Average Error: 25.6 → 6.1
Time: 5.5s
Precision: binary64
\[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \leq -2.2541827298019483 \cdot 10^{+147}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{0.5 \cdot \left(t \cdot \frac{a}{z}\right) - z}\right)\\ \mathbf{elif}\;z \leq 4.3109041619914876 \cdot 10^{+103}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{z + \left(t \cdot \frac{a}{z}\right) \cdot -0.5}\right)\\ \end{array}\]
\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}
↓
\begin{array}{l}
\mathbf{if}\;z \leq -2.2541827298019483 \cdot 10^{+147}:\\
\;\;\;\;x \cdot \left(y \cdot \frac{z}{0.5 \cdot \left(t \cdot \frac{a}{z}\right) - z}\right)\\

\mathbf{elif}\;z \leq 4.3109041619914876 \cdot 10^{+103}:\\
\;\;\;\;y \cdot \left(x \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\right)\\

\mathbf{else}:\\
\;\;\;\;x \cdot \left(y \cdot \frac{z}{z + \left(t \cdot \frac{a}{z}\right) \cdot -0.5}\right)\\

\end{array}
double code(double x, double y, double z, double t, double a) {
	return (((double) (((double) (x * y)) * z)) / ((double) sqrt(((double) (((double) (z * z)) - ((double) (t * a)))))));
}
↓
double code(double x, double y, double z, double t, double a) {
	double VAR;
	if ((z <= -2.2541827298019483e+147)) {
		VAR = ((double) (x * ((double) (y * (z / ((double) (((double) (0.5 * ((double) (t * (a / z))))) - z)))))));
	} else {
		double VAR_1;
		if ((z <= 4.3109041619914876e+103)) {
			VAR_1 = ((double) (y * ((double) (x * (z / ((double) sqrt(((double) (((double) (z * z)) - ((double) (t * a)))))))))));
		} else {
			VAR_1 = ((double) (x * ((double) (y * (z / ((double) (z + ((double) (((double) (t * (a / z))) * -0.5)))))))));
		}
		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

Original25.6
Target7.8
Herbie6.1
\[\begin{array}{l} \mathbf{if}\;z < -3.1921305903852764 \cdot 10^{+46}:\\ \;\;\;\;-y \cdot x\\ \mathbf{elif}\;z < 5.976268120920894 \cdot 10^{+90}:\\ \;\;\;\;\frac{x \cdot z}{\frac{\sqrt{z \cdot z - a \cdot t}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -2.2541827298019483e147

    1. Initial program 53.3

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

    2. Simplified52.4

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

    3. Taylor expanded around -inf 5.8

      \[\leadsto x \cdot \left(y \cdot \frac{z}{\color{blue}{0.5 \cdot \frac{a \cdot t}{z} - z}}\right)\]

    4. Simplified1.2

      \[\leadsto x \cdot \left(y \cdot \frac{z}{\color{blue}{0.5 \cdot \left(t \cdot \frac{a}{z}\right) - z}}\right)\]

    if -2.2541827298019483e147 < z < 4.31090416199148756e103

    1. Initial program 11.7

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

    2. Simplified9.1

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

    4. Applied pow19.1

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

    5. Applied pow19.1

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

    6. Applied pow-prod-down9.1

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

    7. Applied pow19.1

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

    8. Applied pow-prod-down9.1

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

    9. Simplified8.9

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

    if 4.31090416199148756e103 < z

    1. Initial program 44.8

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

    2. Simplified42.3

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

    3. Taylor expanded around inf 5.6

      \[\leadsto x \cdot \left(y \cdot \frac{z}{\color{blue}{z - 0.5 \cdot \frac{a \cdot t}{z}}}\right)\]

    4. Simplified1.6

      \[\leadsto x \cdot \left(y \cdot \frac{z}{\color{blue}{z + \left(t \cdot \frac{a}{z}\right) \cdot -0.5}}\right)\]
  3. Recombined 3 regimes into one program.

  4. Final simplification6.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.2541827298019483 \cdot 10^{+147}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{0.5 \cdot \left(t \cdot \frac{a}{z}\right) - z}\right)\\ \mathbf{elif}\;z \leq 4.3109041619914876 \cdot 10^{+103}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{z + \left(t \cdot \frac{a}{z}\right) \cdot -0.5}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020196 
(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)))))