Average Error: 10.7 → 0.3
Time: 7.3s
Precision: binary64
\[x + \frac{y \cdot \left(z - t\right)}{z - a} \]
↓
\[\begin{array}{l} t_1 := \frac{y \cdot \left(z - t\right)}{z - a}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{z - a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_2 := \frac{z - a}{z}\\ \mathbf{if}\;t_1 \leq 5.413814096002667 \cdot 10^{+297}:\\ \;\;\;\;\left(x + \frac{y}{t_2}\right) - \frac{y \cdot t}{z - a}\\ \mathbf{else}:\\ \;\;\;\;\left(x + y \cdot \frac{1}{t_2}\right) - \frac{y}{\frac{z - a}{t}}\\ \end{array}\\ \end{array} \]
x + \frac{y \cdot \left(z - t\right)}{z - a}
↓
\begin{array}{l}
t_1 := \frac{y \cdot \left(z - t\right)}{z - a}\\
\mathbf{if}\;t_1 \leq -\infty:\\
\;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{z - a}, x\right)\\

\mathbf{else}:\\
\;\;\;\;\begin{array}{l}
t_2 := \frac{z - a}{z}\\
\mathbf{if}\;t_1 \leq 5.413814096002667 \cdot 10^{+297}:\\
\;\;\;\;\left(x + \frac{y}{t_2}\right) - \frac{y \cdot t}{z - a}\\

\mathbf{else}:\\
\;\;\;\;\left(x + y \cdot \frac{1}{t_2}\right) - \frac{y}{\frac{z - a}{t}}\\


\end{array}\\


\end{array}

(FPCore (x y z t a) :precision binary64 (+ x (/ (* y (- z t)) (- z a))))
↓
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* y (- z t)) (- z a))))
   (if (<= t_1 (- INFINITY))
     (fma y (/ (- z t) (- z a)) x)
     (let* ((t_2 (/ (- z a) z)))
       (if (<= t_1 5.413814096002667e+297)
         (- (+ x (/ y t_2)) (/ (* y t) (- z a)))
         (- (+ x (* y (/ 1.0 t_2))) (/ y (/ (- z a) 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 t_1 = (y * (z - t)) / (z - a);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(y, ((z - t) / (z - a)), x);
	} else {
		double t_2 = (z - a) / z;
		double tmp_1;
		if (t_1 <= 5.413814096002667e+297) {
			tmp_1 = (x + (y / t_2)) - ((y * t) / (z - a));
		} else {
			tmp_1 = (x + (y * (1.0 / t_2))) - (y / ((z - a) / t));
		}
		tmp = tmp_1;
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original10.7
Target1.4
Herbie0.3
\[x + \frac{y}{\frac{z - a}{z - t}} \]

Derivation

  1. Split input into 3 regimes

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

    1. Initial program 64.0

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

    2. Simplified0.1

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

    if -inf.0 < (/.f64 (*.f64 y (-.f64 z t)) (-.f64 z a)) < 5.4138140960026674e297

    1. Initial program 0.2

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

    2. Simplified1.8

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

    3. Taylor expanded in y around 0 0.2

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

    4. Applied associate-/l*_binary640.4

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

    if 5.4138140960026674e297 < (/.f64 (*.f64 y (-.f64 z t)) (-.f64 z a))

    1. Initial program 62.7

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

    2. Simplified0.4

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

    3. Taylor expanded in y around 0 62.7

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

    4. Applied add-cube-cbrt_binary6462.7

      \[\leadsto \left(\frac{y \cdot z}{\color{blue}{\left(\sqrt[3]{z - a} \cdot \sqrt[3]{z - a}\right) \cdot \sqrt[3]{z - a}}} + x\right) - \frac{y \cdot t}{z - a} \]

    5. Applied times-frac_binary6427.1

      \[\leadsto \left(\color{blue}{\frac{y}{\sqrt[3]{z - a} \cdot \sqrt[3]{z - a}} \cdot \frac{z}{\sqrt[3]{z - a}}} + x\right) - \frac{y \cdot t}{z - a} \]

    6. Applied associate-/l*_binary641.4

      \[\leadsto \left(\frac{y}{\sqrt[3]{z - a} \cdot \sqrt[3]{z - a}} \cdot \frac{z}{\sqrt[3]{z - a}} + x\right) - \color{blue}{\frac{y}{\frac{z - a}{t}}} \]

    7. Applied div-inv_binary641.4

      \[\leadsto \left(\color{blue}{\left(y \cdot \frac{1}{\sqrt[3]{z - a} \cdot \sqrt[3]{z - a}}\right)} \cdot \frac{z}{\sqrt[3]{z - a}} + x\right) - \frac{y}{\frac{z - a}{t}} \]

    8. Applied associate-*l*_binary641.1

      \[\leadsto \left(\color{blue}{y \cdot \left(\frac{1}{\sqrt[3]{z - a} \cdot \sqrt[3]{z - a}} \cdot \frac{z}{\sqrt[3]{z - a}}\right)} + x\right) - \frac{y}{\frac{z - a}{t}} \]

    9. Simplified0.4

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{z - a} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{z - a}, x\right)\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{z - a} \leq 5.413814096002667 \cdot 10^{+297}:\\ \;\;\;\;\left(x + \frac{y}{\frac{z - a}{z}}\right) - \frac{y \cdot t}{z - a}\\ \mathbf{else}:\\ \;\;\;\;\left(x + y \cdot \frac{1}{\frac{z - a}{z}}\right) - \frac{y}{\frac{z - a}{t}}\\ \end{array} \]

Reproduce

herbie shell --seed 2021340 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTicks from plot-0.2.3.4, A"
  :precision binary64

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

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