Average Error: 10.5 → 1.2
Time: 22.8s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{z - a}\]
↓
\[y \cdot \frac{z - t}{z - a} + x\]
x + \frac{y \cdot \left(z - t\right)}{z - a}
↓
y \cdot \frac{z - t}{z - a} + x
double f(double x, double y, double z, double t, double a) {
        double r294201 = x;
        double r294202 = y;
        double r294203 = z;
        double r294204 = t;
        double r294205 = r294203 - r294204;
        double r294206 = r294202 * r294205;
        double r294207 = a;
        double r294208 = r294203 - r294207;
        double r294209 = r294206 / r294208;
        double r294210 = r294201 + r294209;
        return r294210;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r294211 = y;
        double r294212 = z;
        double r294213 = t;
        double r294214 = r294212 - r294213;
        double r294215 = a;
        double r294216 = r294212 - r294215;
        double r294217 = r294214 / r294216;
        double r294218 = r294211 * r294217;
        double r294219 = x;
        double r294220 = r294218 + r294219;
        return r294220;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original10.5
Target1.1
Herbie1.2
\[x + \frac{y}{\frac{z - a}{z - t}}\]

Derivation

  1. Initial program 10.5

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

  2. Simplified2.8

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

  4. Applied fma-udef2.8

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

  6. Applied div-inv2.8

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

  7. Applied associate-*l*1.3

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

  8. Simplified1.2

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

  9. Final simplification1.2

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

Reproduce

herbie shell --seed 2019212 +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))))