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

Average Error: 20.7 → 4.7

Time: 7.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;c \le -3.71250790597694037 \cdot 10^{58}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{t}{\sqrt[3]{c} \cdot \sqrt[3]{c}} \cdot \frac{a}{\sqrt[3]{c}}, \frac{1}{z} \cdot \mathsf{fma}\left(9, x \cdot \frac{y}{c}, \frac{b}{c}\right)\right)\\ \mathbf{elif}\;c \le 2.9909017744909091 \cdot 10^{25}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{t \cdot a}{c}, \frac{\frac{\mathsf{fma}\left(9 \cdot x, y, b\right)}{z}}{c}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{t}{\sqrt[3]{c} \cdot \sqrt[3]{c}} \cdot \frac{a}{\sqrt[3]{c}}, \frac{1}{z} \cdot \mathsf{fma}\left(9, \frac{x}{\frac{c}{y}}, \frac{b}{c}\right)\right)\\ \end{array}\]

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 <= -3.7125079059769404e+58)) {
		VAR = fma(-4.0, ((t / (cbrt(c) * cbrt(c))) * (a / cbrt(c))), ((1.0 / z) * fma(9.0, (x * (y / c)), (b / c))));
	} else {
		double VAR_1;
		if ((c <= 2.990901774490909e+25)) {
			VAR_1 = fma(-4.0, ((t * a) / c), ((fma((9.0 * x), y, b) / z) / c));
		} else {
			VAR_1 = fma(-4.0, ((t / (cbrt(c) * cbrt(c))) * (a / cbrt(c))), ((1.0 / z) * fma(9.0, (x / (c / y)), (b / c))));
		}
		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	14.4
Herbie	4.7

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

Derivation

1. Split input into 3 regimes

2. if c < -3.7125079059769404e+58

   1. Initial program 24.1

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

   2. Simplified 16.7

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

   4. Applied add-cube-cbrt 17.0

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

   5. Applied times-frac 12.1

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

   7. Applied *-un-lft-identity 12.1

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

   8. Applied times-frac 9.1

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

   9. Taylor expanded around 0 9.0

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

   10. Simplified 9.0

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

   12. Applied *-un-lft-identity 9.0

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

   13. Applied times-frac 5.8

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

   14. Simplified 5.8

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

   if -3.7125079059769404e+58 < c < 2.990901774490909e+25

   1. Initial program 14.9

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

   2. Simplified 5.1

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

   4. Applied associate-/r* 3.3

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

   5. Simplified 3.3

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

   if 2.990901774490909e+25 < c

   1. Initial program 24.6

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

   2. Simplified 15.4

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

   4. Applied add-cube-cbrt 15.7

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

   5. Applied times-frac 11.8

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

   7. Applied *-un-lft-identity 11.8

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

   8. Applied times-frac 9.0

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

   9. Taylor expanded around 0 8.9

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

   10. Simplified 8.9

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

   12. Applied associate-/l* 5.6

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

4. Final simplification 4.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \le -3.71250790597694037 \cdot 10^{58}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{t}{\sqrt[3]{c} \cdot \sqrt[3]{c}} \cdot \frac{a}{\sqrt[3]{c}}, \frac{1}{z} \cdot \mathsf{fma}\left(9, x \cdot \frac{y}{c}, \frac{b}{c}\right)\right)\\ \mathbf{elif}\;c \le 2.9909017744909091 \cdot 10^{25}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{t \cdot a}{c}, \frac{\frac{\mathsf{fma}\left(9 \cdot x, y, b\right)}{z}}{c}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{t}{\sqrt[3]{c} \cdot \sqrt[3]{c}} \cdot \frac{a}{\sqrt[3]{c}}, \frac{1}{z} \cdot \mathsf{fma}\left(9, \frac{x}{\frac{c}{y}}, \frac{b}{c}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020079 +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)))