Average Error: 24.7 → 6.4
Time: 8.8s
Precision: binary64
\[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
↓
\[\begin{array}{l} \mathbf{if}\;z \leq -7.914973368704496 \cdot 10^{+71}:\\ \;\;\;\;-y \cdot x\\ \mathbf{elif}\;z \leq 4.1905476995839096 \cdot 10^{+145}:\\ \;\;\;\;\frac{y \cdot x}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}
↓
\begin{array}{l}
\mathbf{if}\;z \leq -7.914973368704496 \cdot 10^{+71}:\\
\;\;\;\;-y \cdot x\\

\mathbf{elif}\;z \leq 4.1905476995839096 \cdot 10^{+145}:\\
\;\;\;\;\frac{y \cdot x}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}\\

\mathbf{else}:\\
\;\;\;\;y \cdot x\\


\end{array}

(FPCore (x y z t a)
 :precision binary64
 (/ (* (* x y) z) (sqrt (- (* z z) (* t a)))))
↓
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -7.914973368704496e+71)
   (- (* y x))
   (if (<= z 4.1905476995839096e+145)
     (/ (* y x) (/ (sqrt (- (* z z) (* t a))) z))
     (* y x))))
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 tmp;
	if (z <= -7.914973368704496e+71) {
		tmp = -(y * x);
	} else if (z <= 4.1905476995839096e+145) {
		tmp = (y * x) / (sqrt((z * z) - (t * a)) / z);
	} else {
		tmp = y * x;
	}
	return tmp;
}

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

Original24.7
Target7.8
Herbie6.4
\[\begin{array}{l} \mathbf{if}\;z < -3.1921305903852764 \cdot 10^{+46}:\\ \;\;\;\;-y \cdot x\\ \mathbf{elif}\;z < 5.976268120920894 \cdot 10^{+90}:\\ \;\;\;\;\frac{x \cdot z}{\frac{\sqrt{z \cdot z - a \cdot t}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

  2. if z < -7.914973368704496e71

    1. Initial program 40.7

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

    2. Taylor expanded in z around -inf 3.2

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

    3. Simplified3.2

      \[\leadsto \color{blue}{-y \cdot x} \]

    if -7.914973368704496e71 < z < 4.1905476995839096e145

    1. Initial program 11.0

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

    2. Applied associate-/l*_binary649.2

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

    3. Applied *-un-lft-identity_binary649.2

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

    4. Applied *-un-lft-identity_binary649.2

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

    5. Applied sqrt-prod_binary649.2

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

    6. Applied times-frac_binary649.2

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

    7. Applied associate-/r*_binary649.2

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

    8. Simplified9.2

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

    if 4.1905476995839096e145 < z

    1. Initial program 51.3

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

    2. Taylor expanded in z around inf 1.2

      \[\leadsto \color{blue}{y \cdot x} \]
  3. Recombined 3 regimes into one program.

  4. Final simplification6.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.914973368704496 \cdot 10^{+71}:\\ \;\;\;\;-y \cdot x\\ \mathbf{elif}\;z \leq 4.1905476995839096 \cdot 10^{+145}:\\ \;\;\;\;\frac{y \cdot x}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Reproduce

herbie shell --seed 2021313 
(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)))))