Average Error: 27.0 → 22.1
Time: 7.1s
Precision: binary64
\[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \leq -6.056383831737611 \cdot 10^{+74}:\\ \;\;\;\;a - \frac{y}{\sqrt[3]{x + \left(y + t\right)} \cdot \sqrt[3]{x + \left(y + t\right)}} \cdot \frac{b}{\sqrt[3]{x + \left(y + t\right)}}\\ \mathbf{elif}\;a \leq -1.5917280185250292 \cdot 10^{-152}:\\ \;\;\;\;\left(\left(\left(y + x\right) \cdot z + a \cdot \left(y + t\right)\right) - y \cdot b\right) \cdot \frac{1}{x + \left(y + t\right)}\\ \mathbf{elif}\;a \leq 1.0121751081902255 \cdot 10^{-217}:\\ \;\;\;\;z - \frac{y}{\sqrt[3]{x + \left(y + t\right)} \cdot \sqrt[3]{x + \left(y + t\right)}} \cdot \frac{b}{\sqrt[3]{x + \left(y + t\right)}}\\ \mathbf{elif}\;a \leq 1.826313548779271 \cdot 10^{+62}:\\ \;\;\;\;\left(\left(\left(y + x\right) \cdot z + a \cdot \left(y + t\right)\right) - y \cdot b\right) \cdot \frac{1}{x + \left(y + t\right)}\\ \mathbf{else}:\\ \;\;\;\;a - \frac{y}{\sqrt[3]{x + \left(y + t\right)} \cdot \sqrt[3]{x + \left(y + t\right)}} \cdot \frac{b}{\sqrt[3]{x + \left(y + t\right)}}\\ \end{array}\]
\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}
↓
\begin{array}{l}
\mathbf{if}\;a \leq -6.056383831737611 \cdot 10^{+74}:\\
\;\;\;\;a - \frac{y}{\sqrt[3]{x + \left(y + t\right)} \cdot \sqrt[3]{x + \left(y + t\right)}} \cdot \frac{b}{\sqrt[3]{x + \left(y + t\right)}}\\

\mathbf{elif}\;a \leq -1.5917280185250292 \cdot 10^{-152}:\\
\;\;\;\;\left(\left(\left(y + x\right) \cdot z + a \cdot \left(y + t\right)\right) - y \cdot b\right) \cdot \frac{1}{x + \left(y + t\right)}\\

\mathbf{elif}\;a \leq 1.0121751081902255 \cdot 10^{-217}:\\
\;\;\;\;z - \frac{y}{\sqrt[3]{x + \left(y + t\right)} \cdot \sqrt[3]{x + \left(y + t\right)}} \cdot \frac{b}{\sqrt[3]{x + \left(y + t\right)}}\\

\mathbf{elif}\;a \leq 1.826313548779271 \cdot 10^{+62}:\\
\;\;\;\;\left(\left(\left(y + x\right) \cdot z + a \cdot \left(y + t\right)\right) - y \cdot b\right) \cdot \frac{1}{x + \left(y + t\right)}\\

\mathbf{else}:\\
\;\;\;\;a - \frac{y}{\sqrt[3]{x + \left(y + t\right)} \cdot \sqrt[3]{x + \left(y + t\right)}} \cdot \frac{b}{\sqrt[3]{x + \left(y + t\right)}}\\

\end{array}
(FPCore (x y z t a b)
 :precision binary64
 (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)))
↓
(FPCore (x y z t a b)
 :precision binary64
 (if (<= a -6.056383831737611e+74)
   (-
    a
    (*
     (/ y (* (cbrt (+ x (+ y t))) (cbrt (+ x (+ y t)))))
     (/ b (cbrt (+ x (+ y t))))))
   (if (<= a -1.5917280185250292e-152)
     (* (- (+ (* (+ y x) z) (* a (+ y t))) (* y b)) (/ 1.0 (+ x (+ y t))))
     (if (<= a 1.0121751081902255e-217)
       (-
        z
        (*
         (/ y (* (cbrt (+ x (+ y t))) (cbrt (+ x (+ y t)))))
         (/ b (cbrt (+ x (+ y t))))))
       (if (<= a 1.826313548779271e+62)
         (* (- (+ (* (+ y x) z) (* a (+ y t))) (* y b)) (/ 1.0 (+ x (+ y t))))
         (-
          a
          (*
           (/ y (* (cbrt (+ x (+ y t))) (cbrt (+ x (+ y t)))))
           (/ b (cbrt (+ x (+ y t)))))))))))
double code(double x, double y, double z, double t, double a, double b) {
	return (((double) (((double) (((double) (((double) (x + y)) * z)) + ((double) (((double) (t + y)) * a)))) - ((double) (y * b)))) / ((double) (((double) (x + t)) + y)));
}
↓
double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a <= -6.056383831737611e+74)) {
		tmp = ((double) (a - ((double) ((y / ((double) (((double) cbrt(((double) (x + ((double) (y + t)))))) * ((double) cbrt(((double) (x + ((double) (y + t))))))))) * (b / ((double) cbrt(((double) (x + ((double) (y + t)))))))))));
	} else {
		double tmp_1;
		if ((a <= -1.5917280185250292e-152)) {
			tmp_1 = ((double) (((double) (((double) (((double) (((double) (y + x)) * z)) + ((double) (a * ((double) (y + t)))))) - ((double) (y * b)))) * (1.0 / ((double) (x + ((double) (y + t)))))));
		} else {
			double tmp_2;
			if ((a <= 1.0121751081902255e-217)) {
				tmp_2 = ((double) (z - ((double) ((y / ((double) (((double) cbrt(((double) (x + ((double) (y + t)))))) * ((double) cbrt(((double) (x + ((double) (y + t))))))))) * (b / ((double) cbrt(((double) (x + ((double) (y + t)))))))))));
			} else {
				double tmp_3;
				if ((a <= 1.826313548779271e+62)) {
					tmp_3 = ((double) (((double) (((double) (((double) (((double) (y + x)) * z)) + ((double) (a * ((double) (y + t)))))) - ((double) (y * b)))) * (1.0 / ((double) (x + ((double) (y + t)))))));
				} else {
					tmp_3 = ((double) (a - ((double) ((y / ((double) (((double) cbrt(((double) (x + ((double) (y + t)))))) * ((double) cbrt(((double) (x + ((double) (y + t))))))))) * (b / ((double) cbrt(((double) (x + ((double) (y + t)))))))))));
				}
				tmp_2 = tmp_3;
			}
			tmp_1 = tmp_2;
		}
		tmp = tmp_1;
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original27.0
Target11.7
Herbie22.1
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} < -3.5813117084150564 \cdot 10^{+153}:\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{elif}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} < 1.2285964308315609 \cdot 10^{+82}:\\ \;\;\;\;\frac{1}{\frac{\left(x + t\right) + y}{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\left(z + a\right) - b\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if a < -6.05638383173761113e74 or 1.826313548779271e62 < a

    1. Initial program 38.0

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

    3. Applied div-sub_binary6438.0

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

    4. Simplified38.0

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

    5. Simplified38.0

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

    7. Applied add-cube-cbrt_binary6438.1

      \[\leadsto \frac{\left(x + y\right) \cdot z + \left(y + t\right) \cdot a}{x + \left(y + t\right)} - \frac{y \cdot b}{\color{blue}{\left(\sqrt[3]{x + \left(y + t\right)} \cdot \sqrt[3]{x + \left(y + t\right)}\right) \cdot \sqrt[3]{x + \left(y + t\right)}}}\]

    8. Applied times-frac_binary6437.3

      \[\leadsto \frac{\left(x + y\right) \cdot z + \left(y + t\right) \cdot a}{x + \left(y + t\right)} - \color{blue}{\frac{y}{\sqrt[3]{x + \left(y + t\right)} \cdot \sqrt[3]{x + \left(y + t\right)}} \cdot \frac{b}{\sqrt[3]{x + \left(y + t\right)}}}\]

    9. Taylor expanded around 0 25.8

      \[\leadsto \color{blue}{a} - \frac{y}{\sqrt[3]{x + \left(y + t\right)} \cdot \sqrt[3]{x + \left(y + t\right)}} \cdot \frac{b}{\sqrt[3]{x + \left(y + t\right)}}\]

    if -6.05638383173761113e74 < a < -1.59172801852502918e-152 or 1.0121751081902255e-217 < a < 1.826313548779271e62

    1. Initial program 20.4

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

    3. Applied div-inv_binary6420.5

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

    4. Simplified20.5

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

    if -1.59172801852502918e-152 < a < 1.0121751081902255e-217

    1. Initial program 19.2

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

    3. Applied div-sub_binary6419.2

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

    4. Simplified19.2

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

    5. Simplified19.2

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

    7. Applied add-cube-cbrt_binary6419.4

      \[\leadsto \frac{\left(x + y\right) \cdot z + \left(y + t\right) \cdot a}{x + \left(y + t\right)} - \frac{y \cdot b}{\color{blue}{\left(\sqrt[3]{x + \left(y + t\right)} \cdot \sqrt[3]{x + \left(y + t\right)}\right) \cdot \sqrt[3]{x + \left(y + t\right)}}}\]

    8. Applied times-frac_binary6416.2

      \[\leadsto \frac{\left(x + y\right) \cdot z + \left(y + t\right) \cdot a}{x + \left(y + t\right)} - \color{blue}{\frac{y}{\sqrt[3]{x + \left(y + t\right)} \cdot \sqrt[3]{x + \left(y + t\right)}} \cdot \frac{b}{\sqrt[3]{x + \left(y + t\right)}}}\]

    9. Taylor expanded around inf 18.2

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

  4. Final simplification22.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6.056383831737611 \cdot 10^{+74}:\\ \;\;\;\;a - \frac{y}{\sqrt[3]{x + \left(y + t\right)} \cdot \sqrt[3]{x + \left(y + t\right)}} \cdot \frac{b}{\sqrt[3]{x + \left(y + t\right)}}\\ \mathbf{elif}\;a \leq -1.5917280185250292 \cdot 10^{-152}:\\ \;\;\;\;\left(\left(\left(y + x\right) \cdot z + a \cdot \left(y + t\right)\right) - y \cdot b\right) \cdot \frac{1}{x + \left(y + t\right)}\\ \mathbf{elif}\;a \leq 1.0121751081902255 \cdot 10^{-217}:\\ \;\;\;\;z - \frac{y}{\sqrt[3]{x + \left(y + t\right)} \cdot \sqrt[3]{x + \left(y + t\right)}} \cdot \frac{b}{\sqrt[3]{x + \left(y + t\right)}}\\ \mathbf{elif}\;a \leq 1.826313548779271 \cdot 10^{+62}:\\ \;\;\;\;\left(\left(\left(y + x\right) \cdot z + a \cdot \left(y + t\right)\right) - y \cdot b\right) \cdot \frac{1}{x + \left(y + t\right)}\\ \mathbf{else}:\\ \;\;\;\;a - \frac{y}{\sqrt[3]{x + \left(y + t\right)} \cdot \sqrt[3]{x + \left(y + t\right)}} \cdot \frac{b}{\sqrt[3]{x + \left(y + t\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020210 
(FPCore (x y z t a b)
  :name "AI.Clustering.Hierarchical.Internal:ward from clustering-0.2.1"
  :precision binary64

  :herbie-target
  (if (< (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)) -3.5813117084150564e+153) (- (+ z a) b) (if (< (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)) 1.2285964308315609e+82) (/ 1.0 (/ (+ (+ x t) y) (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)))) (- (+ z a) b)))

  (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)))