Average Error: 11.1 → 1.2
Time: 3.4s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{z - a}\]
↓
\[\mathsf{fma}\left(\frac{z - t}{z - a}, y, x\right)\]
x + \frac{y \cdot \left(z - t\right)}{z - a}
↓
\mathsf{fma}\left(\frac{z - t}{z - a}, y, x\right)
double f(double x, double y, double z, double t, double a) {
        double r589303 = x;
        double r589304 = y;
        double r589305 = z;
        double r589306 = t;
        double r589307 = r589305 - r589306;
        double r589308 = r589304 * r589307;
        double r589309 = a;
        double r589310 = r589305 - r589309;
        double r589311 = r589308 / r589310;
        double r589312 = r589303 + r589311;
        return r589312;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r589313 = z;
        double r589314 = t;
        double r589315 = r589313 - r589314;
        double r589316 = a;
        double r589317 = r589313 - r589316;
        double r589318 = r589315 / r589317;
        double r589319 = y;
        double r589320 = x;
        double r589321 = fma(r589318, r589319, r589320);
        return r589321;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original11.1
Target1.1
Herbie1.2
\[x + \frac{y}{\frac{z - a}{z - t}}\]

Derivation

  1. Initial program 11.1

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

  2. Simplified3.1

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

  4. Applied clear-num3.2

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

  6. Applied fma-udef3.2

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

  7. Simplified3.0

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

  9. Applied *-un-lft-identity3.0

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

  10. Applied *-un-lft-identity3.0

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

  11. Applied distribute-lft-out3.0

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

  12. Simplified1.2

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

  13. Final simplification1.2

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

Reproduce

herbie shell --seed 2020020 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTicks from plot-0.2.3.4, A"
  :precision binary64

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

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