Average Error: 2.2 → 1.5
Time: 3.8s
Precision: 64
\[x + \left(y - x\right) \cdot \frac{z}{t}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{z}{t} \le 3.2897350745799706 \cdot 10^{215}:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(y - x\right) \cdot z}{t} + x\\ \end{array}\]
x + \left(y - x\right) \cdot \frac{z}{t}
↓
\begin{array}{l}
\mathbf{if}\;\frac{z}{t} \le 3.2897350745799706 \cdot 10^{215}:\\
\;\;\;\;\mathsf{fma}\left(y - x, \frac{z}{t}, 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 r585255 = x;
        double r585256 = y;
        double r585257 = r585256 - r585255;
        double r585258 = z;
        double r585259 = t;
        double r585260 = r585258 / r585259;
        double r585261 = r585257 * r585260;
        double r585262 = r585255 + r585261;
        return r585262;
}
↓
double f(double x, double y, double z, double t) {
        double r585263 = z;
        double r585264 = t;
        double r585265 = r585263 / r585264;
        double r585266 = 3.2897350745799706e+215;
        bool r585267 = r585265 <= r585266;
        double r585268 = y;
        double r585269 = x;
        double r585270 = r585268 - r585269;
        double r585271 = fma(r585270, r585265, r585269);
        double r585272 = r585270 * r585263;
        double r585273 = r585272 / r585264;
        double r585274 = r585273 + r585269;
        double r585275 = r585267 ? r585271 : r585274;
        return r585275;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original2.2
Target2.5
Herbie1.5
\[\begin{array}{l} \mathbf{if}\;\left(y - x\right) \cdot \frac{z}{t} \lt -1013646692435.887:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \mathbf{elif}\;\left(y - x\right) \cdot \frac{z}{t} \lt -0.0:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ z t) < 3.2897350745799706e+215

    1. Initial program 1.5

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

    2. Simplified1.5

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

    if 3.2897350745799706e+215 < (/ z t)

    1. Initial program 25.9

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

    2. Simplified25.9

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

    4. Applied fma-udef25.9

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

    6. Applied associate-*r/0.5

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

  4. Final simplification1.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z}{t} \le 3.2897350745799706 \cdot 10^{215}:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(y - x\right) \cdot z}{t} + x\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(FPCore (x y z t)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:tickPosition from plot-0.2.3.4"
  :precision binary64

  :herbie-target
  (if (< (* (- y x) (/ z t)) -1013646692435.887) (+ x (/ (- y x) (/ t z))) (if (< (* (- y x) (/ z t)) -0.0) (+ x (/ (* (- y x) z) t)) (+ x (/ (- y x) (/ t z)))))

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