Average Error: 11.0 → 1.3
Time: 45.3s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{z - a}\]
↓
\[\frac{z - t}{z - a} \cdot y + x\]
x + \frac{y \cdot \left(z - t\right)}{z - a}
↓
\frac{z - t}{z - a} \cdot y + x
double f(double x, double y, double z, double t, double a) {
        double r25698398 = x;
        double r25698399 = y;
        double r25698400 = z;
        double r25698401 = t;
        double r25698402 = r25698400 - r25698401;
        double r25698403 = r25698399 * r25698402;
        double r25698404 = a;
        double r25698405 = r25698400 - r25698404;
        double r25698406 = r25698403 / r25698405;
        double r25698407 = r25698398 + r25698406;
        return r25698407;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r25698408 = z;
        double r25698409 = t;
        double r25698410 = r25698408 - r25698409;
        double r25698411 = a;
        double r25698412 = r25698408 - r25698411;
        double r25698413 = r25698410 / r25698412;
        double r25698414 = y;
        double r25698415 = r25698413 * r25698414;
        double r25698416 = x;
        double r25698417 = r25698415 + r25698416;
        return r25698417;
}

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

Original11.0
Target1.2
Herbie1.3
\[x + \frac{y}{\frac{z - a}{z - t}}\]

Derivation

  1. Initial program 11.0

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

  2. Simplified2.6

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

  4. Applied div-inv2.7

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

  6. Applied fma-udef2.7

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

  7. Simplified2.7

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

  9. Applied associate-/r/1.3

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

  10. Final simplification1.3

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

Reproduce

herbie shell --seed 2019165 +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))))