Average Error: 1.4 → 0.9
Time: 5.3s
Precision: binary64
\[x + y \cdot \frac{z - t}{a - t} \]
↓
\[\begin{array}{l} \mathbf{if}\;y \leq -8.181875273801221 \cdot 10^{+64}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{1}{\frac{a - t}{z}} - \frac{t}{a - t}, x\right)\\ \mathbf{elif}\;y \leq 3.208986056831293 \cdot 10^{-122}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a - t}, x\right)\\ \end{array} \]
x + y \cdot \frac{z - t}{a - t}
↓
\begin{array}{l}
\mathbf{if}\;y \leq -8.181875273801221 \cdot 10^{+64}:\\
\;\;\;\;\mathsf{fma}\left(y, \frac{1}{\frac{a - t}{z}} - \frac{t}{a - t}, x\right)\\

\mathbf{elif}\;y \leq 3.208986056831293 \cdot 10^{-122}:\\
\;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a - t}\\

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


\end{array}

(FPCore (x y z t a) :precision binary64 (+ x (* y (/ (- z t) (- a t)))))
↓
(FPCore (x y z t a)
 :precision binary64
 (if (<= y -8.181875273801221e+64)
   (fma y (- (/ 1.0 (/ (- a t) z)) (/ t (- a t))) x)
   (if (<= y 3.208986056831293e-122)
     (+ x (/ (* y (- z t)) (- a t)))
     (fma y (/ (- z t) (- a t)) x))))
double code(double x, double y, double z, double t, double a) {
	return x + (y * ((z - t) / (a - t)));
}
↓
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (y <= -8.181875273801221e+64) {
		tmp = fma(y, ((1.0 / ((a - t) / z)) - (t / (a - t))), x);
	} else if (y <= 3.208986056831293e-122) {
		tmp = x + ((y * (z - t)) / (a - t));
	} else {
		tmp = fma(y, ((z - t) / (a - t)), x);
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original1.4
Target0.5
Herbie0.9
\[\begin{array}{l} \mathbf{if}\;y < -8.508084860551241 \cdot 10^{-17}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;y < 2.894426862792089 \cdot 10^{-49}:\\ \;\;\;\;x + \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{a - t}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

  2. if y < -8.1818752738012213e64

    1. Initial program 0.9

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

    2. Simplified0.9

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

    3. Taylor expanded in z around 0 0.9

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

    4. Applied clear-num_binary641.1

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

    if -8.1818752738012213e64 < y < 3.208986056831293e-122

    1. Initial program 2.0

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

    2. Taylor expanded in y around 0 0.9

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

    if 3.208986056831293e-122 < y

    1. Initial program 0.8

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

    2. Simplified0.8

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

    3. Applied *-un-lft-identity_binary640.8

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

    4. Applied associate-/r*_binary640.8

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.181875273801221 \cdot 10^{+64}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{1}{\frac{a - t}{z}} - \frac{t}{a - t}, x\right)\\ \mathbf{elif}\;y \leq 3.208986056831293 \cdot 10^{-122}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a - t}, x\right)\\ \end{array} \]

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, B"
  :precision binary64

  :herbie-target
  (if (< y -8.508084860551241e-17) (+ x (* y (/ (- z t) (- a t)))) (if (< y 2.894426862792089e-49) (+ x (* (* y (- z t)) (/ 1.0 (- a t)))) (+ x (* y (/ (- z t) (- a t))))))

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