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

↓

\[\begin{array}{l} \mathbf{if}\;a \le -2.145869478761536680500210929593317634387 \cdot 10^{-21}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\frac{a - z}{t - x}}, y - z, x\right)\\ \mathbf{elif}\;a \le 9.245984676792925009764001853570344430799 \cdot 10^{-181}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, t - \frac{y \cdot t}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t - x}{\frac{a - z}{y - z}} + x\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;a \le -2.145869478761536680500210929593317634387 \cdot 10^{-21}:\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{\frac{a - z}{t - x}}, y - z, x\right)\\

\mathbf{elif}\;a \le 9.245984676792925009764001853570344430799 \cdot 10^{-181}:\\
\;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, t - \frac{y \cdot t}{z}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{t - x}{\frac{a - z}{y - z}} + x\\

\end{array}

double f(double x, double y, double z, double t, double a) {
        double r119589 = x;
        double r119590 = y;
        double r119591 = z;
        double r119592 = r119590 - r119591;
        double r119593 = t;
        double r119594 = r119593 - r119589;
        double r119595 = a;
        double r119596 = r119595 - r119591;
        double r119597 = r119594 / r119596;
        double r119598 = r119592 * r119597;
        double r119599 = r119589 + r119598;
        return r119599;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r119600 = a;
        double r119601 = -2.1458694787615367e-21;
        bool r119602 = r119600 <= r119601;
        double r119603 = 1.0;
        double r119604 = z;
        double r119605 = r119600 - r119604;
        double r119606 = t;
        double r119607 = x;
        double r119608 = r119606 - r119607;
        double r119609 = r119605 / r119608;
        double r119610 = r119603 / r119609;
        double r119611 = y;
        double r119612 = r119611 - r119604;
        double r119613 = fma(r119610, r119612, r119607);
        double r119614 = 9.245984676792925e-181;
        bool r119615 = r119600 <= r119614;
        double r119616 = r119607 / r119604;
        double r119617 = r119611 * r119606;
        double r119618 = r119617 / r119604;
        double r119619 = r119606 - r119618;
        double r119620 = fma(r119616, r119611, r119619);
        double r119621 = r119605 / r119612;
        double r119622 = r119608 / r119621;
        double r119623 = r119622 + r119607;
        double r119624 = r119615 ? r119620 : r119623;
        double r119625 = r119602 ? r119613 : r119624;
        return r119625;
}

Derivation

Split input into 3 regimes
if a < -2.1458694787615367e-21
1. Initial program 8.6
  \[x + \left(y - z\right) \cdot \frac{t - x}{a - z}\]
2. Simplified8.6
  \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - x}{a - z}, y - z, x\right)}\]
3. Using strategy rm
4. Applied clear-num8.9
  \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{\frac{a - z}{t - x}}}, y - z, x\right)\]
if -2.1458694787615367e-21 < a < 9.245984676792925e-181
1. Initial program 23.0
  \[x + \left(y - z\right) \cdot \frac{t - x}{a - z}\]
2. Simplified23.0
  \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - x}{a - z}, y - z, x\right)}\]
3. Using strategy rm
4. Applied fma-udef23.0
  \[\leadsto \color{blue}{\frac{t - x}{a - z} \cdot \left(y - z\right) + x}\]
5. Using strategy rm
6. Applied div-inv23.1
  \[\leadsto \color{blue}{\left(\left(t - x\right) \cdot \frac{1}{a - z}\right)} \cdot \left(y - z\right) + x\]
7. Applied associate-*l*18.8
  \[\leadsto \color{blue}{\left(t - x\right) \cdot \left(\frac{1}{a - z} \cdot \left(y - z\right)\right)} + x\]
8. Simplified18.7
  \[\leadsto \left(t - x\right) \cdot \color{blue}{\frac{y - z}{a - z}} + x\]
9. Taylor expanded around inf 18.0
  \[\leadsto \color{blue}{\left(t + \frac{x \cdot y}{z}\right) - \frac{t \cdot y}{z}}\]
10. Simplified16.9
  \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{z}, y, t - \frac{y \cdot t}{z}\right)}\]
if 9.245984676792925e-181 < a
1. Initial program 11.5
  \[x + \left(y - z\right) \cdot \frac{t - x}{a - z}\]
2. Simplified11.5
  \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - x}{a - z}, y - z, x\right)}\]
3. Using strategy rm
4. Applied fma-udef11.5
  \[\leadsto \color{blue}{\frac{t - x}{a - z} \cdot \left(y - z\right) + x}\]
5. Using strategy rm
6. Applied div-inv11.6
  \[\leadsto \color{blue}{\left(\left(t - x\right) \cdot \frac{1}{a - z}\right)} \cdot \left(y - z\right) + x\]
7. Applied associate-*l*9.1
  \[\leadsto \color{blue}{\left(t - x\right) \cdot \left(\frac{1}{a - z} \cdot \left(y - z\right)\right)} + x\]
8. Simplified9.0
  \[\leadsto \left(t - x\right) \cdot \color{blue}{\frac{y - z}{a - z}} + x\]
9. Using strategy rm
10. Applied clear-num9.2
  \[\leadsto \left(t - x\right) \cdot \color{blue}{\frac{1}{\frac{a - z}{y - z}}} + x\]
11. Using strategy rm
12. Applied *-un-lft-identity9.2
  \[\leadsto \color{blue}{\left(1 \cdot \left(t - x\right)\right)} \cdot \frac{1}{\frac{a - z}{y - z}} + x\]
13. Applied associate-*l*9.2
  \[\leadsto \color{blue}{1 \cdot \left(\left(t - x\right) \cdot \frac{1}{\frac{a - z}{y - z}}\right)} + x\]
14. Simplified9.1
  \[\leadsto 1 \cdot \color{blue}{\frac{t - x}{\frac{a - z}{y - z}}} + x\]
Recombined 3 regimes into one program.
Final simplification11.4
\[\leadsto \begin{array}{l} \mathbf{if}\;a \le -2.145869478761536680500210929593317634387 \cdot 10^{-21}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\frac{a - z}{t - x}}, y - z, x\right)\\ \mathbf{elif}\;a \le 9.245984676792925009764001853570344430799 \cdot 10^{-181}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, t - \frac{y \cdot t}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t - x}{\frac{a - z}{y - z}} + x\\ \end{array}\]

Error

Derivation

`if a < -2.1458694787615367e-21`

`if -2.1458694787615367e-21 < a < 9.245984676792925e-181`

`if 9.245984676792925e-181 < a`

Reproduce