Average Error: 10.3 → 1.3
Time: 18.6s
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 r24781818 = x;
        double r24781819 = y;
        double r24781820 = z;
        double r24781821 = t;
        double r24781822 = r24781820 - r24781821;
        double r24781823 = r24781819 * r24781822;
        double r24781824 = a;
        double r24781825 = r24781820 - r24781824;
        double r24781826 = r24781823 / r24781825;
        double r24781827 = r24781818 + r24781826;
        return r24781827;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r24781828 = z;
        double r24781829 = t;
        double r24781830 = r24781828 - r24781829;
        double r24781831 = a;
        double r24781832 = r24781828 - r24781831;
        double r24781833 = r24781830 / r24781832;
        double r24781834 = y;
        double r24781835 = x;
        double r24781836 = fma(r24781833, r24781834, r24781835);
        return r24781836;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original10.3
Target1.2
Herbie1.3
\[x + \frac{y}{\frac{z - a}{z - t}}\]

Derivation

  1. Initial program 10.3

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

  2. Simplified3.1

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

  4. Applied clear-num3.3

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

  6. Applied fma-udef3.3

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

  7. Simplified3.1

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

  9. Applied associate-/r/1.3

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

  10. Applied fma-def1.3

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

  11. Final simplification1.3

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

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTicks from plot-0.2.3.4, A"

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

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