Average Error: 6.0 → 3.2
Time: 4.4s
Precision: binary64
\[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 3.956437111800793 \cdot 10^{+303}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + \left(y \cdot \left(4 \cdot \left(z + \sqrt{t}\right)\right)\right) \cdot \left(\sqrt{t} - z\right)\\ \end{array}\]
x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)
↓
\begin{array}{l}
\mathbf{if}\;z \cdot z \leq 3.956437111800793 \cdot 10^{+303}:\\
\;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\

\mathbf{else}:\\
\;\;\;\;x \cdot x + \left(y \cdot \left(4 \cdot \left(z + \sqrt{t}\right)\right)\right) \cdot \left(\sqrt{t} - z\right)\\

\end{array}
(FPCore (x y z t) :precision binary64 (- (* x x) (* (* y 4.0) (- (* z z) t))))
↓
(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 3.956437111800793e+303)
   (+ (* x x) (* (* y 4.0) (- t (* z z))))
   (+ (* x x) (* (* y (* 4.0 (+ z (sqrt t)))) (- (sqrt t) z)))))
double code(double x, double y, double z, double t) {
	return ((double) (((double) (x * x)) - ((double) (((double) (y * 4.0)) * ((double) (((double) (z * z)) - t))))));
}
↓
double code(double x, double y, double z, double t) {
	double tmp;
	if ((((double) (z * z)) <= 3.956437111800793e+303)) {
		tmp = ((double) (((double) (x * x)) + ((double) (((double) (y * 4.0)) * ((double) (t - ((double) (z * z))))))));
	} else {
		tmp = ((double) (((double) (x * x)) + ((double) (((double) (y * ((double) (4.0 * ((double) (z + ((double) sqrt(t)))))))) * ((double) (((double) sqrt(t)) - z))))));
	}
	return tmp;
}

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

Original6.0
Target6.0
Herbie3.2
\[x \cdot x - 4 \cdot \left(y \cdot \left(z \cdot z - t\right)\right)\]

Derivation

  1. Split input into 2 regimes

  2. if (* z z) < 3.95643711180079285e303

    1. Initial program Error: 0.1 bits

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\]

    if 3.95643711180079285e303 < (* z z)

    1. Initial program Error: 62.2 bits

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\]
    2. Using strategy rm

    3. Applied add-sqr-sqrtError: 63.1 bits

      \[\leadsto x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - \color{blue}{\sqrt{t} \cdot \sqrt{t}}\right)\]

    4. Applied difference-of-squaresError: 63.1 bits

      \[\leadsto x \cdot x - \left(y \cdot 4\right) \cdot \color{blue}{\left(\left(z + \sqrt{t}\right) \cdot \left(z - \sqrt{t}\right)\right)}\]

    5. Applied associate-*r*Error: 32.8 bits

      \[\leadsto x \cdot x - \color{blue}{\left(\left(y \cdot 4\right) \cdot \left(z + \sqrt{t}\right)\right) \cdot \left(z - \sqrt{t}\right)}\]

    6. SimplifiedError: 32.8 bits

      \[\leadsto x \cdot x - \color{blue}{\left(y \cdot \left(4 \cdot \left(z + \sqrt{t}\right)\right)\right)} \cdot \left(z - \sqrt{t}\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplificationError: 3.2 bits

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 3.956437111800793 \cdot 10^{+303}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + \left(y \cdot \left(4 \cdot \left(z + \sqrt{t}\right)\right)\right) \cdot \left(\sqrt{t} - z\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020204 
(FPCore (x y z t)
  :name "Graphics.Rasterific.Shading:$sradialGradientWithFocusShader from Rasterific-0.6.1, B"
  :precision binary64

  :herbie-target
  (- (* x x) (* 4.0 (* y (- (* z z) t))))

  (- (* x x) (* (* y 4.0) (- (* z z) t))))