Average Error: 16.6 → 7.4
Time: 8.6s
Precision: binary64
\[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{y \cdot \left(z - t\right)}{a - t} \leq -6.947203039761551 \cdot 10^{-276} \lor \neg \left(\left(x + y\right) - \frac{y \cdot \left(z - t\right)}{a - t} \leq 0\right):\\ \;\;\;\;\left(x + y\right) - \left(\left(z - t\right) \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{a - t}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot z}{t}\\ \end{array}\]
\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}
↓
\begin{array}{l}
\mathbf{if}\;\left(x + y\right) - \frac{y \cdot \left(z - t\right)}{a - t} \leq -6.947203039761551 \cdot 10^{-276} \lor \neg \left(\left(x + y\right) - \frac{y \cdot \left(z - t\right)}{a - t} \leq 0\right):\\
\;\;\;\;\left(x + y\right) - \left(\left(z - t\right) \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{a - t}}\\

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

\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 (<= (- (+ x y) (/ (* y (- z t)) (- a t))) -6.947203039761551e-276)
         (not (<= (- (+ x y) (/ (* y (- z t)) (- a t))) 0.0)))
   (-
    (+ x y)
    (*
     (* (- z t) (/ (* (cbrt y) (cbrt y)) (* (cbrt (- a t)) (cbrt (- a t)))))
     (/ (cbrt y) (cbrt (- a t)))))
   (+ x (/ (* y z) t))))
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 ((((x + y) - ((y * (z - t)) / (a - t))) <= -6.947203039761551e-276) || !(((x + y) - ((y * (z - t)) / (a - t))) <= 0.0)) {
		tmp = (x + y) - (((z - t) * ((cbrt(y) * cbrt(y)) / (cbrt(a - t) * cbrt(a - t)))) * (cbrt(y) / cbrt(a - t)));
	} else {
		tmp = x + ((y * 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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original16.6
Target8.2
Herbie7.4
\[\begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} < -1.3664970889390727 \cdot 10^{-07}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} < 1.4754293444577233 \cdot 10^{-239}:\\ \;\;\;\;\frac{y \cdot \left(a - z\right) - x \cdot t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t))) < -6.9472030397615508e-276 or 0.0 < (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t)))

    1. Initial program 12.7

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\]
    2. Using strategy rm

    3. Applied *-un-lft-identity_binary64_1746912.7

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

    4. Applied times-frac_binary64_174757.7

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

    5. Simplified7.7

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

    7. Applied add-cube-cbrt_binary64_175047.9

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

    8. Applied add-cube-cbrt_binary64_175048.0

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

    9. Applied times-frac_binary64_174758.0

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

    10. Applied associate-*r*_binary64_174096.7

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

    if -6.9472030397615508e-276 < (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t))) < 0.0

    1. Initial program 60.8

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

    2. Taylor expanded around inf 16.2

      \[\leadsto \color{blue}{\frac{z \cdot y}{t} + x}\]

    3. Simplified16.2

      \[\leadsto \color{blue}{x + \frac{z \cdot y}{t}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification7.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{y \cdot \left(z - t\right)}{a - t} \leq -6.947203039761551 \cdot 10^{-276} \lor \neg \left(\left(x + y\right) - \frac{y \cdot \left(z - t\right)}{a - t} \leq 0\right):\\ \;\;\;\;\left(x + y\right) - \left(\left(z - t\right) \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{a - t}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot z}{t}\\ \end{array}\]

Reproduce

herbie shell --seed 2020295 
(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-07) (- (+ 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))))