Average Error: 10.3 → 0.7
Time: 7.1s
Precision: binary64
\[x + \frac{y \cdot \left(z - t\right)}{a - t} \]
↓
\[\begin{array}{l} t_1 := \frac{y \cdot \left(z - t\right)}{a - t}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a - t}, x\right)\\ \mathbf{elif}\;t_1 \leq -6.911634280618228 \cdot 10^{-289}:\\ \;\;\;\;t_1 + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{1}{\frac{a - t}{z - t}}, x\right)\\ \end{array} \]
x + \frac{y \cdot \left(z - t\right)}{a - t}
↓
\begin{array}{l}
t_1 := \frac{y \cdot \left(z - t\right)}{a - t}\\
\mathbf{if}\;t_1 \leq -\infty:\\
\;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a - t}, x\right)\\

\mathbf{elif}\;t_1 \leq -6.911634280618228 \cdot 10^{-289}:\\
\;\;\;\;t_1 + x\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(y, \frac{1}{\frac{a - t}{z - 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
 (let* ((t_1 (/ (* y (- z t)) (- a t))))
   (if (<= t_1 (- INFINITY))
     (fma y (/ (- z t) (- a t)) x)
     (if (<= t_1 -6.911634280618228e-289)
       (+ t_1 x)
       (fma y (/ 1.0 (/ (- a t) (- z 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 t_1 = (y * (z - t)) / (a - t);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(y, ((z - t) / (a - t)), x);
	} else if (t_1 <= -6.911634280618228e-289) {
		tmp = t_1 + x;
	} else {
		tmp = fma(y, (1.0 / ((a - t) / (z - 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

Original10.3
Target1.2
Herbie0.7
\[x + \frac{y}{\frac{a - t}{z - t}} \]

Derivation

  1. Split input into 3 regimes

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

    1. Initial program 64.0

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

    2. Simplified0.1

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

    if -inf.0 < (/.f64 (*.f64 y (-.f64 z t)) (-.f64 a t)) < -6.9116342806182276e-289

    1. Initial program 0.1

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

    2. Simplified2.0

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

    3. Taylor expanded in y around 0 0.1

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

    4. Simplified3.5

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

    5. Applied *-un-lft-identity_binary643.5

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

    6. Applied associate-*l*_binary643.5

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

    7. Simplified0.1

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

    if -6.9116342806182276e-289 < (/.f64 (*.f64 y (-.f64 z t)) (-.f64 a t))

    1. Initial program 9.1

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

    2. Simplified1.0

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

    3. Applied clear-num_binary641.1

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{a - t} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a - t}, x\right)\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{a - t} \leq -6.911634280618228 \cdot 10^{-289}:\\ \;\;\;\;\frac{y \cdot \left(z - t\right)}{a - t} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{1}{\frac{a - t}{z - t}}, x\right)\\ \end{array} \]

Reproduce

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

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

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