Average Error: 10.9 → 1.3
Time: 17.9s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{a - t}\]
↓
\[\mathsf{fma}\left(\frac{z - t}{a - t}, y, x\right)\]
x + \frac{y \cdot \left(z - t\right)}{a - t}
↓
\mathsf{fma}\left(\frac{z - t}{a - t}, y, x\right)
double f(double x, double y, double z, double t, double a) {
        double r372386 = x;
        double r372387 = y;
        double r372388 = z;
        double r372389 = t;
        double r372390 = r372388 - r372389;
        double r372391 = r372387 * r372390;
        double r372392 = a;
        double r372393 = r372392 - r372389;
        double r372394 = r372391 / r372393;
        double r372395 = r372386 + r372394;
        return r372395;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r372396 = z;
        double r372397 = t;
        double r372398 = r372396 - r372397;
        double r372399 = a;
        double r372400 = r372399 - r372397;
        double r372401 = r372398 / r372400;
        double r372402 = y;
        double r372403 = x;
        double r372404 = fma(r372401, r372402, r372403);
        return r372404;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original10.9
Target1.2
Herbie1.3
\[x + \frac{y}{\frac{a - t}{z - t}}\]

Derivation

  1. Initial program 10.9

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

  2. Simplified2.9

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

  4. Applied div-inv2.9

    \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot \frac{1}{a - t}}, z - t, x\right)\]
  5. Using strategy rm

  6. Applied add-cube-cbrt3.4

    \[\leadsto \mathsf{fma}\left(y \cdot \frac{1}{\color{blue}{\left(\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}\right) \cdot \sqrt[3]{a - t}}}, z - t, x\right)\]

  7. Applied associate-/r*3.4

    \[\leadsto \mathsf{fma}\left(y \cdot \color{blue}{\frac{\frac{1}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}{\sqrt[3]{a - t}}}, z - t, x\right)\]
  8. Using strategy rm

  9. Applied fma-udef3.4

    \[\leadsto \color{blue}{\left(y \cdot \frac{\frac{1}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}{\sqrt[3]{a - t}}\right) \cdot \left(z - t\right) + x}\]

  10. Simplified1.3

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

  12. Applied fma-def1.3

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

  13. Final simplification1.3

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

Reproduce

herbie shell --seed 2019303 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTicks from plot-0.2.3.4, B"
  :precision binary64

  :herbie-target
  (+ x (/ y (/ (- a t) (- z t))))

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