Average Error: 6.3 → 0.3
Time: 14.7s
Precision: 64
\[x - \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(z - t\right) \cdot y \le -1.056329197442560351249015028822880627264 \cdot 10^{281}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right) - \frac{z}{\frac{a}{y}}\\ \mathbf{elif}\;\left(z - t\right) \cdot y \le 1.301385111956770635951737168355727076246 \cdot 10^{272}:\\ \;\;\;\;x - \frac{\left(z - t\right) \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\ \end{array}\]
x - \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;\left(z - t\right) \cdot y \le -1.056329197442560351249015028822880627264 \cdot 10^{281}:\\
\;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right) - \frac{z}{\frac{a}{y}}\\

\mathbf{elif}\;\left(z - t\right) \cdot y \le 1.301385111956770635951737168355727076246 \cdot 10^{272}:\\
\;\;\;\;x - \frac{\left(z - t\right) \cdot y}{a}\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r13595021 = x;
        double r13595022 = y;
        double r13595023 = z;
        double r13595024 = t;
        double r13595025 = r13595023 - r13595024;
        double r13595026 = r13595022 * r13595025;
        double r13595027 = a;
        double r13595028 = r13595026 / r13595027;
        double r13595029 = r13595021 - r13595028;
        return r13595029;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r13595030 = z;
        double r13595031 = t;
        double r13595032 = r13595030 - r13595031;
        double r13595033 = y;
        double r13595034 = r13595032 * r13595033;
        double r13595035 = -1.0563291974425604e+281;
        bool r13595036 = r13595034 <= r13595035;
        double r13595037 = a;
        double r13595038 = r13595031 / r13595037;
        double r13595039 = x;
        double r13595040 = fma(r13595038, r13595033, r13595039);
        double r13595041 = r13595037 / r13595033;
        double r13595042 = r13595030 / r13595041;
        double r13595043 = r13595040 - r13595042;
        double r13595044 = 1.3013851119567706e+272;
        bool r13595045 = r13595034 <= r13595044;
        double r13595046 = r13595034 / r13595037;
        double r13595047 = r13595039 - r13595046;
        double r13595048 = r13595033 / r13595037;
        double r13595049 = r13595031 - r13595030;
        double r13595050 = fma(r13595048, r13595049, r13595039);
        double r13595051 = r13595045 ? r13595047 : r13595050;
        double r13595052 = r13595036 ? r13595043 : r13595051;
        return r13595052;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original6.3
Target0.6
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753216593153715602325729 \cdot 10^{-10}:\\ \;\;\;\;x - \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.894426862792089097262541964056085749132 \cdot 10^{-49}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (* y (- z t)) < -1.0563291974425604e+281

    1. Initial program 49.5

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

    2. Taylor expanded around 0 49.5

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

    3. Simplified0.2

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

    if -1.0563291974425604e+281 < (* y (- z t)) < 1.3013851119567706e+272

    1. Initial program 0.3

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

    if 1.3013851119567706e+272 < (* y (- z t))

    1. Initial program 48.9

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

    2. Simplified0.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z - t\right) \cdot y \le -1.056329197442560351249015028822880627264 \cdot 10^{281}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right) - \frac{z}{\frac{a}{y}}\\ \mathbf{elif}\;\left(z - t\right) \cdot y \le 1.301385111956770635951737168355727076246 \cdot 10^{272}:\\ \;\;\;\;x - \frac{\left(z - t\right) \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, F"

  :herbie-target
  (if (< y -1.0761266216389975e-10) (- x (/ 1.0 (/ (/ a (- z t)) y))) (if (< y 2.894426862792089e-49) (- x (/ (* y (- z t)) a)) (- x (/ y (/ a (- z t))))))

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