Average Error: 11.1 → 1.4
Time: 7.0s
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 r1979 = x;
        double r1980 = y;
        double r1981 = z;
        double r1982 = t;
        double r1983 = r1981 - r1982;
        double r1984 = r1980 * r1983;
        double r1985 = a;
        double r1986 = r1981 - r1985;
        double r1987 = r1984 / r1986;
        double r1988 = r1979 + r1987;
        return r1988;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r1989 = z;
        double r1990 = t;
        double r1991 = r1989 - r1990;
        double r1992 = a;
        double r1993 = r1989 - r1992;
        double r1994 = r1991 / r1993;
        double r1995 = y;
        double r1996 = x;
        double r1997 = fma(r1994, r1995, r1996);
        return r1997;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original11.1
Target1.3
Herbie1.4
\[x + \frac{y}{\frac{z - a}{z - t}}\]

Derivation

  1. Initial program 11.1

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

  2. Simplified2.7

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

  4. Applied clear-num3.0

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

  6. Applied fma-udef3.0

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

  7. Simplified2.8

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

  9. Applied associate-/r/1.4

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

  10. Applied fma-def1.4

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

  11. Final simplification1.4

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

Reproduce

herbie shell --seed 2020025 +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))))