Average Error: 16.3 → 6.8
Time: 7.9s
Precision: binary64
\[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
↓
\[\begin{array}{l} \mathbf{if}\;t \leq -2.874394328703431 \cdot 10^{+162} \lor \neg \left(t \leq 5.485833698658141 \cdot 10^{+80}\right):\\ \;\;\;\;x + \frac{y}{t} \cdot \left(z - a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y + \left(x + \frac{t \cdot y}{a - t}\right)\right) - \frac{y}{\frac{a - t}{z}}\\ \end{array} \]
\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}
↓
\begin{array}{l}
\mathbf{if}\;t \leq -2.874394328703431 \cdot 10^{+162} \lor \neg \left(t \leq 5.485833698658141 \cdot 10^{+80}\right):\\
\;\;\;\;x + \frac{y}{t} \cdot \left(z - a\right)\\

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


\end{array}

(FPCore (x y z t a) :precision binary64 (- (+ x y) (/ (* (- z t) y) (- a t))))
↓
(FPCore (x y z t a)
 :precision binary64
 (if (or (<= t -2.874394328703431e+162) (not (<= t 5.485833698658141e+80)))
   (+ x (* (/ y t) (- z a)))
   (- (+ y (+ x (/ (* t y) (- a t)))) (/ y (/ (- a t) z)))))
double code(double x, double y, double z, double t, double a) {
	return (x + y) - (((z - t) * y) / (a - t));
}
↓
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= -2.874394328703431e+162) || !(t <= 5.485833698658141e+80)) {
		tmp = x + ((y / t) * (z - a));
	} else {
		tmp = (y + (x + ((t * y) / (a - t)))) - (y / ((a - t) / z));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original16.3
Target8.4
Herbie6.8
\[\begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} < -1.3664970889390727 \cdot 10^{-7}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} < 1.4754293444577233 \cdot 10^{-239}:\\ \;\;\;\;\frac{y \cdot \left(a - z\right) - x \cdot t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \end{array} \]

Derivation

  1. Split input into 2 regimes

  2. if t < -2.874394328703431e162 or 5.4858336986581407e80 < t

    1. Initial program 31.1

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

    2. Simplified21.5

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

    3. Taylor expanded in t around inf 14.9

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

    4. Simplified7.9

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

    if -2.874394328703431e162 < t < 5.4858336986581407e80

    1. Initial program 9.3

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

    2. Simplified6.7

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

    3. Taylor expanded in y around 0 8.0

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

    4. Applied associate-/l*_binary646.3

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

  4. Final simplification6.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.874394328703431 \cdot 10^{+162} \lor \neg \left(t \leq 5.485833698658141 \cdot 10^{+80}\right):\\ \;\;\;\;x + \frac{y}{t} \cdot \left(z - a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y + \left(x + \frac{t \cdot y}{a - t}\right)\right) - \frac{y}{\frac{a - t}{z}}\\ \end{array} \]

Reproduce

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

  :herbie-target
  (if (< (- (+ x y) (/ (* (- z t) y) (- a t))) -1.3664970889390727e-7) (- (+ y x) (* (* (- z t) (/ 1.0 (- a t))) y)) (if (< (- (+ x y) (/ (* (- z t) y) (- a t))) 1.4754293444577233e-239) (/ (- (* y (- a z)) (* x t)) (- a t)) (- (+ y x) (* (* (- z t) (/ 1.0 (- a t))) y))))

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