Average Error: 6.3 → 2.3
Time: 4.4s
Precision: 64
\[x + \frac{\left(y - x\right) \cdot z}{t}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.33529079990606012 \cdot 10^{-79} \lor \neg \left(z \le 2.779801418179183 \cdot 10^{-143}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y - x}{t}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sqrt[3]{-1}}{t}, x \cdot z - z \cdot y, x\right)\\ \end{array}\]
x + \frac{\left(y - x\right) \cdot z}{t}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.33529079990606012 \cdot 10^{-79} \lor \neg \left(z \le 2.779801418179183 \cdot 10^{-143}\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{y - x}{t}, z, x\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{\sqrt[3]{-1}}{t}, x \cdot z - z \cdot y, x\right)\\

\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 <= -1.3352907999060601e-79) || !(z <= 2.779801418179183e-143))) {
		VAR = fma(((y - x) / t), z, x);
	} else {
		VAR = fma((cbrt(-1.0) / t), ((x * z) - (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

Original6.3
Target2.0
Herbie2.3
\[\begin{array}{l} \mathbf{if}\;x \lt -9.0255111955330046 \cdot 10^{-135}:\\ \;\;\;\;x - \frac{z}{t} \cdot \left(x - y\right)\\ \mathbf{elif}\;x \lt 4.2750321637007147 \cdot 10^{-250}:\\ \;\;\;\;x + \frac{y - x}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -1.3352907999060601e-79 or 2.779801418179183e-143 < z

    1. Initial program 10.0

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

    2. Simplified3.0

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

    if -1.3352907999060601e-79 < z < 2.779801418179183e-143

    1. Initial program 1.1

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

    2. Simplified11.2

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

    4. Applied add-cube-cbrt11.4

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\sqrt[3]{\frac{y - x}{t}} \cdot \sqrt[3]{\frac{y - x}{t}}\right) \cdot \sqrt[3]{\frac{y - x}{t}}}, z, x\right)\]
    5. Using strategy rm

    6. Applied cbrt-div11.4

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

    7. Taylor expanded around -inf 1.1

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

    8. Simplified1.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\sqrt[3]{-1}}{t}, x \cdot z - z \cdot y, x\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification2.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.33529079990606012 \cdot 10^{-79} \lor \neg \left(z \le 2.779801418179183 \cdot 10^{-143}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y - x}{t}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sqrt[3]{-1}}{t}, x \cdot z - z \cdot y, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020078 +o rules:numerics
(FPCore (x y z t)
  :name "Numeric.Histogram:binBounds from Chart-1.5.3"
  :precision binary64

  :herbie-target
  (if (< x -9.025511195533005e-135) (- x (* (/ z t) (- x y))) (if (< x 4.275032163700715e-250) (+ x (* (/ (- y x) t) z)) (+ x (/ (- y x) (/ t z)))))

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