Average Error: 6.1 → 3.1
Time: 11.3s
Precision: binary64
\[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 1.1176245341763669 \cdot 10^{+297}:\\ \;\;\;\;x \cdot x - \left(4 \cdot \left({z}^{2} \cdot y\right) - 4 \cdot \left(y \cdot t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - \left(\left(4 \cdot y\right) \cdot \left(z + \sqrt{t}\right)\right) \cdot \left(z - \sqrt{t}\right)\\ \end{array}\]
x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)
↓
\begin{array}{l}
\mathbf{if}\;z \cdot z \leq 1.1176245341763669 \cdot 10^{+297}:\\
\;\;\;\;x \cdot x - \left(4 \cdot \left({z}^{2} \cdot y\right) - 4 \cdot \left(y \cdot t\right)\right)\\

\mathbf{else}:\\
\;\;\;\;x \cdot x - \left(\left(4 \cdot y\right) \cdot \left(z + \sqrt{t}\right)\right) \cdot \left(z - \sqrt{t}\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) 1.1176245341763669e+297)
   (- (* x x) (- (* 4.0 (* (pow z 2.0) y)) (* 4.0 (* y t))))
   (- (* x x) (* (* (* 4.0 y) (+ z (sqrt t))) (- z (sqrt t))))))
double code(double x, double y, double z, double t) {
	return (x * x) - ((y * 4.0) * ((z * z) - t));
}
↓
double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 1.1176245341763669e+297) {
		tmp = (x * x) - ((4.0 * (pow(z, 2.0) * y)) - (4.0 * (y * t)));
	} else {
		tmp = (x * x) - (((4.0 * y) * (z + sqrt(t))) * (z - sqrt(t)));
	}
	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

Original6.1
Target6.1
Herbie3.1
\[x \cdot x - 4 \cdot \left(y \cdot \left(z \cdot z - t\right)\right)\]

Derivation

  1. Split input into 2 regimes

  2. if (*.f64 z z) < 1.11762453417636693e297

    1. Initial program 0.1

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

    2. Taylor expanded around 0 0.1

      \[\leadsto x \cdot x - \color{blue}{\left(4 \cdot \left({z}^{2} \cdot y\right) - 4 \cdot \left(t \cdot y\right)\right)}\]

    if 1.11762453417636693e297 < (*.f64 z z)

    1. Initial program 59.3

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

    3. Applied add-sqr-sqrt_binary64_1510461.3

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

    4. Applied difference-of-squares_binary64_1505161.3

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

    5. Applied associate-*r*_binary64_1502230.2

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

  4. Final simplification3.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 1.1176245341763669 \cdot 10^{+297}:\\ \;\;\;\;x \cdot x - \left(4 \cdot \left({z}^{2} \cdot y\right) - 4 \cdot \left(y \cdot t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - \left(\left(4 \cdot y\right) \cdot \left(z + \sqrt{t}\right)\right) \cdot \left(z - \sqrt{t}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021044 
(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))))