Average Error: 11.0 → 1.5
Time: 27.8s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{a - t}\]
↓
\[\mathsf{fma}\left(y, \frac{1}{\frac{a - t}{z - t}}, x\right)\]
x + \frac{y \cdot \left(z - t\right)}{a - t}
↓
\mathsf{fma}\left(y, \frac{1}{\frac{a - t}{z - t}}, x\right)
double f(double x, double y, double z, double t, double a) {
        double r23396189 = x;
        double r23396190 = y;
        double r23396191 = z;
        double r23396192 = t;
        double r23396193 = r23396191 - r23396192;
        double r23396194 = r23396190 * r23396193;
        double r23396195 = a;
        double r23396196 = r23396195 - r23396192;
        double r23396197 = r23396194 / r23396196;
        double r23396198 = r23396189 + r23396197;
        return r23396198;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r23396199 = y;
        double r23396200 = 1.0;
        double r23396201 = a;
        double r23396202 = t;
        double r23396203 = r23396201 - r23396202;
        double r23396204 = z;
        double r23396205 = r23396204 - r23396202;
        double r23396206 = r23396203 / r23396205;
        double r23396207 = r23396200 / r23396206;
        double r23396208 = x;
        double r23396209 = fma(r23396199, r23396207, r23396208);
        return r23396209;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original11.0
Target1.3
Herbie1.5
\[x + \frac{y}{\frac{a - t}{z - t}}\]

Derivation

  1. Initial program 11.0

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

  2. Simplified1.5

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

  4. Applied clear-num1.5

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

  5. Final simplification1.5

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

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTicks from plot-0.2.3.4, B"

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

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