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

Average Error: 20.8 → 9.0

Time: 8.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -2.7030268561822704 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{t \cdot a}{c}, \frac{1}{c} \cdot \frac{\mathsf{fma}\left(x, 9 \cdot y, b\right)}{z}\right)\\ \mathbf{elif}\;z \le -7.080579750204059 \cdot 10^{-309}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{a}{\frac{c}{t}}, \frac{\frac{\mathsf{fma}\left(x, 9 \cdot y, b\right)}{c}}{z}\right)\\ \mathbf{elif}\;z \le 1.0887209274456963 \cdot 10^{179}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{a}{-c} \cdot \left(-t\right), \mathsf{fma}\left(9, y \cdot \frac{x}{z \cdot c}, \frac{b}{z \cdot c}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{a}{\frac{c}{t}}, \mathsf{fma}\left(9, \frac{y}{c} \cdot \frac{x}{z}, \frac{b}{z \cdot c}\right)\right)\\ \end{array}\]

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 temp;
	if ((z <= -2.7030268561822704e-10)) {
		temp = fma(-4.0, ((t * a) / c), ((1.0 / c) * (fma(x, (9.0 * y), b) / z)));
	} else {
		double temp_1;
		if ((z <= -7.08057975020406e-309)) {
			temp_1 = fma(-4.0, (a / (c / t)), ((fma(x, (9.0 * y), b) / c) / z));
		} else {
			double temp_2;
			if ((z <= 1.0887209274456963e+179)) {
				temp_2 = fma(-4.0, ((a / -c) * -t), fma(9.0, (y * (x / (z * c))), (b / (z * c))));
			} else {
				temp_2 = fma(-4.0, (a / (c / t)), fma(9.0, ((y / c) * (x / z)), (b / (z * c))));
			}
			temp_1 = temp_2;
		}
		temp = temp_1;
	}
	return temp;
}

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	20.8
Target	14.7
Herbie	9.0

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

Derivation

1. Split input into 4 regimes

2. if z < -2.7030268561822704e-10

   1. Initial program 29.1

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

   2. Simplified 13.4

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

   4. Applied *-commutative 13.4

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

   5. Applied *-un-lft-identity 13.4

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

   6. Applied times-frac 9.1

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

   if -2.7030268561822704e-10 < z < -7.08057975020406e-309

   1. Initial program 7.5

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

   2. Simplified 9.8

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

   4. Applied *-commutative 9.8

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

   5. Applied associate-/l* 7.3

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

   7. Applied *-commutative 7.3

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

   8. Applied associate-/r* 6.2

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

   if -7.08057975020406e-309 < z < 1.0887209274456963e+179

   1. Initial program 13.8

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

   2. Simplified 11.2

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

   4. Applied *-commutative 11.2

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

   5. Applied associate-/l* 10.0

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

   6. Taylor expanded around 0 9.9

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

   7. Simplified 9.9

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

   9. Applied *-un-lft-identity 9.9

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

   10. Applied *-commutative 9.9

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

   11. Applied times-frac 9.7

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

   12. Simplified 9.7

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

   14. Applied frac-2neg 9.7

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

   15. Applied associate-/r/ 9.3

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

   if 1.0887209274456963e+179 < z

   1. Initial program 39.2

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

   2. Simplified 14.4

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

   4. Applied *-commutative 14.4

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

   5. Applied associate-/l* 16.0

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

   6. Taylor expanded around 0 16.0

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

   7. Simplified 16.0

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

   9. Applied *-commutative 16.0

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

   10. Applied *-commutative 16.0

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

   11. Applied times-frac 11.8

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

4. Final simplification 9.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -2.7030268561822704 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{t \cdot a}{c}, \frac{1}{c} \cdot \frac{\mathsf{fma}\left(x, 9 \cdot y, b\right)}{z}\right)\\ \mathbf{elif}\;z \le -7.080579750204059 \cdot 10^{-309}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{a}{\frac{c}{t}}, \frac{\frac{\mathsf{fma}\left(x, 9 \cdot y, b\right)}{c}}{z}\right)\\ \mathbf{elif}\;z \le 1.0887209274456963 \cdot 10^{179}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{a}{-c} \cdot \left(-t\right), \mathsf{fma}\left(9, y \cdot \frac{x}{z \cdot c}, \frac{b}{z \cdot c}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{a}{\frac{c}{t}}, \mathsf{fma}\left(9, \frac{y}{c} \cdot \frac{x}{z}, \frac{b}{z \cdot c}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020066 +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)))