Average Error: 10.9 → 1.3
Time: 18.2s
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 r374273 = x;
        double r374274 = y;
        double r374275 = z;
        double r374276 = t;
        double r374277 = r374275 - r374276;
        double r374278 = r374274 * r374277;
        double r374279 = a;
        double r374280 = r374279 - r374276;
        double r374281 = r374278 / r374280;
        double r374282 = r374273 + r374281;
        return r374282;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r374283 = z;
        double r374284 = t;
        double r374285 = r374283 - r374284;
        double r374286 = a;
        double r374287 = r374286 - r374284;
        double r374288 = r374285 / r374287;
        double r374289 = y;
        double r374290 = x;
        double r374291 = fma(r374288, r374289, r374290);
        return r374291;
}

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))))