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

Average Error: 20.4 → 6.4

Time: 10.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \leq -\infty:\\ \;\;\;\;\left(\frac{\left(x \cdot 9\right) \cdot y + b}{z} - 4 \cdot \left(t \cdot a\right)\right) \cdot \frac{1}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \leq -3.012632262259407 \cdot 10^{-111}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + \sqrt{9} \cdot \left(\sqrt{9} \cdot \frac{x \cdot y}{z \cdot c}\right)\right) - \frac{4 \cdot \left(t \cdot a\right)}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \leq 5.206155083483664 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{1}{\frac{z}{\left(x \cdot 9\right) \cdot y + b}} - 4 \cdot \left(t \cdot a\right)}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \leq 8.198015276865958 \cdot 10^{+304}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \left(\frac{x}{z} \cdot \frac{y}{c}\right)\right) - \frac{4 \cdot \left(t \cdot a\right)}{c}\\ \end{array}\]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)))

↓

(FPCore (x y z t a b c)
 :precision binary64
 (if (<=
      (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c))
      (- INFINITY))
   (* (- (/ (+ (* (* x 9.0) y) b) z) (* 4.0 (* t a))) (/ 1.0 c))
   (if (<=
        (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c))
        -3.012632262259407e-111)
     (-
      (+ (/ b (* z c)) (* (sqrt 9.0) (* (sqrt 9.0) (/ (* x y) (* z c)))))
      (/ (* 4.0 (* t a)) c))
     (if (<=
          (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c))
          5.206155083483664e-52)
       (/ (- (/ 1.0 (/ z (+ (* (* x 9.0) y) b))) (* 4.0 (* t a))) c)
       (if (<=
            (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c))
            8.198015276865958e+304)
         (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c))
         (-
          (+ (/ b (* z c)) (* 9.0 (* (/ x z) (/ y c))))
          (/ (* 4.0 (* t a)) c)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	return (((double) (((double) (((double) (((double) (x * 9.0)) * y)) - ((double) (((double) (((double) (z * 4.0)) * t)) * a)))) + b)) / ((double) (z * c)));
}

↓

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if (((((double) (((double) (((double) (((double) (x * 9.0)) * y)) - ((double) (((double) (((double) (z * 4.0)) * t)) * a)))) + b)) / ((double) (z * c))) <= ((double) -(((double) INFINITY))))) {
		tmp = ((double) (((double) ((((double) (((double) (((double) (x * 9.0)) * y)) + b)) / z) - ((double) (4.0 * ((double) (t * a)))))) * (1.0 / c)));
	} else {
		double tmp_1;
		if (((((double) (((double) (((double) (((double) (x * 9.0)) * y)) - ((double) (((double) (((double) (z * 4.0)) * t)) * a)))) + b)) / ((double) (z * c))) <= -3.012632262259407e-111)) {
			tmp_1 = ((double) (((double) ((b / ((double) (z * c))) + ((double) (((double) sqrt(9.0)) * ((double) (((double) sqrt(9.0)) * (((double) (x * y)) / ((double) (z * c))))))))) - (((double) (4.0 * ((double) (t * a)))) / c)));
		} else {
			double tmp_2;
			if (((((double) (((double) (((double) (((double) (x * 9.0)) * y)) - ((double) (((double) (((double) (z * 4.0)) * t)) * a)))) + b)) / ((double) (z * c))) <= 5.206155083483664e-52)) {
				tmp_2 = (((double) ((1.0 / (z / ((double) (((double) (((double) (x * 9.0)) * y)) + b)))) - ((double) (4.0 * ((double) (t * a)))))) / c);
			} else {
				double tmp_3;
				if (((((double) (((double) (((double) (((double) (x * 9.0)) * y)) - ((double) (((double) (((double) (z * 4.0)) * t)) * a)))) + b)) / ((double) (z * c))) <= 8.198015276865958e+304)) {
					tmp_3 = (((double) (((double) (((double) (((double) (x * 9.0)) * y)) - ((double) (((double) (((double) (z * 4.0)) * t)) * a)))) + b)) / ((double) (z * c)));
				} else {
					tmp_3 = ((double) (((double) ((b / ((double) (z * c))) + ((double) (9.0 * ((double) ((x / z) * (y / c))))))) - (((double) (4.0 * ((double) (t * a)))) / c)));
				}
				tmp_2 = tmp_3;
			}
			tmp_1 = tmp_2;
		}
		tmp = tmp_1;
	}
	return tmp;
}

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	20.4
Target	14.5
Herbie	6.4

\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} < -1.1001567408041051 \cdot 10^{-171}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} < -0:\\ \;\;\;\;\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} < 1.1708877911747488 \cdot 10^{-53}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} < 2.876823679546137 \cdot 10^{+130}:\\ \;\;\;\;\left(\left(9 \cdot \frac{y}{c}\right) \cdot \frac{x}{z} + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} < 1.3838515042456319 \cdot 10^{+158}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\left(9 \cdot \left(\frac{y}{c \cdot z} \cdot x\right) + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \end{array}\]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x 9.0) y) (*.f64 (*.f64 (*.f64 z 4.0) t) a)) b) (*.f64 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 26.7

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

   4. Applied div-inv_binary64 26.8

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

   if -inf.0 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x 9.0) y) (*.f64 (*.f64 (*.f64 z 4.0) t) a)) b) (*.f64 z c)) < -3.01263226225940707e-111

   1. Initial program 0.7

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

   2. Simplified 11.6

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

   3. Taylor expanded around 0 2.9

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

   4. Simplified 3.0

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

   6. Applied add-sqr-sqrt_binary64 3.0

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

   7. Applied associate-*l*_binary64 3.0

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

   8. Simplified 3.0

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

   if -3.01263226225940707e-111 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x 9.0) y) (*.f64 (*.f64 (*.f64 z 4.0) t) a)) b) (*.f64 z c)) < 5.2061550834836637e-52

   1. Initial program 19.2

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

   2. Simplified 0.9

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

   4. Applied clear-num_binary64 1.1

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

   if 5.2061550834836637e-52 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x 9.0) y) (*.f64 (*.f64 (*.f64 z 4.0) t) a)) b) (*.f64 z c)) < 8.19801527686595761e304

   1. Initial program 0.7

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

   if 8.19801527686595761e304 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x 9.0) y) (*.f64 (*.f64 (*.f64 z 4.0) t) a)) b) (*.f64 z c))

   1. Initial program 63.4

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

   2. Simplified 27.2

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

   3. Taylor expanded around 0 31.3

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

   4. Simplified 31.2

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

   6. Applied times-frac_binary64 17.9

      \[\leadsto \left(\frac{b}{z \cdot c} + 9 \cdot \color{blue}{\left(\frac{x}{z} \cdot \frac{y}{c}\right)}\right) - \frac{4 \cdot \left(t \cdot a\right)}{c}\]
3. Recombined 5 regimes into one program.

4. Final simplification 6.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \leq -\infty:\\ \;\;\;\;\left(\frac{\left(x \cdot 9\right) \cdot y + b}{z} - 4 \cdot \left(t \cdot a\right)\right) \cdot \frac{1}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \leq -3.012632262259407 \cdot 10^{-111}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + \sqrt{9} \cdot \left(\sqrt{9} \cdot \frac{x \cdot y}{z \cdot c}\right)\right) - \frac{4 \cdot \left(t \cdot a\right)}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \leq 5.206155083483664 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{1}{\frac{z}{\left(x \cdot 9\right) \cdot y + b}} - 4 \cdot \left(t \cdot a\right)}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \leq 8.198015276865958 \cdot 10^{+304}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \left(\frac{x}{z} \cdot \frac{y}{c}\right)\right) - \frac{4 \cdot \left(t \cdot a\right)}{c}\\ \end{array}\]

Reproduce

herbie shell --seed 2020210 
(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 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) -1.1001567408041051e-171) (/ (+ (- (* (* x 9.0) y) (* (* z 4.0) (* t a))) b) (* z c)) (if (< (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) -0.0) (/ (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) z) c) (if (< (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) 1.1708877911747488e-53) (/ (+ (- (* (* x 9.0) y) (* (* z 4.0) (* t a))) b) (* z c)) (if (< (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) 2.876823679546137e+130) (- (+ (* (* 9.0 (/ y c)) (/ x z)) (/ b (* c z))) (* 4.0 (/ (* a t) c))) (if (< (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) 1.3838515042456319e+158) (/ (+ (- (* (* x 9.0) y) (* (* z 4.0) (* t a))) b) (* z c)) (- (+ (* 9.0 (* (/ y (* c z)) x)) (/ b (* c z))) (* 4.0 (/ (* a t) c))))))))

  (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)))