Average Error: 1.3 → 1.3
Time: 4.2s
Precision: 64
\[x + y \cdot \frac{z - t}{a - t}\]
↓
\[\mathsf{fma}\left(y, 1 \cdot \frac{z - t}{a - t}, x\right)\]
x + y \cdot \frac{z - t}{a - t}
↓
\mathsf{fma}\left(y, 1 \cdot \frac{z - t}{a - t}, x\right)
double f(double x, double y, double z, double t, double a) {
        double r550436 = x;
        double r550437 = y;
        double r550438 = z;
        double r550439 = t;
        double r550440 = r550438 - r550439;
        double r550441 = a;
        double r550442 = r550441 - r550439;
        double r550443 = r550440 / r550442;
        double r550444 = r550437 * r550443;
        double r550445 = r550436 + r550444;
        return r550445;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r550446 = y;
        double r550447 = 1.0;
        double r550448 = z;
        double r550449 = t;
        double r550450 = r550448 - r550449;
        double r550451 = a;
        double r550452 = r550451 - r550449;
        double r550453 = r550450 / r550452;
        double r550454 = r550447 * r550453;
        double r550455 = x;
        double r550456 = fma(r550446, r550454, r550455);
        return r550456;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original1.3
Target0.3
Herbie1.3
\[\begin{array}{l} \mathbf{if}\;y \lt -8.50808486055124107 \cdot 10^{-17}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;y \lt 2.8944268627920891 \cdot 10^{-49}:\\ \;\;\;\;x + \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{a - t}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \end{array}\]

Derivation

  1. Initial program 1.3

    \[x + y \cdot \frac{z - t}{a - t}\]

  2. Simplified1.3

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

  4. Applied clear-num1.3

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

  6. Applied *-un-lft-identity1.3

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

  7. Applied *-un-lft-identity1.3

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

  8. Applied times-frac1.3

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

  9. Applied add-cube-cbrt1.3

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

  10. Applied times-frac1.3

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

  11. Simplified1.3

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

  12. Simplified1.3

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

  13. Final simplification1.3

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

Reproduce

herbie shell --seed 2020046 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, B"
  :precision binary64

  :herbie-target
  (if (< y -8.508084860551241e-17) (+ x (* y (/ (- z t) (- a t)))) (if (< y 2.894426862792089e-49) (+ x (* (* y (- z t)) (/ 1 (- a t)))) (+ x (* y (/ (- z t) (- a t))))))

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