Average Error: 10.8 → 0.3
Time: 4.4s
Precision: binary64
\[x + \frac{\left(y - z\right) \cdot t}{a - z} \]
↓
\[\begin{array}{l} t_1 := \frac{\left(y - z\right) \cdot t}{a - z}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t}{a - z}, x\right)\\ \mathbf{elif}\;t_1 \leq 8.055890673554846 \cdot 10^{+307}:\\ \;\;\;\;t_1 + x\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - z}{\frac{a - z}{t}}\\ \end{array} \]
x + \frac{\left(y - z\right) \cdot t}{a - z}
↓
\begin{array}{l}
t_1 := \frac{\left(y - z\right) \cdot t}{a - z}\\
\mathbf{if}\;t_1 \leq -\infty:\\
\;\;\;\;\mathsf{fma}\left(y - z, \frac{t}{a - z}, x\right)\\

\mathbf{elif}\;t_1 \leq 8.055890673554846 \cdot 10^{+307}:\\
\;\;\;\;t_1 + x\\

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


\end{array}

(FPCore (x y z t a) :precision binary64 (+ x (/ (* (- y z) t) (- a z))))
↓
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* (- y z) t) (- a z))))
   (if (<= t_1 (- INFINITY))
     (fma (- y z) (/ t (- a z)) x)
     (if (<= t_1 8.055890673554846e+307)
       (+ t_1 x)
       (+ x (/ (- y z) (/ (- a z) t)))))))
double code(double x, double y, double z, double t, double a) {
	return x + (((y - z) * t) / (a - z));
}
↓
double code(double x, double y, double z, double t, double a) {
	double t_1 = ((y - z) * t) / (a - z);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma((y - z), (t / (a - z)), x);
	} else if (t_1 <= 8.055890673554846e+307) {
		tmp = t_1 + x;
	} else {
		tmp = x + ((y - z) / ((a - z) / 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

Original10.8
Target0.6
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;t < -1.0682974490174067 \cdot 10^{-39}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \mathbf{elif}\;t < 3.9110949887586375 \cdot 10^{-141}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

  2. if (/.f64 (*.f64 (-.f64 y z) t) (-.f64 a z)) < -inf.0

    1. Initial program 64.0

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

    2. Simplified0.1

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

    if -inf.0 < (/.f64 (*.f64 (-.f64 y z) t) (-.f64 a z)) < 8.0558906735548456e307

    1. Initial program 0.3

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

    if 8.0558906735548456e307 < (/.f64 (*.f64 (-.f64 y z) t) (-.f64 a z))

    1. Initial program 64.0

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

    2. Applied associate-/l*_binary640.1

      \[\leadsto x + \color{blue}{\frac{y - z}{\frac{a - z}{t}}} \]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(y - z\right) \cdot t}{a - z} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t}{a - z}, x\right)\\ \mathbf{elif}\;\frac{\left(y - z\right) \cdot t}{a - z} \leq 8.055890673554846 \cdot 10^{+307}:\\ \;\;\;\;\frac{\left(y - z\right) \cdot t}{a - z} + x\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - z}{\frac{a - z}{t}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022077 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, A"
  :precision binary64

  :herbie-target
  (if (< t -1.0682974490174067e-39) (+ x (* (/ (- y z) (- a z)) t)) (if (< t 3.9110949887586375e-141) (+ x (/ (* (- y z) t) (- a z))) (+ x (* (/ (- y z) (- a z)) t))))

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