Average Error: 10.3 → 1.3
Time: 2.3m
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{z - a}\]
↓
\[x + \left(\mathsf{fma}\left(\frac{-1}{z - a}, t, t \cdot \frac{1}{z - a}\right) \cdot y + \mathsf{fma}\left(1, \frac{1}{\frac{z - a}{z}}, t \cdot \frac{-1}{z - a}\right) \cdot y\right)\]
x + \frac{y \cdot \left(z - t\right)}{z - a}
↓
x + \left(\mathsf{fma}\left(\frac{-1}{z - a}, t, t \cdot \frac{1}{z - a}\right) \cdot y + \mathsf{fma}\left(1, \frac{1}{\frac{z - a}{z}}, t \cdot \frac{-1}{z - a}\right) \cdot y\right)
double f(double x, double y, double z, double t, double a) {
        double r21237475 = x;
        double r21237476 = y;
        double r21237477 = z;
        double r21237478 = t;
        double r21237479 = r21237477 - r21237478;
        double r21237480 = r21237476 * r21237479;
        double r21237481 = a;
        double r21237482 = r21237477 - r21237481;
        double r21237483 = r21237480 / r21237482;
        double r21237484 = r21237475 + r21237483;
        return r21237484;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r21237485 = x;
        double r21237486 = -1.0;
        double r21237487 = z;
        double r21237488 = a;
        double r21237489 = r21237487 - r21237488;
        double r21237490 = r21237486 / r21237489;
        double r21237491 = t;
        double r21237492 = 1.0;
        double r21237493 = r21237492 / r21237489;
        double r21237494 = r21237491 * r21237493;
        double r21237495 = fma(r21237490, r21237491, r21237494);
        double r21237496 = y;
        double r21237497 = r21237495 * r21237496;
        double r21237498 = r21237489 / r21237487;
        double r21237499 = r21237492 / r21237498;
        double r21237500 = r21237491 * r21237490;
        double r21237501 = fma(r21237492, r21237499, r21237500);
        double r21237502 = r21237501 * r21237496;
        double r21237503 = r21237497 + r21237502;
        double r21237504 = r21237485 + r21237503;
        return r21237504;
}

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. Simplified1.3

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

  4. Applied div-sub1.3

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

  6. Applied fma-udef1.3

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

  8. Applied clear-num1.3

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

  10. Applied div-inv1.3

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

  11. Applied *-un-lft-identity1.3

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

  12. Applied prod-diff1.3

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

  13. Applied distribute-rgt-in1.3

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

  14. Final simplification1.3

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

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))))