Average Error: 14.7 → 13.3
Time: 21.9s
Precision: 64
\[x + \left(y - z\right) \cdot \frac{t - x}{a - z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -2.682196460897034293396785828877679215877 \cdot 10^{143} \lor \neg \left(z \le 4.016279813338475745054191319638190832337 \cdot 10^{120}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, t\right) - \frac{t \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\frac{a - z}{t - x}}, y - z, x\right)\\ \end{array}\]
x + \left(y - z\right) \cdot \frac{t - x}{a - z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -2.682196460897034293396785828877679215877 \cdot 10^{143} \lor \neg \left(z \le 4.016279813338475745054191319638190832337 \cdot 10^{120}\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, t\right) - \frac{t \cdot y}{z}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{\frac{a - z}{t - x}}, y - z, x\right)\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r94288 = x;
        double r94289 = y;
        double r94290 = z;
        double r94291 = r94289 - r94290;
        double r94292 = t;
        double r94293 = r94292 - r94288;
        double r94294 = a;
        double r94295 = r94294 - r94290;
        double r94296 = r94293 / r94295;
        double r94297 = r94291 * r94296;
        double r94298 = r94288 + r94297;
        return r94298;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r94299 = z;
        double r94300 = -2.6821964608970343e+143;
        bool r94301 = r94299 <= r94300;
        double r94302 = 4.016279813338476e+120;
        bool r94303 = r94299 <= r94302;
        double r94304 = !r94303;
        bool r94305 = r94301 || r94304;
        double r94306 = x;
        double r94307 = r94306 / r94299;
        double r94308 = y;
        double r94309 = t;
        double r94310 = fma(r94307, r94308, r94309);
        double r94311 = r94309 * r94308;
        double r94312 = r94311 / r94299;
        double r94313 = r94310 - r94312;
        double r94314 = 1.0;
        double r94315 = a;
        double r94316 = r94315 - r94299;
        double r94317 = r94309 - r94306;
        double r94318 = r94316 / r94317;
        double r94319 = r94314 / r94318;
        double r94320 = r94308 - r94299;
        double r94321 = fma(r94319, r94320, r94306);
        double r94322 = r94305 ? r94313 : r94321;
        return r94322;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Derivation

  1. Split input into 2 regimes

  2. if z < -2.6821964608970343e+143 or 4.016279813338476e+120 < z

    1. Initial program 27.3

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

    2. Simplified27.3

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

    4. Applied div-inv27.3

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

    6. Applied fma-udef27.4

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

    7. Simplified22.4

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

    8. Taylor expanded around inf 25.5

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

    9. Simplified22.7

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

    if -2.6821964608970343e+143 < z < 4.016279813338476e+120

    1. Initial program 8.7

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

    2. Simplified8.7

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

    4. Applied clear-num8.8

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{\frac{a - z}{t - x}}}, y - z, x\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification13.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -2.682196460897034293396785828877679215877 \cdot 10^{143} \lor \neg \left(z \le 4.016279813338475745054191319638190832337 \cdot 10^{120}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, t\right) - \frac{t \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\frac{a - z}{t - x}}, y - z, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019212 +o rules:numerics
(FPCore (x y z t a)
  :name "Numeric.Signal:interpolate   from hsignal-0.2.7.1"
  :precision binary64
  (+ x (* (- y z) (/ (- t x) (- a z)))))