Average Error: 1.1 → 1.9
Time: 5.3s
Precision: binary64
\[x + y \cdot \frac{z - t}{z - a} \]
↓
\[\begin{array}{l} \mathbf{if}\;z \leq -1.6518303730418872 \cdot 10^{-206} \lor \neg \left(z \leq 1.9393432326971347 \cdot 10^{-171}\right):\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{z - a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z - a}, z, x\right) - \frac{1}{\frac{z - a}{y \cdot t}}\\ \end{array} \]
x + y \cdot \frac{z - t}{z - a}
↓
\begin{array}{l}
\mathbf{if}\;z \leq -1.6518303730418872 \cdot 10^{-206} \lor \neg \left(z \leq 1.9393432326971347 \cdot 10^{-171}\right):\\
\;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{z - a}, x\right)\\

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


\end{array}

(FPCore (x y z t a) :precision binary64 (+ x (* y (/ (- z t) (- z a)))))
↓
(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -1.6518303730418872e-206) (not (<= z 1.9393432326971347e-171)))
   (fma y (/ (- z t) (- z a)) x)
   (- (fma (/ y (- z a)) z x) (/ 1.0 (/ (- z a) (* y t))))))
double code(double x, double y, double z, double t, double a) {
	return x + (y * ((z - t) / (z - a)));
}
↓
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -1.6518303730418872e-206) || !(z <= 1.9393432326971347e-171)) {
		tmp = fma(y, ((z - t) / (z - a)), x);
	} else {
		tmp = fma((y / (z - a)), z, x) - (1.0 / ((z - a) / (y * t)));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original1.1
Target1.1
Herbie1.9
\[x + \frac{y}{\frac{z - a}{z - t}} \]

Derivation

  1. Split input into 2 regimes

  2. if z < -1.65183037304188715e-206 or 1.9393432326971347e-171 < z

    1. Initial program 0.7

      \[x + y \cdot \frac{z - t}{z - a} \]

    2. Simplified0.7

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

    if -1.65183037304188715e-206 < z < 1.9393432326971347e-171

    1. Initial program 3.1

      \[x + y \cdot \frac{z - t}{z - a} \]

    2. Simplified3.1

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

    3. Taylor expanded in y around 0 3.0

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

    4. Simplified7.2

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

    5. Applied clear-num_binary647.3

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

  4. Final simplification1.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.6518303730418872 \cdot 10^{-206} \lor \neg \left(z \leq 1.9393432326971347 \cdot 10^{-171}\right):\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{z - a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z - a}, z, x\right) - \frac{1}{\frac{z - a}{y \cdot t}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022068 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, A"
  :precision binary64

  :herbie-target
  (+ x (/ y (/ (- z a) (- z t))))

  (+ x (* y (/ (- z t) (- z a)))))