Average Error: 2.0 → 2.0
Time: 28.7s
Precision: 64
\[x + \left(y - x\right) \cdot \frac{z}{t}\]
↓
\[\mathsf{fma}\left(\frac{z}{t}, y - x, x\right)\]
x + \left(y - x\right) \cdot \frac{z}{t}
↓
\mathsf{fma}\left(\frac{z}{t}, y - x, x\right)
double f(double x, double y, double z, double t) {
        double r454016 = x;
        double r454017 = y;
        double r454018 = r454017 - r454016;
        double r454019 = z;
        double r454020 = t;
        double r454021 = r454019 / r454020;
        double r454022 = r454018 * r454021;
        double r454023 = r454016 + r454022;
        return r454023;
}
↓
double f(double x, double y, double z, double t) {
        double r454024 = z;
        double r454025 = t;
        double r454026 = r454024 / r454025;
        double r454027 = y;
        double r454028 = x;
        double r454029 = r454027 - r454028;
        double r454030 = fma(r454026, r454029, r454028);
        return r454030;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original2.0
Target2.2
Herbie2.0
\[\begin{array}{l} \mathbf{if}\;\left(y - x\right) \cdot \frac{z}{t} \lt -1013646692435.88671875:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \mathbf{elif}\;\left(y - x\right) \cdot \frac{z}{t} \lt -0.0:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array}\]

Derivation

  1. Initial program 2.0

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

  2. Simplified2.0

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

  3. Final simplification2.0

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

Reproduce

herbie shell --seed 2019196 +o rules:numerics
(FPCore (x y z t)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:tickPosition from plot-0.2.3.4"

  :herbie-target
  (if (< (* (- y x) (/ z t)) -1013646692435.8867) (+ 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))))