Average Error: 14.7 → 4.8
Time: 7.5s
Precision: binary64
\[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
↓
\[\begin{array}{l} \mathbf{if}\;\begin{array}{l} t_1 := x + \left(y - z\right) \cdot \frac{t - x}{a - z}\\ t_1 \leq -2.268598199978211 \cdot 10^{-299} \lor \neg \left(t_1 \leq 0\right) \end{array}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x \cdot y}{z} + \left(t + \frac{t \cdot a}{z}\right)\right) - \left(\frac{y \cdot t}{z} + \frac{x \cdot a}{z}\right)\\ \end{array} \]
x + \left(y - z\right) \cdot \frac{t - x}{a - z}
↓
\begin{array}{l}
\mathbf{if}\;\begin{array}{l}
t_1 := x + \left(y - z\right) \cdot \frac{t - x}{a - z}\\
t_1 \leq -2.268598199978211 \cdot 10^{-299} \lor \neg \left(t_1 \leq 0\right)
\end{array}:\\
\;\;\;\;\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)\\

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


\end{array}

(FPCore (x y z t a) :precision binary64 (+ x (* (- y z) (/ (- t x) (- a z)))))
↓
(FPCore (x y z t a)
 :precision binary64
 (if (let* ((t_1 (+ x (* (- y z) (/ (- t x) (- a z))))))
       (or (<= t_1 -2.268598199978211e-299) (not (<= t_1 0.0))))
   (fma (/ (- y z) (- a z)) (- t x) x)
   (- (+ (/ (* x y) z) (+ t (/ (* t a) z))) (+ (/ (* y t) z) (/ (* x a) z)))))
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 t_1 = x + ((y - z) * ((t - x) / (a - z)));
	double tmp;
	if ((t_1 <= -2.268598199978211e-299) || !(t_1 <= 0.0)) {
		tmp = fma(((y - z) / (a - z)), (t - x), x);
	} else {
		tmp = (((x * y) / z) + (t + ((t * a) / z))) - (((y * t) / z) + ((x * a) / z));
	}
	return tmp;
}

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 (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < -2.2685981999782111e-299 or 0.0 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z))))

    1. Initial program 7.4

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

    2. Simplified7.3

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

    3. Applied clear-num_binary647.6

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

    4. Applied fma-udef_binary647.6

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

    5. Simplified7.3

      \[\leadsto \color{blue}{\frac{y - z}{\frac{a - z}{t - x}}} + x \]

    6. Applied associate-/r/_binary643.9

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

    7. Applied fma-def_binary643.9

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

    if -2.2685981999782111e-299 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < 0.0

    1. Initial program 61.6

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

    2. Simplified61.3

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

    3. Taylor expanded in z around inf 10.7

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

  4. Final simplification4.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x + \left(y - z\right) \cdot \frac{t - x}{a - z} \leq -2.268598199978211 \cdot 10^{-299} \lor \neg \left(x + \left(y - z\right) \cdot \frac{t - x}{a - z} \leq 0\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x \cdot y}{z} + \left(t + \frac{t \cdot a}{z}\right)\right) - \left(\frac{y \cdot t}{z} + \frac{x \cdot a}{z}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022081 
(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)))))