Average Error: 24.2 → 6.7
Time: 4.2s
Precision: 64
\[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -2.46601267578198948 \cdot 10^{117}:\\ \;\;\;\;x \cdot \left(-1 \cdot y\right)\\ \mathbf{elif}\;z \le 1.153415909331297 \cdot 10^{100}:\\ \;\;\;\;\frac{x \cdot y}{\sqrt{\sqrt{z \cdot z - t \cdot a}}} \cdot \frac{z}{\sqrt{\sqrt{z \cdot z - t \cdot a}}}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot 1\\ \end{array}\]
\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}
↓
\begin{array}{l}
\mathbf{if}\;z \le -2.46601267578198948 \cdot 10^{117}:\\
\;\;\;\;x \cdot \left(-1 \cdot y\right)\\

\mathbf{elif}\;z \le 1.153415909331297 \cdot 10^{100}:\\
\;\;\;\;\frac{x \cdot y}{\sqrt{\sqrt{z \cdot z - t \cdot a}}} \cdot \frac{z}{\sqrt{\sqrt{z \cdot z - t \cdot a}}}\\

\mathbf{else}:\\
\;\;\;\;\left(x \cdot y\right) \cdot 1\\

\end{array}
double code(double x, double y, double z, double t, double a) {
	return (((x * y) * z) / sqrt(((z * z) - (t * a))));
}
↓
double code(double x, double y, double z, double t, double a) {
	double VAR;
	if ((z <= -2.4660126757819895e+117)) {
		VAR = (x * (-1.0 * y));
	} else {
		double VAR_1;
		if ((z <= 1.1534159093312974e+100)) {
			VAR_1 = (((x * y) / sqrt(sqrt(((z * z) - (t * a))))) * (z / sqrt(sqrt(((z * z) - (t * a))))));
		} else {
			VAR_1 = ((x * y) * 1.0);
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original24.2
Target7.5
Herbie6.7
\[\begin{array}{l} \mathbf{if}\;z \lt -3.1921305903852764 \cdot 10^{46}:\\ \;\;\;\;-y \cdot x\\ \mathbf{elif}\;z \lt 5.9762681209208942 \cdot 10^{90}:\\ \;\;\;\;\frac{x \cdot z}{\frac{\sqrt{z \cdot z - a \cdot t}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -2.4660126757819895e+117

    1. Initial program 46.3

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

    3. Applied *-un-lft-identity46.3

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

    4. Applied sqrt-prod46.3

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

    5. Applied times-frac44.3

      \[\leadsto \color{blue}{\frac{x \cdot y}{\sqrt{1}} \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}}\]

    6. Simplified44.3

      \[\leadsto \color{blue}{\left(x \cdot y\right)} \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\]
    7. Using strategy rm

    8. Applied associate-*l*44.3

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

    9. Taylor expanded around -inf 2.2

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

    if -2.4660126757819895e+117 < z < 1.1534159093312974e+100

    1. Initial program 10.7

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

    3. Applied add-sqr-sqrt10.7

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

    4. Applied sqrt-prod10.9

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

    5. Applied times-frac9.7

      \[\leadsto \color{blue}{\frac{x \cdot y}{\sqrt{\sqrt{z \cdot z - t \cdot a}}} \cdot \frac{z}{\sqrt{\sqrt{z \cdot z - t \cdot a}}}}\]

    if 1.1534159093312974e+100 < z

    1. Initial program 42.5

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

    3. Applied *-un-lft-identity42.5

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

    4. Applied sqrt-prod42.5

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

    5. Applied times-frac40.6

      \[\leadsto \color{blue}{\frac{x \cdot y}{\sqrt{1}} \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}}\]

    6. Simplified40.6

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

    7. Taylor expanded around inf 2.0

      \[\leadsto \left(x \cdot y\right) \cdot \color{blue}{1}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification6.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -2.46601267578198948 \cdot 10^{117}:\\ \;\;\;\;x \cdot \left(-1 \cdot y\right)\\ \mathbf{elif}\;z \le 1.153415909331297 \cdot 10^{100}:\\ \;\;\;\;\frac{x \cdot y}{\sqrt{\sqrt{z \cdot z - t \cdot a}}} \cdot \frac{z}{\sqrt{\sqrt{z \cdot z - t \cdot a}}}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot 1\\ \end{array}\]

Reproduce

herbie shell --seed 2020078 +o rules:numerics
(FPCore (x y z t a)
  :name "Statistics.Math.RootFinding:ridders from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (if (< z -3.1921305903852764e+46) (- (* y x)) (if (< z 5.976268120920894e+90) (/ (* x z) (/ (sqrt (- (* z z) (* a t))) y)) (* y x)))

  (/ (* (* x y) z) (sqrt (- (* z z) (* t a)))))