Diagrams.Solve.Polynomial:cubForm from diagrams-solve-0.1, J

Average Error: 20.7 → 8.9

Time: 5.6s

Precision: 64

Math

\[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\]

↓

\[\begin{array}{l} \mathbf{if}\;z \le -4.4627057489340555 \cdot 10^{-108}:\\ \;\;\;\;\left(\frac{\sqrt[3]{b} \cdot \sqrt[3]{b}}{z} \cdot \frac{\sqrt[3]{b}}{c} + 9 \cdot \frac{1}{\frac{\frac{z \cdot c}{y}}{x}}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \mathbf{elif}\;z \le 2.29026724389609061 \cdot 10^{-68}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c}\\ \mathbf{elif}\;z \le 2.90195016732761552 \cdot 10^{126}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{1}{\frac{\frac{z \cdot c}{y}}{x}}\right) - 4 \cdot \left(a \cdot \frac{t}{c}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{1}{\frac{z \cdot \frac{c}{y}}{x}}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \end{array}\]

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	return (((((x * 9.0) * y) - (((z * 4.0) * t) * a)) + b) / (z * c));
}

↓

double code(double x, double y, double z, double t, double a, double b, double c) {
	double VAR;
	if ((z <= -4.4627057489340555e-108)) {
		VAR = (((((cbrt(b) * cbrt(b)) / z) * (cbrt(b) / c)) + (9.0 * (1.0 / (((z * c) / y) / x)))) - (4.0 * ((a * t) / c)));
	} else {
		double VAR_1;
		if ((z <= 2.2902672438960906e-68)) {
			VAR_1 = ((1.0 / z) * (((((x * 9.0) * y) - (((z * 4.0) * t) * a)) + b) / c));
		} else {
			double VAR_2;
			if ((z <= 2.9019501673276155e+126)) {
				VAR_2 = (((b / (z * c)) + (9.0 * (1.0 / (((z * c) / y) / x)))) - (4.0 * (a * (t / c))));
			} else {
				VAR_2 = (((b / (z * c)) + (9.0 * (1.0 / ((z * (c / y)) / x)))) - (4.0 * ((a * t) / c)));
			}
			VAR_1 = VAR_2;
		}
		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

Bits error versus b

Bits error versus c

Target

Original	20.7
Target	14.5
Herbie	8.9

\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt -1.10015674080410512 \cdot 10^{-171}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt -0.0:\\ \;\;\;\;\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 1.17088779117474882 \cdot 10^{-53}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 2.8768236795461372 \cdot 10^{130}:\\ \;\;\;\;\left(\left(9 \cdot \frac{y}{c}\right) \cdot \frac{x}{z} + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 1.3838515042456319 \cdot 10^{158}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\left(9 \cdot \left(\frac{y}{c \cdot z} \cdot x\right) + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \end{array}\]

Derivation

1. Split input into 4 regimes

2. if z < -4.4627057489340555e-108

   1. Initial program 25.3

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\]

   2. Taylor expanded around 0 12.4

      \[\leadsto \color{blue}{\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x \cdot y}{z \cdot c}\right) - 4 \cdot \frac{a \cdot t}{c}}\]
   3. Using strategy rm

   4. Applied associate-/l* 10.8

      \[\leadsto \left(\frac{b}{z \cdot c} + 9 \cdot \color{blue}{\frac{x}{\frac{z \cdot c}{y}}}\right) - 4 \cdot \frac{a \cdot t}{c}\]
   5. Using strategy rm

   6. Applied clear-num 10.8

      \[\leadsto \left(\frac{b}{z \cdot c} + 9 \cdot \color{blue}{\frac{1}{\frac{\frac{z \cdot c}{y}}{x}}}\right) - 4 \cdot \frac{a \cdot t}{c}\]
   7. Using strategy rm

   8. Applied add-cube-cbrt 11.0

      \[\leadsto \left(\frac{\color{blue}{\left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right) \cdot \sqrt[3]{b}}}{z \cdot c} + 9 \cdot \frac{1}{\frac{\frac{z \cdot c}{y}}{x}}\right) - 4 \cdot \frac{a \cdot t}{c}\]

   9. Applied times-frac 9.3

      \[\leadsto \left(\color{blue}{\frac{\sqrt[3]{b} \cdot \sqrt[3]{b}}{z} \cdot \frac{\sqrt[3]{b}}{c}} + 9 \cdot \frac{1}{\frac{\frac{z \cdot c}{y}}{x}}\right) - 4 \cdot \frac{a \cdot t}{c}\]

   if -4.4627057489340555e-108 < z < 2.2902672438960906e-68

   1. Initial program 6.0

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\]
   2. Using strategy rm

   3. Applied *-un-lft-identity 6.0

      \[\leadsto \frac{\color{blue}{1 \cdot \left(\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b\right)}}{z \cdot c}\]

   4. Applied times-frac 6.0

      \[\leadsto \color{blue}{\frac{1}{z} \cdot \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c}}\]

   if 2.2902672438960906e-68 < z < 2.9019501673276155e+126

   1. Initial program 14.6

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\]

   2. Taylor expanded around 0 10.1

      \[\leadsto \color{blue}{\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x \cdot y}{z \cdot c}\right) - 4 \cdot \frac{a \cdot t}{c}}\]
   3. Using strategy rm

   4. Applied associate-/l* 9.0

      \[\leadsto \left(\frac{b}{z \cdot c} + 9 \cdot \color{blue}{\frac{x}{\frac{z \cdot c}{y}}}\right) - 4 \cdot \frac{a \cdot t}{c}\]
   5. Using strategy rm

   6. Applied clear-num 9.0

      \[\leadsto \left(\frac{b}{z \cdot c} + 9 \cdot \color{blue}{\frac{1}{\frac{\frac{z \cdot c}{y}}{x}}}\right) - 4 \cdot \frac{a \cdot t}{c}\]
   7. Using strategy rm

   8. Applied *-un-lft-identity 9.0

      \[\leadsto \left(\frac{b}{z \cdot c} + 9 \cdot \frac{1}{\frac{\frac{z \cdot c}{y}}{x}}\right) - 4 \cdot \frac{a \cdot t}{\color{blue}{1 \cdot c}}\]

   9. Applied times-frac 8.9

      \[\leadsto \left(\frac{b}{z \cdot c} + 9 \cdot \frac{1}{\frac{\frac{z \cdot c}{y}}{x}}\right) - 4 \cdot \color{blue}{\left(\frac{a}{1} \cdot \frac{t}{c}\right)}\]

   10. Simplified 8.9

       \[\leadsto \left(\frac{b}{z \cdot c} + 9 \cdot \frac{1}{\frac{\frac{z \cdot c}{y}}{x}}\right) - 4 \cdot \left(\color{blue}{a} \cdot \frac{t}{c}\right)\]

   if 2.9019501673276155e+126 < z

   1. Initial program 37.2

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\]

   2. Taylor expanded around 0 15.3

      \[\leadsto \color{blue}{\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x \cdot y}{z \cdot c}\right) - 4 \cdot \frac{a \cdot t}{c}}\]
   3. Using strategy rm

   4. Applied associate-/l* 12.7

      \[\leadsto \left(\frac{b}{z \cdot c} + 9 \cdot \color{blue}{\frac{x}{\frac{z \cdot c}{y}}}\right) - 4 \cdot \frac{a \cdot t}{c}\]
   5. Using strategy rm

   6. Applied clear-num 12.7

      \[\leadsto \left(\frac{b}{z \cdot c} + 9 \cdot \color{blue}{\frac{1}{\frac{\frac{z \cdot c}{y}}{x}}}\right) - 4 \cdot \frac{a \cdot t}{c}\]
   7. Using strategy rm

   8. Applied *-un-lft-identity 12.7

      \[\leadsto \left(\frac{b}{z \cdot c} + 9 \cdot \frac{1}{\frac{\frac{z \cdot c}{\color{blue}{1 \cdot y}}}{x}}\right) - 4 \cdot \frac{a \cdot t}{c}\]

   9. Applied times-frac 12.3

      \[\leadsto \left(\frac{b}{z \cdot c} + 9 \cdot \frac{1}{\frac{\color{blue}{\frac{z}{1} \cdot \frac{c}{y}}}{x}}\right) - 4 \cdot \frac{a \cdot t}{c}\]

   10. Simplified 12.3

       \[\leadsto \left(\frac{b}{z \cdot c} + 9 \cdot \frac{1}{\frac{\color{blue}{z} \cdot \frac{c}{y}}{x}}\right) - 4 \cdot \frac{a \cdot t}{c}\]
3. Recombined 4 regimes into one program.

4. Final simplification 8.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -4.4627057489340555 \cdot 10^{-108}:\\ \;\;\;\;\left(\frac{\sqrt[3]{b} \cdot \sqrt[3]{b}}{z} \cdot \frac{\sqrt[3]{b}}{c} + 9 \cdot \frac{1}{\frac{\frac{z \cdot c}{y}}{x}}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \mathbf{elif}\;z \le 2.29026724389609061 \cdot 10^{-68}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c}\\ \mathbf{elif}\;z \le 2.90195016732761552 \cdot 10^{126}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{1}{\frac{\frac{z \cdot c}{y}}{x}}\right) - 4 \cdot \left(a \cdot \frac{t}{c}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{1}{\frac{z \cdot \frac{c}{y}}{x}}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \end{array}\]

Reproduce

herbie shell --seed 2020078 
(FPCore (x y z t a b c)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, J"
  :precision binary64

  :herbie-target
  (if (< (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) (* z c)) -1.1001567408041051e-171) (/ (+ (- (* (* x 9) y) (* (* z 4) (* t a))) b) (* z c)) (if (< (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) (* z c)) -0.0) (/ (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) z) c) (if (< (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) (* z c)) 1.1708877911747488e-53) (/ (+ (- (* (* x 9) y) (* (* z 4) (* t a))) b) (* z c)) (if (< (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) (* z c)) 2.876823679546137e+130) (- (+ (* (* 9 (/ y c)) (/ x z)) (/ b (* c z))) (* 4 (/ (* a t) c))) (if (< (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) (* z c)) 1.3838515042456319e+158) (/ (+ (- (* (* x 9) y) (* (* z 4) (* t a))) b) (* z c)) (- (+ (* 9 (* (/ y (* c z)) x)) (/ b (* c z))) (* 4 (/ (* a t) c))))))))

  (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) (* z c)))