Average Error: 1.4 → 1.4
Time: 1.1m
Precision: 64
\[x + y \cdot \frac{z - t}{z - a}\]
↓
\[\mathsf{fma}\left(\frac{z}{z - a} - \frac{1}{z - a} \cdot t, y, x\right)\]
x + y \cdot \frac{z - t}{z - a}
↓
\mathsf{fma}\left(\frac{z}{z - a} - \frac{1}{z - a} \cdot t, y, x\right)
double f(double x, double y, double z, double t, double a) {
        double r29655196 = x;
        double r29655197 = y;
        double r29655198 = z;
        double r29655199 = t;
        double r29655200 = r29655198 - r29655199;
        double r29655201 = a;
        double r29655202 = r29655198 - r29655201;
        double r29655203 = r29655200 / r29655202;
        double r29655204 = r29655197 * r29655203;
        double r29655205 = r29655196 + r29655204;
        return r29655205;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r29655206 = z;
        double r29655207 = a;
        double r29655208 = r29655206 - r29655207;
        double r29655209 = r29655206 / r29655208;
        double r29655210 = 1.0;
        double r29655211 = r29655210 / r29655208;
        double r29655212 = t;
        double r29655213 = r29655211 * r29655212;
        double r29655214 = r29655209 - r29655213;
        double r29655215 = y;
        double r29655216 = x;
        double r29655217 = fma(r29655214, r29655215, r29655216);
        return r29655217;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original1.4
Target1.3
Herbie1.4
\[x + \frac{y}{\frac{z - a}{z - t}}\]

Derivation

  1. Initial program 1.4

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

  2. Simplified1.4

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

  4. Applied div-sub1.4

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

  6. Applied div-inv1.4

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

  7. Final simplification1.4

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

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, A"

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

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