Average Error: 6.3 → 1.7
Time: 13.9s
Precision: 64
\[x + \frac{y \cdot \left(z - x\right)}{t}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -8.363664958439672667278554160764100153889 \cdot 10^{69}:\\ \;\;\;\;\mathsf{fma}\left(z - x, \frac{y}{t}, x\right)\\ \mathbf{elif}\;t \le 7.486542709900388243398285213925344255536 \cdot 10^{-169}:\\ \;\;\;\;\frac{\left(z - x\right) \cdot y}{t} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z - x, \frac{y}{t}, x\right)\\ \end{array}\]
x + \frac{y \cdot \left(z - x\right)}{t}
↓
\begin{array}{l}
\mathbf{if}\;t \le -8.363664958439672667278554160764100153889 \cdot 10^{69}:\\
\;\;\;\;\mathsf{fma}\left(z - x, \frac{y}{t}, x\right)\\

\mathbf{elif}\;t \le 7.486542709900388243398285213925344255536 \cdot 10^{-169}:\\
\;\;\;\;\frac{\left(z - x\right) \cdot y}{t} + x\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(z - x, \frac{y}{t}, x\right)\\

\end{array}
double f(double x, double y, double z, double t) {
        double r12651754 = x;
        double r12651755 = y;
        double r12651756 = z;
        double r12651757 = r12651756 - r12651754;
        double r12651758 = r12651755 * r12651757;
        double r12651759 = t;
        double r12651760 = r12651758 / r12651759;
        double r12651761 = r12651754 + r12651760;
        return r12651761;
}
↓
double f(double x, double y, double z, double t) {
        double r12651762 = t;
        double r12651763 = -8.363664958439673e+69;
        bool r12651764 = r12651762 <= r12651763;
        double r12651765 = z;
        double r12651766 = x;
        double r12651767 = r12651765 - r12651766;
        double r12651768 = y;
        double r12651769 = r12651768 / r12651762;
        double r12651770 = fma(r12651767, r12651769, r12651766);
        double r12651771 = 7.486542709900388e-169;
        bool r12651772 = r12651762 <= r12651771;
        double r12651773 = r12651767 * r12651768;
        double r12651774 = r12651773 / r12651762;
        double r12651775 = r12651774 + r12651766;
        double r12651776 = r12651772 ? r12651775 : r12651770;
        double r12651777 = r12651764 ? r12651770 : r12651776;
        return r12651777;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original6.3
Target2.0
Herbie1.7
\[x - \left(x \cdot \frac{y}{t} + \left(-z\right) \cdot \frac{y}{t}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if t < -8.363664958439673e+69 or 7.486542709900388e-169 < t

    1. Initial program 8.1

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

    2. Simplified1.3

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

    if -8.363664958439673e+69 < t < 7.486542709900388e-169

    1. Initial program 2.7

      \[x + \frac{y \cdot \left(z - x\right)}{t}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -8.363664958439672667278554160764100153889 \cdot 10^{69}:\\ \;\;\;\;\mathsf{fma}\left(z - x, \frac{y}{t}, x\right)\\ \mathbf{elif}\;t \le 7.486542709900388243398285213925344255536 \cdot 10^{-169}:\\ \;\;\;\;\frac{\left(z - x\right) \cdot y}{t} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z - x, \frac{y}{t}, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(FPCore (x y z t)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, D"

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

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