Average Error: 15.3 → 11.6
Time: 4.3s
Precision: 64
\[x + \left(y - z\right) \cdot \frac{t - x}{a - z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.8711755082866498 \cdot 10^{149} \lor \neg \left(z \le 2.2746469954276827 \cdot 10^{114}\right):\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x}{z} - \frac{t}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t - x}{a - z}, x\right)\\ \end{array}\]
x + \left(y - z\right) \cdot \frac{t - x}{a - z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.8711755082866498 \cdot 10^{149} \lor \neg \left(z \le 2.2746469954276827 \cdot 10^{114}\right):\\
\;\;\;\;\mathsf{fma}\left(y, \frac{x}{z} - \frac{t}{z}, t\right)\\

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

\end{array}
double code(double x, double y, double z, double t, double a) {
	return (x + ((y - z) * ((t - x) / (a - z))));
}
↓
double code(double x, double y, double z, double t, double a) {
	double temp;
	if (((z <= -1.8711755082866498e+149) || !(z <= 2.2746469954276827e+114))) {
		temp = fma(y, ((x / z) - (t / z)), t);
	} else {
		temp = fma((y - z), ((t - x) / (a - z)), x);
	}
	return temp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if z < -1.8711755082866498e+149 or 2.2746469954276827e+114 < z

    1. Initial program 28.2

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

    2. Simplified28.1

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

    3. Taylor expanded around inf 24.9

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

    4. Simplified16.8

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

    if -1.8711755082866498e+149 < z < 2.2746469954276827e+114

    1. Initial program 9.0

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

    2. Simplified9.0

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

    4. Applied fma-udef9.0

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

    6. Applied fma-def9.0

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

  4. Final simplification11.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.8711755082866498 \cdot 10^{149} \lor \neg \left(z \le 2.2746469954276827 \cdot 10^{114}\right):\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x}{z} - \frac{t}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t - x}{a - z}, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020049 +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)))))