Average Error: 25.1 → 6.5
Time: 13.5s
Precision: binary64
\[[x, y] = \mathsf{sort}([x, y]) \\]
\[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
↓
\[\begin{array}{l} \mathbf{if}\;z \leq -6.796797640097235 \cdot 10^{+161}:\\ \;\;\;\;-y \cdot x\\ \mathbf{elif}\;z \leq 1.870606781150024 \cdot 10^{+79}:\\ \;\;\;\;x \cdot \frac{y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}
↓
\begin{array}{l}
\mathbf{if}\;z \leq -6.796797640097235 \cdot 10^{+161}:\\
\;\;\;\;-y \cdot x\\

\mathbf{elif}\;z \leq 1.870606781150024 \cdot 10^{+79}:\\
\;\;\;\;x \cdot \frac{y}{\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 -6.796797640097235e+161)
   (- (* y x))
   (if (<= z 1.870606781150024e+79)
     (* x (/ y (/ (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 <= -6.796797640097235e+161) {
		tmp = -(y * x);
	} else if (z <= 1.870606781150024e+79) {
		tmp = x * (y / (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

Original25.1
Target7.3
Herbie6.5
\[\begin{array}{l} \mathbf{if}\;z < -3.1921305903852764 \cdot 10^{+46}:\\ \;\;\;\;-y \cdot x\\ \mathbf{elif}\;z < 5.976268120920894 \cdot 10^{+90}:\\ \;\;\;\;\frac{x \cdot z}{\frac{\sqrt{z \cdot z - a \cdot t}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

  2. if z < -6.7967976400972349e161

    1. Initial program 55.5

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

    2. Taylor expanded in z around -inf 1.0

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

    3. Simplified1.0

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

    if -6.7967976400972349e161 < z < 1.87060678115002416e79

    1. Initial program 11.1

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

    2. Applied associate-/l*_binary649.6

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

    3. Applied *-un-lft-identity_binary649.6

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

    4. Applied times-frac_binary649.4

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

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

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

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

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

    7. Applied sqrt-prod_binary649.4

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

    8. Applied times-frac_binary649.4

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

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

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

    10. Applied times-frac_binary649.4

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

    11. Applied associate-*r*_binary649.4

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

    12. Simplified9.4

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

    if 1.87060678115002416e79 < z

    1. Initial program 41.4

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

    2. Taylor expanded in z around inf 2.6

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

  4. Final simplification6.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.796797640097235 \cdot 10^{+161}:\\ \;\;\;\;-y \cdot x\\ \mathbf{elif}\;z \leq 1.870606781150024 \cdot 10^{+79}:\\ \;\;\;\;x \cdot \frac{y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Reproduce

herbie shell --seed 2022068 
(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)))))