Average Error: 2.2 → 0.4
Time: 16.1s
Precision: 64
\[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b = -\infty \lor \neg \left(\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \le 5.116019414478908543360651182801591114306 \cdot 10^{293}\right):\\ \;\;\;\;\mathsf{fma}\left(z, y, \mathsf{fma}\left(\mathsf{fma}\left(z, b, t\right), a, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b\\ \end{array}\]
\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b
↓
\begin{array}{l}
\mathbf{if}\;\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b = -\infty \lor \neg \left(\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \le 5.116019414478908543360651182801591114306 \cdot 10^{293}\right):\\
\;\;\;\;\mathsf{fma}\left(z, y, \mathsf{fma}\left(\mathsf{fma}\left(z, b, t\right), a, x\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r484560 = x;
        double r484561 = y;
        double r484562 = z;
        double r484563 = r484561 * r484562;
        double r484564 = r484560 + r484563;
        double r484565 = t;
        double r484566 = a;
        double r484567 = r484565 * r484566;
        double r484568 = r484564 + r484567;
        double r484569 = r484566 * r484562;
        double r484570 = b;
        double r484571 = r484569 * r484570;
        double r484572 = r484568 + r484571;
        return r484572;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r484573 = x;
        double r484574 = y;
        double r484575 = z;
        double r484576 = r484574 * r484575;
        double r484577 = r484573 + r484576;
        double r484578 = t;
        double r484579 = a;
        double r484580 = r484578 * r484579;
        double r484581 = r484577 + r484580;
        double r484582 = r484579 * r484575;
        double r484583 = b;
        double r484584 = r484582 * r484583;
        double r484585 = r484581 + r484584;
        double r484586 = -inf.0;
        bool r484587 = r484585 <= r484586;
        double r484588 = 5.1160194144789085e+293;
        bool r484589 = r484585 <= r484588;
        double r484590 = !r484589;
        bool r484591 = r484587 || r484590;
        double r484592 = fma(r484575, r484583, r484578);
        double r484593 = fma(r484592, r484579, r484573);
        double r484594 = fma(r484575, r484574, r484593);
        double r484595 = r484591 ? r484594 : r484585;
        return r484595;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Target

Original2.2
Target0.4
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;z \lt -11820553527347888128:\\ \;\;\;\;z \cdot \left(b \cdot a + y\right) + \left(x + t \cdot a\right)\\ \mathbf{elif}\;z \lt 4.758974318836428710669076838657752600596 \cdot 10^{-122}:\\ \;\;\;\;\left(b \cdot z + t\right) \cdot a + \left(z \cdot y + x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(b \cdot a + y\right) + \left(x + t \cdot a\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (+ (+ (+ x (* y z)) (* t a)) (* (* a z) b)) < -inf.0 or 5.1160194144789085e+293 < (+ (+ (+ x (* y z)) (* t a)) (* (* a z) b))

    1. Initial program 34.0

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b\]

    2. Simplified2.1

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

    if -inf.0 < (+ (+ (+ x (* y z)) (* t a)) (* (* a z) b)) < 5.1160194144789085e+293

    1. Initial program 0.3

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b = -\infty \lor \neg \left(\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \le 5.116019414478908543360651182801591114306 \cdot 10^{293}\right):\\ \;\;\;\;\mathsf{fma}\left(z, y, \mathsf{fma}\left(\mathsf{fma}\left(z, b, t\right), a, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 +o rules:numerics
(FPCore (x y z t a b)
  :name "Graphics.Rasterific.CubicBezier:cachedBezierAt from Rasterific-0.6.1"
  :precision binary64

  :herbie-target
  (if (< z -11820553527347888000) (+ (* z (+ (* b a) y)) (+ x (* t a))) (if (< z 4.75897431883642871e-122) (+ (* (+ (* b z) t) a) (+ (* z y) x)) (+ (* z (+ (* b a) y)) (+ x (* t a)))))

  (+ (+ (+ x (* y z)) (* t a)) (* (* a z) b)))