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

Average Error: 20.1 → 7.7

Time: 8.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} = -\infty:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{t}{\sqrt[3]{c} \cdot \sqrt[3]{c}} \cdot \frac{a}{\sqrt[3]{c}}, 0\right)\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le -2.9908337504604717 \cdot 10^{-226}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le 4.86960761957445033 \cdot 10^{-212}:\\ \;\;\;\;\mathsf{fma}\left(-4, 1 \cdot \frac{a \cdot t}{c}, \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1} \cdot \frac{\frac{\mathsf{fma}\left(x, 9 \cdot y, b\right)}{c}}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{t}{\sqrt[3]{c} \cdot \left(\left(\sqrt[3]{\sqrt[3]{c}} \cdot \sqrt[3]{\sqrt[3]{c}}\right) \cdot \sqrt[3]{\sqrt[3]{c}}\right)} \cdot \frac{a}{\sqrt[3]{c}}, \frac{\mathsf{fma}\left(x, 9 \cdot y, b\right)}{z \cdot c}\right)\\ \end{array}\]

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 (((((((x * 9.0) * y) - (((z * 4.0) * t) * a)) + b) / (z * c)) <= -inf.0)) {
		temp = fma(-4.0, ((t / (cbrt(c) * cbrt(c))) * (a / cbrt(c))), 0.0);
	} else {
		double temp_1;
		if (((((((x * 9.0) * y) - (((z * 4.0) * t) * a)) + b) / (z * c)) <= -2.9908337504604717e-226)) {
			temp_1 = (((((x * 9.0) * y) - (((z * 4.0) * t) * a)) + b) / (z * c));
		} else {
			double temp_2;
			if (((((((x * 9.0) * y) - (((z * 4.0) * t) * a)) + b) / (z * c)) <= 4.86960761957445e-212)) {
				temp_2 = fma(-4.0, (1.0 * ((a * t) / c)), (((cbrt(1.0) * cbrt(1.0)) / 1.0) * ((fma(x, (9.0 * y), b) / c) / z)));
			} else {
				temp_2 = fma(-4.0, ((t / (cbrt(c) * ((cbrt(cbrt(c)) * cbrt(cbrt(c))) * cbrt(cbrt(c))))) * (a / cbrt(c))), (fma(x, (9.0 * y), 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	20.1
Target	14.6
Herbie	7.7

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

Derivation

1. Split input into 4 regimes

2. if (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < -inf.0

   1. Initial program 64.0

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

   2. Simplified 32.5

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

   4. Applied add-cube-cbrt 33.0

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

   5. Applied times-frac 27.2

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

   6. Taylor expanded around inf 27.6

      \[\leadsto \mathsf{fma}\left(-4, \frac{t}{\sqrt[3]{c} \cdot \sqrt[3]{c}} \cdot \frac{a}{\sqrt[3]{c}}, \color{blue}{0}\right)\]

   if -inf.0 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < -2.9908337504604717e-226

   1. Initial program 0.6

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

   if -2.9908337504604717e-226 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < 4.86960761957445e-212

   1. Initial program 30.1

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

   2. Simplified 19.1

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

   4. Applied add-cube-cbrt 19.3

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

   5. Applied times-frac 22.4

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

   7. Applied *-un-lft-identity 22.4

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

   8. Applied times-frac 6.4

      \[\leadsto \mathsf{fma}\left(-4, \frac{t}{\sqrt[3]{c} \cdot \sqrt[3]{c}} \cdot \frac{a}{\sqrt[3]{c}}, \color{blue}{\frac{1}{z} \cdot \frac{\mathsf{fma}\left(x, 9 \cdot y, b\right)}{c}}\right)\]
   9. Using strategy rm

   10. Applied *-un-lft-identity 6.4

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

   11. Applied add-cube-cbrt 6.4

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

   12. Applied times-frac 6.4

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

   13. Applied associate-*l* 6.4

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

   14. Simplified 6.4

       \[\leadsto \mathsf{fma}\left(-4, \frac{t}{\sqrt[3]{c} \cdot \sqrt[3]{c}} \cdot \frac{a}{\sqrt[3]{c}}, \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1} \cdot \color{blue}{\frac{\frac{\mathsf{fma}\left(x, 9 \cdot y, b\right)}{c}}{z}}\right)\]
   15. Using strategy rm

   16. Applied *-un-lft-identity 6.4

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

   17. Applied associate-*l* 6.4

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

   18. Simplified 3.0

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

   if 4.86960761957445e-212 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c))

   1. Initial program 23.4

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

   2. Simplified 13.1

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

   4. Applied add-cube-cbrt 13.4

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

   5. Applied times-frac 10.4

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

   7. Applied add-cube-cbrt 10.6

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

4. Final simplification 7.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} = -\infty:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{t}{\sqrt[3]{c} \cdot \sqrt[3]{c}} \cdot \frac{a}{\sqrt[3]{c}}, 0\right)\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le -2.9908337504604717 \cdot 10^{-226}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le 4.86960761957445033 \cdot 10^{-212}:\\ \;\;\;\;\mathsf{fma}\left(-4, 1 \cdot \frac{a \cdot t}{c}, \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1} \cdot \frac{\frac{\mathsf{fma}\left(x, 9 \cdot y, b\right)}{c}}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{t}{\sqrt[3]{c} \cdot \left(\left(\sqrt[3]{\sqrt[3]{c}} \cdot \sqrt[3]{\sqrt[3]{c}}\right) \cdot \sqrt[3]{\sqrt[3]{c}}\right)} \cdot \frac{a}{\sqrt[3]{c}}, \frac{\mathsf{fma}\left(x, 9 \cdot y, b\right)}{z \cdot c}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020058 +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)))