Average Error: 14.7 → 10.4
Time: 5.2s
Precision: 64
\[x + \left(y - z\right) \cdot \frac{t - x}{a - z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -3.2165983826730228 \cdot 10^{246} \lor \neg \left(z \le 9.10019436889470365 \cdot 10^{191}\right):\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x}{z} - \frac{t}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{x} \cdot \sqrt[3]{x}, \sqrt[3]{x}, \left(\left(y - z\right) \cdot \frac{\sqrt[3]{t - x} \cdot \sqrt[3]{t - x}}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}\right) \cdot \frac{\sqrt[3]{t - x}}{\sqrt[3]{a - z}}\right)\\ \end{array}\]
x + \left(y - z\right) \cdot \frac{t - x}{a - z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -3.2165983826730228 \cdot 10^{246} \lor \neg \left(z \le 9.10019436889470365 \cdot 10^{191}\right):\\
\;\;\;\;\mathsf{fma}\left(y, \frac{x}{z} - \frac{t}{z}, t\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\sqrt[3]{x} \cdot \sqrt[3]{x}, \sqrt[3]{x}, \left(\left(y - z\right) \cdot \frac{\sqrt[3]{t - x} \cdot \sqrt[3]{t - x}}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}\right) \cdot \frac{\sqrt[3]{t - x}}{\sqrt[3]{a - z}}\right)\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r126166 = x;
        double r126167 = y;
        double r126168 = z;
        double r126169 = r126167 - r126168;
        double r126170 = t;
        double r126171 = r126170 - r126166;
        double r126172 = a;
        double r126173 = r126172 - r126168;
        double r126174 = r126171 / r126173;
        double r126175 = r126169 * r126174;
        double r126176 = r126166 + r126175;
        return r126176;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r126177 = z;
        double r126178 = -3.216598382673023e+246;
        bool r126179 = r126177 <= r126178;
        double r126180 = 9.100194368894704e+191;
        bool r126181 = r126177 <= r126180;
        double r126182 = !r126181;
        bool r126183 = r126179 || r126182;
        double r126184 = y;
        double r126185 = x;
        double r126186 = r126185 / r126177;
        double r126187 = t;
        double r126188 = r126187 / r126177;
        double r126189 = r126186 - r126188;
        double r126190 = fma(r126184, r126189, r126187);
        double r126191 = cbrt(r126185);
        double r126192 = r126191 * r126191;
        double r126193 = r126184 - r126177;
        double r126194 = r126187 - r126185;
        double r126195 = cbrt(r126194);
        double r126196 = r126195 * r126195;
        double r126197 = a;
        double r126198 = r126197 - r126177;
        double r126199 = cbrt(r126198);
        double r126200 = r126199 * r126199;
        double r126201 = r126196 / r126200;
        double r126202 = r126193 * r126201;
        double r126203 = r126195 / r126199;
        double r126204 = r126202 * r126203;
        double r126205 = fma(r126192, r126191, r126204);
        double r126206 = r126183 ? r126190 : r126205;
        return r126206;
}

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 < -3.216598382673023e+246 or 9.100194368894704e+191 < z

    1. Initial program 31.4

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

    2. Simplified31.3

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

    3. Taylor expanded around inf 23.5

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

    4. Simplified13.6

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

    if -3.216598382673023e+246 < z < 9.100194368894704e+191

    1. Initial program 11.4

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

    3. Applied add-cube-cbrt11.9

      \[\leadsto x + \left(y - z\right) \cdot \frac{t - x}{\color{blue}{\left(\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}\right) \cdot \sqrt[3]{a - z}}}\]

    4. Applied add-cube-cbrt12.1

      \[\leadsto x + \left(y - z\right) \cdot \frac{\color{blue}{\left(\sqrt[3]{t - x} \cdot \sqrt[3]{t - x}\right) \cdot \sqrt[3]{t - x}}}{\left(\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}\right) \cdot \sqrt[3]{a - z}}\]

    5. Applied times-frac12.1

      \[\leadsto x + \left(y - z\right) \cdot \color{blue}{\left(\frac{\sqrt[3]{t - x} \cdot \sqrt[3]{t - x}}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \frac{\sqrt[3]{t - x}}{\sqrt[3]{a - z}}\right)}\]

    6. Applied associate-*r*9.4

      \[\leadsto x + \color{blue}{\left(\left(y - z\right) \cdot \frac{\sqrt[3]{t - x} \cdot \sqrt[3]{t - x}}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}\right) \cdot \frac{\sqrt[3]{t - x}}{\sqrt[3]{a - z}}}\]
    7. Using strategy rm

    8. Applied add-cube-cbrt9.8

      \[\leadsto \color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}} + \left(\left(y - z\right) \cdot \frac{\sqrt[3]{t - x} \cdot \sqrt[3]{t - x}}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}\right) \cdot \frac{\sqrt[3]{t - x}}{\sqrt[3]{a - z}}\]

    9. Applied fma-def9.8

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

  4. Final simplification10.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -3.2165983826730228 \cdot 10^{246} \lor \neg \left(z \le 9.10019436889470365 \cdot 10^{191}\right):\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x}{z} - \frac{t}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{x} \cdot \sqrt[3]{x}, \sqrt[3]{x}, \left(\left(y - z\right) \cdot \frac{\sqrt[3]{t - x} \cdot \sqrt[3]{t - x}}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}\right) \cdot \frac{\sqrt[3]{t - x}}{\sqrt[3]{a - z}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020100 +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)))))