Average Error: 11.0 → 1.4
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 r375721 = x;
        double r375722 = y;
        double r375723 = z;
        double r375724 = t;
        double r375725 = r375723 - r375724;
        double r375726 = r375722 * r375725;
        double r375727 = a;
        double r375728 = r375727 - r375724;
        double r375729 = r375726 / r375728;
        double r375730 = r375721 + r375729;
        return r375730;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r375731 = z;
        double r375732 = t;
        double r375733 = r375731 - r375732;
        double r375734 = a;
        double r375735 = r375734 - r375732;
        double r375736 = r375733 / r375735;
        double r375737 = y;
        double r375738 = x;
        double r375739 = fma(r375736, r375737, r375738);
        return r375739;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original11.0
Target1.3
Herbie1.4
\[x + \frac{y}{\frac{a - t}{z - t}}\]

Derivation

  1. Initial program 11.0

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

  2. Simplified3.0

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

  4. Applied clear-num3.3

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

  6. Applied fma-udef3.3

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

  7. Simplified3.1

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

  9. Applied associate-/r/1.4

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

  10. Applied fma-def1.4

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

  11. Final simplification1.4

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

Reproduce

herbie shell --seed 2019235 +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))))