Average Error: 24.3 → 6.5
Time: 10.8s
Precision: binary64
Cost: 7874
\[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \leq -1.7958128377052614 \cdot 10^{+121}:\\ \;\;\;\;-x \cdot y\\ \mathbf{elif}\;z \leq 3.422582606751965 \cdot 10^{+100}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]
\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}
↓
\begin{array}{l}
\mathbf{if}\;z \leq -1.7958128377052614 \cdot 10^{+121}:\\
\;\;\;\;-x \cdot y\\

\mathbf{elif}\;z \leq 3.422582606751965 \cdot 10^{+100}:\\
\;\;\;\;\left(x \cdot y\right) \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\\

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

\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 -1.7958128377052614e+121)
   (- (* x y))
   (if (<= z 3.422582606751965e+100)
     (* (* x y) (/ z (sqrt (- (* z z) (* t a)))))
     (* x y))))
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 <= -1.7958128377052614e+121) {
		tmp = -(x * y);
	} else if (z <= 3.422582606751965e+100) {
		tmp = (x * y) * (z / sqrt((z * z) - (t * a)));
	} else {
		tmp = x * y;
	}
	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

Original24.3
Target7.7
Herbie6.5
\[\begin{array}{l} \mathbf{if}\;z < -3.1921305903852764 \cdot 10^{+46}:\\ \;\;\;\;-y \cdot x\\ \mathbf{elif}\;z < 5.976268120920894 \cdot 10^{+90}:\\ \;\;\;\;\frac{x \cdot z}{\frac{\sqrt{z \cdot z - a \cdot t}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array}\]

Alternatives

Alternative 1
Error12.5
Cost7682
\[\begin{array}{l} \mathbf{if}\;z \leq -6.787769142766883 \cdot 10^{-92}:\\ \;\;\;\;-x \cdot y\\ \mathbf{elif}\;z \leq 1.2568411837978658 \cdot 10^{-54}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{z}{\sqrt{-t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]
Alternative 2
Error16.9
Cost1281
\[\begin{array}{l} \mathbf{if}\;z \leq -2.602311216522087 \cdot 10^{-150}:\\ \;\;\;\;-x \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{z - 0.5 \cdot \frac{t \cdot a}{z}}\right)\\ \end{array}\]
Alternative 3
Error17.5
Cost834
\[\begin{array}{l} \mathbf{if}\;z \leq -4.369096939137676 \cdot 10^{-149}:\\ \;\;\;\;-x \cdot y\\ \mathbf{elif}\;z \leq 9.163533629704878 \cdot 10^{-172}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]
Alternative 4
Error33.0
Cost513
\[\begin{array}{l} \mathbf{if}\;z \leq 1.4585272085326925 \cdot 10^{-170}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]
Alternative 5
Error49.0
Cost64
\[0\]
Alternative 6
Error61.7
Cost64
\[1\]

Error

Derivation

  1. Split input into 3 regimes

  2. if z < -1.7958128377052614e121

    1. Initial program 47.1

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

    2. Taylor expanded around -inf 2.2

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

    3. Simplified2.2

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

    4. Simplified2.2

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

    if -1.7958128377052614e121 < z < 3.4225826067519651e100

    1. Initial program 10.5

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

    3. Applied *-un-lft-identity_binary64_655710.5

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

    4. Applied sqrt-prod_binary64_657310.5

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

    5. Applied times-frac_binary64_65639.1

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

    6. Simplified9.1

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

    7. Simplified9.1

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

    if 3.4225826067519651e100 < z

    1. Initial program 43.8

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

    2. Taylor expanded around inf 2.8

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

    3. Simplified2.8

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

  4. Final simplification6.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.7958128377052614 \cdot 10^{+121}:\\ \;\;\;\;-x \cdot y\\ \mathbf{elif}\;z \leq 3.422582606751965 \cdot 10^{+100}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]

Reproduce

herbie shell --seed 2021044 
(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)))))