Average Error: 2.0 → 2.1
Time: 2.2s
Precision: 64
\[x + \left(y - x\right) \cdot \frac{z}{t}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{z}{t} \le -354.15624360949397 \lor \neg \left(\frac{z}{t} \le -1.3295372768341099 \cdot 10^{-243} \lor \neg \left(\frac{z}{t} \le 8.4862425903035375 \cdot 10^{-308}\right)\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t} - \frac{x}{t}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{\frac{\frac{t}{z}}{y - x}}\\ \end{array}\]
x + \left(y - x\right) \cdot \frac{z}{t}
↓
\begin{array}{l}
\mathbf{if}\;\frac{z}{t} \le -354.15624360949397 \lor \neg \left(\frac{z}{t} \le -1.3295372768341099 \cdot 10^{-243} \lor \neg \left(\frac{z}{t} \le 8.4862425903035375 \cdot 10^{-308}\right)\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{t} - \frac{x}{t}, z, x\right)\\

\mathbf{else}:\\
\;\;\;\;x + \frac{1}{\frac{\frac{t}{z}}{y - x}}\\

\end{array}
double code(double x, double y, double z, double t) {
	return (x + ((y - x) * (z / t)));
}
↓
double code(double x, double y, double z, double t) {
	double VAR;
	if ((((z / t) <= -354.15624360949397) || !(((z / t) <= -1.32953727683411e-243) || !((z / t) <= 8.486242590303537e-308)))) {
		VAR = fma(((y / t) - (x / t)), z, x);
	} else {
		VAR = (x + (1.0 / ((t / z) / (y - x))));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original2.0
Target2.2
Herbie2.1
\[\begin{array}{l} \mathbf{if}\;\left(y - x\right) \cdot \frac{z}{t} \lt -1013646692435.887:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \mathbf{elif}\;\left(y - x\right) \cdot \frac{z}{t} \lt -0.0:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ z t) < -354.15624360949397 or -1.32953727683411e-243 < (/ z t) < 8.486242590303537e-308

    1. Initial program 3.8

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

    3. Applied clear-num4.0

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

    4. Applied un-div-inv3.6

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

    5. Taylor expanded around 0 4.0

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

    6. Simplified4.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - x}{t}, z, x\right)}\]
    7. Using strategy rm

    8. Applied div-sub4.0

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

    if -354.15624360949397 < (/ z t) < -1.32953727683411e-243 or 8.486242590303537e-308 < (/ z t)

    1. Initial program 1.1

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

    3. Applied clear-num1.1

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

    4. Applied un-div-inv0.9

      \[\leadsto x + \color{blue}{\frac{y - x}{\frac{t}{z}}}\]
    5. Using strategy rm

    6. Applied clear-num1.0

      \[\leadsto x + \color{blue}{\frac{1}{\frac{\frac{t}{z}}{y - x}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification2.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z}{t} \le -354.15624360949397 \lor \neg \left(\frac{z}{t} \le -1.3295372768341099 \cdot 10^{-243} \lor \neg \left(\frac{z}{t} \le 8.4862425903035375 \cdot 10^{-308}\right)\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t} - \frac{x}{t}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{\frac{\frac{t}{z}}{y - x}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020078 +o rules:numerics
(FPCore (x y z t)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:tickPosition from plot-0.2.3.4"
  :precision binary64

  :herbie-target
  (if (< (* (- y x) (/ z t)) -1013646692435.887) (+ x (/ (- y x) (/ t z))) (if (< (* (- y x) (/ z t)) -0.0) (+ x (/ (* (- y x) z) t)) (+ x (/ (- y x) (/ t z)))))

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