Average Error: 1.2 → 1.2
Time: 19.5s
Precision: 64
\[x + y \cdot \frac{z - t}{z - a}\]
↓
\[\mathsf{fma}\left(\frac{z}{z - a} - \frac{t}{z - a}, y, x\right)\]
x + y \cdot \frac{z - t}{z - a}
↓
\mathsf{fma}\left(\frac{z}{z - a} - \frac{t}{z - a}, y, x\right)
double f(double x, double y, double z, double t, double a) {
        double r309120 = x;
        double r309121 = y;
        double r309122 = z;
        double r309123 = t;
        double r309124 = r309122 - r309123;
        double r309125 = a;
        double r309126 = r309122 - r309125;
        double r309127 = r309124 / r309126;
        double r309128 = r309121 * r309127;
        double r309129 = r309120 + r309128;
        return r309129;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r309130 = z;
        double r309131 = a;
        double r309132 = r309130 - r309131;
        double r309133 = r309130 / r309132;
        double r309134 = t;
        double r309135 = r309134 / r309132;
        double r309136 = r309133 - r309135;
        double r309137 = y;
        double r309138 = x;
        double r309139 = fma(r309136, r309137, r309138);
        return r309139;
}

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

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

  4. Applied div-sub1.2

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

  5. Final simplification1.2

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

Reproduce

herbie shell --seed 2019212 +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)))))