-1.5.3"

  :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)))