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

Average Error: 20.7 → 9.3

Time: 6.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;c \le -1.690798523259897 \cdot 10^{177}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{t \cdot a}{c}, \frac{1}{z} \cdot \frac{\mathsf{fma}\left(9 \cdot x, y, b\right)}{c}\right)\\ \mathbf{elif}\;c \le -3.4478524687081832 \cdot 10^{-46}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{t}{\frac{c}{a}}, \mathsf{fma}\left(9, \frac{x}{\frac{z \cdot c}{y}}, \frac{b}{z \cdot c}\right)\right)\\ \mathbf{elif}\;c \le 2.0655766129224776 \cdot 10^{-74}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{t \cdot a}{c}, \frac{\frac{\mathsf{fma}\left(9 \cdot x, y, b\right)}{z}}{c}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{t \cdot a}{c}, \mathsf{fma}\left(9, \frac{x}{\frac{z \cdot c}{y}}, \frac{\sqrt[3]{b} \cdot \sqrt[3]{b}}{z} \cdot \frac{\sqrt[3]{b}}{c}\right)\right)\\ \end{array}\]

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 ((c <= -1.6907985232598968e+177)) {
		VAR = fma(-4.0, ((t * a) / c), ((1.0 / z) * (fma((9.0 * x), y, b) / c)));
	} else {
		double VAR_1;
		if ((c <= -3.4478524687081832e-46)) {
			VAR_1 = fma(-4.0, (t / (c / a)), fma(9.0, (x / ((z * c) / y)), (b / (z * c))));
		} else {
			double VAR_2;
			if ((c <= 2.0655766129224776e-74)) {
				VAR_2 = fma(-4.0, ((t * a) / c), ((fma((9.0 * x), y, b) / z) / c));
			} else {
				VAR_2 = fma(-4.0, ((t * a) / c), fma(9.0, (x / ((z * c) / y)), (((cbrt(b) * cbrt(b)) / z) * (cbrt(b) / 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	9.3

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

Derivation

1. Split input into 4 regimes

2. if c < -1.6907985232598968e+177

   1. Initial program 26.7

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

   2. Simplified 18.6

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

   4. Applied *-un-lft-identity 18.6

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

   5. Applied times-frac 14.6

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

   6. Simplified 14.7

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

   if -1.6907985232598968e+177 < c < -3.4478524687081832e-46

   1. Initial program 18.9

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

   2. Simplified 10.7

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

   3. Taylor expanded around 0 10.6

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

   4. Simplified 10.6

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

   6. Applied associate-/l* 8.9

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

   8. Applied associate-/l* 6.5

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

   if -3.4478524687081832e-46 < c < 2.0655766129224776e-74

   1. Initial program 15.7

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

   2. Simplified 5.8

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

   4. Applied associate-/r* 2.3

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

   5. Simplified 2.3

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

   if 2.0655766129224776e-74 < c

   1. Initial program 22.2

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

   2. Simplified 13.5

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

   3. Taylor expanded around 0 13.4

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

   4. Simplified 13.4

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

   6. Applied associate-/l* 12.4

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

   8. Applied add-cube-cbrt 12.7

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

   9. Applied times-frac 12.7

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

4. Final simplification 9.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \le -1.690798523259897 \cdot 10^{177}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{t \cdot a}{c}, \frac{1}{z} \cdot \frac{\mathsf{fma}\left(9 \cdot x, y, b\right)}{c}\right)\\ \mathbf{elif}\;c \le -3.4478524687081832 \cdot 10^{-46}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{t}{\frac{c}{a}}, \mathsf{fma}\left(9, \frac{x}{\frac{z \cdot c}{y}}, \frac{b}{z \cdot c}\right)\right)\\ \mathbf{elif}\;c \le 2.0655766129224776 \cdot 10^{-74}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{t \cdot a}{c}, \frac{\frac{\mathsf{fma}\left(9 \cdot x, y, b\right)}{z}}{c}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{t \cdot a}{c}, \mathsf{fma}\left(9, \frac{x}{\frac{z \cdot c}{y}}, \frac{\sqrt[3]{b} \cdot \sqrt[3]{b}}{z} \cdot \frac{\sqrt[3]{b}}{c}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020078 +o rules:numerics
(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)))