Average Error: 1.2 → 1.2
Time: 6.3s
Precision: 64
\[x + y \cdot \frac{z - t}{z - a}\]
↓
\[\mathsf{fma}\left(y, \frac{z}{z - a} - t \cdot \frac{1}{z - a}, x\right)\]
x + y \cdot \frac{z - t}{z - a}
↓
\mathsf{fma}\left(y, \frac{z}{z - a} - t \cdot \frac{1}{z - a}, x\right)
double f(double x, double y, double z, double t, double a) {
        double r876663 = x;
        double r876664 = y;
        double r876665 = z;
        double r876666 = t;
        double r876667 = r876665 - r876666;
        double r876668 = a;
        double r876669 = r876665 - r876668;
        double r876670 = r876667 / r876669;
        double r876671 = r876664 * r876670;
        double r876672 = r876663 + r876671;
        return r876672;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r876673 = y;
        double r876674 = z;
        double r876675 = a;
        double r876676 = r876674 - r876675;
        double r876677 = r876674 / r876676;
        double r876678 = t;
        double r876679 = 1.0;
        double r876680 = r876679 / r876676;
        double r876681 = r876678 * r876680;
        double r876682 = r876677 - r876681;
        double r876683 = x;
        double r876684 = fma(r876673, r876682, r876683);
        return r876684;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original1.2
Target1.1
Herbie1.2
\[x + \frac{y}{\frac{z - a}{z - t}}\]

Derivation

  1. Initial program 1.2

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

  2. Simplified1.2

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

  4. Applied div-sub1.2

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

  6. Applied div-inv1.2

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

  7. Final simplification1.2

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

Reproduce

herbie shell --seed 2020034 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, A"
  :precision binary64

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

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