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

Average Error: 7.7 → 3.9

Time: 1.3m

Precision: 64

Math

\[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot y \le -7.32093030450982164 \cdot 10^{204}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot \frac{x \cdot 1}{-a}, -y, -\frac{t}{a} \cdot \left(4.5 \cdot \frac{z}{1}\right)\right) + \mathsf{fma}\left(-\frac{t}{a}, 4.5 \cdot \frac{z}{1}, \frac{t}{a} \cdot \left(4.5 \cdot \frac{z}{1}\right)\right)\\ \mathbf{elif}\;x \cdot y \le 7.3955972409760691 \cdot 10^{77}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - \left(\sqrt[3]{4.5} \cdot \sqrt[3]{4.5}\right) \cdot \left(\sqrt[3]{4.5} \cdot \frac{t \cdot z}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot 1}{\frac{a}{y}} - 4.5 \cdot \frac{t \cdot 1}{\frac{a}{z}}\\ \end{array}\]

C

double code(double x, double y, double z, double t, double a) {
	return (((x * y) - ((z * 9.0) * t)) / (a * 2.0));
}

↓

double code(double x, double y, double z, double t, double a) {
	double temp;
	if (((x * y) <= -7.320930304509822e+204)) {
		temp = (fma((0.5 * ((x * 1.0) / -a)), -y, -((t / a) * (4.5 * (z / 1.0)))) + fma(-(t / a), (4.5 * (z / 1.0)), ((t / a) * (4.5 * (z / 1.0)))));
	} else {
		double temp_1;
		if (((x * y) <= 7.395597240976069e+77)) {
			temp_1 = ((0.5 * ((x * y) / a)) - ((cbrt(4.5) * cbrt(4.5)) * (cbrt(4.5) * ((t * z) / a))));
		} else {
			temp_1 = ((0.5 * ((x * 1.0) / (a / y))) - (4.5 * ((t * 1.0) / (a / z))));
		}
		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

Target

Original	7.7
Target	5.5
Herbie	3.9

\[\begin{array}{l} \mathbf{if}\;a \lt -2.090464557976709 \cdot 10^{86}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;a \lt 2.14403070783397609 \cdot 10^{99}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot \left(x \cdot 0.5\right) - \frac{t}{a} \cdot \left(z \cdot 4.5\right)\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if (* x y) < -7.320930304509822e+204

   1. Initial program 32.6

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\]

   2. Taylor expanded around 0 32.5

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \frac{t \cdot z}{a}}\]
   3. Using strategy rm

   4. Applied *-un-lft-identity 32.5

      \[\leadsto 0.5 \cdot \frac{x \cdot \color{blue}{\left(1 \cdot y\right)}}{a} - 4.5 \cdot \frac{t \cdot z}{a}\]

   5. Applied associate-*r* 32.5

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\left(x \cdot 1\right) \cdot y}}{a} - 4.5 \cdot \frac{t \cdot z}{a}\]

   6. Applied associate-/l* 6.6

      \[\leadsto 0.5 \cdot \color{blue}{\frac{x \cdot 1}{\frac{a}{y}}} - 4.5 \cdot \frac{t \cdot z}{a}\]
   7. Using strategy rm

   8. Applied *-un-lft-identity 6.6

      \[\leadsto 0.5 \cdot \frac{x \cdot 1}{\frac{a}{y}} - 4.5 \cdot \frac{t \cdot z}{\color{blue}{1 \cdot a}}\]

   9. Applied *-commutative 6.6

      \[\leadsto 0.5 \cdot \frac{x \cdot 1}{\frac{a}{y}} - 4.5 \cdot \frac{\color{blue}{z \cdot t}}{1 \cdot a}\]

   10. Applied times-frac 0.7

       \[\leadsto 0.5 \cdot \frac{x \cdot 1}{\frac{a}{y}} - 4.5 \cdot \color{blue}{\left(\frac{z}{1} \cdot \frac{t}{a}\right)}\]

   11. Applied associate-*r* 0.8

       \[\leadsto 0.5 \cdot \frac{x \cdot 1}{\frac{a}{y}} - \color{blue}{\left(4.5 \cdot \frac{z}{1}\right) \cdot \frac{t}{a}}\]

   12. Applied frac-2neg 0.8

       \[\leadsto 0.5 \cdot \frac{x \cdot 1}{\color{blue}{\frac{-a}{-y}}} - \left(4.5 \cdot \frac{z}{1}\right) \cdot \frac{t}{a}\]

   13. Applied associate-/r/ 1.2

       \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{x \cdot 1}{-a} \cdot \left(-y\right)\right)} - \left(4.5 \cdot \frac{z}{1}\right) \cdot \frac{t}{a}\]

   14. Applied associate-*r* 1.1

       \[\leadsto \color{blue}{\left(0.5 \cdot \frac{x \cdot 1}{-a}\right) \cdot \left(-y\right)} - \left(4.5 \cdot \frac{z}{1}\right) \cdot \frac{t}{a}\]

   15. Applied prod-diff 1.1

       \[\leadsto \color{blue}{\mathsf{fma}\left(0.5 \cdot \frac{x \cdot 1}{-a}, -y, -\frac{t}{a} \cdot \left(4.5 \cdot \frac{z}{1}\right)\right) + \mathsf{fma}\left(-\frac{t}{a}, 4.5 \cdot \frac{z}{1}, \frac{t}{a} \cdot \left(4.5 \cdot \frac{z}{1}\right)\right)}\]

   if -7.320930304509822e+204 < (* x y) < 7.395597240976069e+77

   1. Initial program 4.1

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\]

   2. Taylor expanded around 0 4.0

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \frac{t \cdot z}{a}}\]
   3. Using strategy rm

   4. Applied add-cube-cbrt 4.0

      \[\leadsto 0.5 \cdot \frac{x \cdot y}{a} - \color{blue}{\left(\left(\sqrt[3]{4.5} \cdot \sqrt[3]{4.5}\right) \cdot \sqrt[3]{4.5}\right)} \cdot \frac{t \cdot z}{a}\]

   5. Applied associate-*l* 4.1

      \[\leadsto 0.5 \cdot \frac{x \cdot y}{a} - \color{blue}{\left(\sqrt[3]{4.5} \cdot \sqrt[3]{4.5}\right) \cdot \left(\sqrt[3]{4.5} \cdot \frac{t \cdot z}{a}\right)}\]

   if 7.395597240976069e+77 < (* x y)

   1. Initial program 15.8

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\]

   2. Taylor expanded around 0 15.7

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \frac{t \cdot z}{a}}\]
   3. Using strategy rm

   4. Applied *-un-lft-identity 15.7

      \[\leadsto 0.5 \cdot \frac{x \cdot \color{blue}{\left(1 \cdot y\right)}}{a} - 4.5 \cdot \frac{t \cdot z}{a}\]

   5. Applied associate-*r* 15.7

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\left(x \cdot 1\right) \cdot y}}{a} - 4.5 \cdot \frac{t \cdot z}{a}\]

   6. Applied associate-/l* 7.8

      \[\leadsto 0.5 \cdot \color{blue}{\frac{x \cdot 1}{\frac{a}{y}}} - 4.5 \cdot \frac{t \cdot z}{a}\]
   7. Using strategy rm

   8. Applied *-un-lft-identity 7.8

      \[\leadsto 0.5 \cdot \frac{x \cdot 1}{\frac{a}{y}} - 4.5 \cdot \frac{t \cdot \color{blue}{\left(1 \cdot z\right)}}{a}\]

   9. Applied associate-*r* 7.8

      \[\leadsto 0.5 \cdot \frac{x \cdot 1}{\frac{a}{y}} - 4.5 \cdot \frac{\color{blue}{\left(t \cdot 1\right) \cdot z}}{a}\]

   10. Applied associate-/l* 4.4

       \[\leadsto 0.5 \cdot \frac{x \cdot 1}{\frac{a}{y}} - 4.5 \cdot \color{blue}{\frac{t \cdot 1}{\frac{a}{z}}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 3.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \le -7.32093030450982164 \cdot 10^{204}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot \frac{x \cdot 1}{-a}, -y, -\frac{t}{a} \cdot \left(4.5 \cdot \frac{z}{1}\right)\right) + \mathsf{fma}\left(-\frac{t}{a}, 4.5 \cdot \frac{z}{1}, \frac{t}{a} \cdot \left(4.5 \cdot \frac{z}{1}\right)\right)\\ \mathbf{elif}\;x \cdot y \le 7.3955972409760691 \cdot 10^{77}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - \left(\sqrt[3]{4.5} \cdot \sqrt[3]{4.5}\right) \cdot \left(\sqrt[3]{4.5} \cdot \frac{t \cdot z}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot 1}{\frac{a}{y}} - 4.5 \cdot \frac{t \cdot 1}{\frac{a}{z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020066 +o rules:numerics
(FPCore (x y z t a)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, I"
  :precision binary64

  :herbie-target
  (if (< a -2.090464557976709e+86) (- (* 0.5 (/ (* y x) a)) (* 4.5 (/ t (/ a z)))) (if (< a 2.144030707833976e+99) (/ (- (* x y) (* z (* 9 t))) (* a 2)) (- (* (/ y a) (* x 0.5)) (* (/ t a) (* z 4.5)))))

  (/ (- (* x y) (* (* z 9) t)) (* a 2)))