Average Error: 6.7 → 2.2
Time: 2.2s
Precision: 64
\[x + \frac{\left(y - x\right) \cdot z}{t}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -5.43286468955906009 \cdot 10^{-191} \lor \neg \left(x \le 8.67583317488940126 \cdot 10^{-93}\right):\\ \;\;\;\;\left(y - x\right) \cdot \frac{z}{t} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(y - x\right) \cdot \frac{1}{t}, z, x\right)\\ \end{array}\]
x + \frac{\left(y - x\right) \cdot z}{t}
↓
\begin{array}{l}
\mathbf{if}\;x \le -5.43286468955906009 \cdot 10^{-191} \lor \neg \left(x \le 8.67583317488940126 \cdot 10^{-93}\right):\\
\;\;\;\;\left(y - x\right) \cdot \frac{z}{t} + x\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\left(y - x\right) \cdot \frac{1}{t}, z, 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 (((x <= -5.43286468955906e-191) || !(x <= 8.675833174889401e-93))) {
		VAR = (((y - x) * (z / t)) + x);
	} else {
		VAR = fma(((y - x) * (1.0 / t)), z, 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

Original6.7
Target2.0
Herbie2.2
\[\begin{array}{l} \mathbf{if}\;x \lt -9.0255111955330046 \cdot 10^{-135}:\\ \;\;\;\;x - \frac{z}{t} \cdot \left(x - y\right)\\ \mathbf{elif}\;x \lt 4.2750321637007147 \cdot 10^{-250}:\\ \;\;\;\;x + \frac{y - x}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -5.43286468955906e-191 or 8.675833174889401e-93 < x

    1. Initial program 7.2

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

    2. Simplified6.8

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

    4. Applied fma-udef6.8

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

    6. Applied div-inv6.9

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

    7. Applied associate-*l*0.9

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

    8. Simplified0.9

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

    if -5.43286468955906e-191 < x < 8.675833174889401e-93

    1. Initial program 5.4

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

    2. Simplified5.2

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

    4. Applied div-inv5.3

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

  4. Final simplification2.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -5.43286468955906009 \cdot 10^{-191} \lor \neg \left(x \le 8.67583317488940126 \cdot 10^{-93}\right):\\ \;\;\;\;\left(y - x\right) \cdot \frac{z}{t} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(y - x\right) \cdot \frac{1}{t}, z, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020103 +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)))