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

Average Error: 20.1 → 8.4

Time: 11.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot 9 \leq -3.261410316015413 \cdot 10^{+70}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{c} \cdot \sqrt[3]{c}} \cdot \frac{\sqrt[3]{x}}{\frac{z}{\frac{y}{\sqrt[3]{c}}}}\right)\right) - 4 \cdot \frac{t \cdot a}{c}\\ \mathbf{elif}\;x \cdot 9 \leq -2.989941699386729 \cdot 10^{-141}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x \cdot y}{z \cdot c}\right) - 4 \cdot \left(\frac{t}{\sqrt[3]{c} \cdot \sqrt[3]{c}} \cdot \frac{a}{\sqrt[3]{c}}\right)\\ \mathbf{elif}\;x \cdot 9 \leq 1.2754336083427134 \cdot 10^{-101}:\\ \;\;\;\;\frac{\left(\frac{b}{z} + 9 \cdot \frac{x \cdot y}{z}\right) + \left(t \cdot a\right) \cdot -4}{c}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{c} \cdot \sqrt[3]{c}} \cdot \frac{\sqrt[3]{x}}{\frac{z}{\frac{y}{\sqrt[3]{c}}}}\right)\right) - 4 \cdot \frac{t \cdot a}{c}\\ \end{array}\]

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) -3.261410316015413e+70)
   (-
    (+
     (/ b (* z c))
     (*
      9.0
      (*
       (/ (* (cbrt x) (cbrt x)) (* (cbrt c) (cbrt c)))
       (/ (cbrt x) (/ z (/ y (cbrt c)))))))
    (* 4.0 (/ (* t a) c)))
   (if (<= (* x 9.0) -2.989941699386729e-141)
     (-
      (+ (/ b (* z c)) (* 9.0 (/ (* x y) (* z c))))
      (* 4.0 (* (/ t (* (cbrt c) (cbrt c))) (/ a (cbrt c)))))
     (if (<= (* x 9.0) 1.2754336083427134e-101)
       (/ (+ (+ (/ b z) (* 9.0 (/ (* x y) z))) (* (* t a) -4.0)) c)
       (-
        (+
         (/ b (* z c))
         (*
          9.0
          (*
           (/ (* (cbrt x) (cbrt x)) (* (cbrt c) (cbrt c)))
           (/ (cbrt x) (/ z (/ y (cbrt c)))))))
        (* 4.0 (/ (* t a) c)))))))

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 tmp;
	if ((x * 9.0) <= -3.261410316015413e+70) {
		tmp = ((b / (z * c)) + (9.0 * (((cbrt(x) * cbrt(x)) / (cbrt(c) * cbrt(c))) * (cbrt(x) / (z / (y / cbrt(c))))))) - (4.0 * ((t * a) / c));
	} else if ((x * 9.0) <= -2.989941699386729e-141) {
		tmp = ((b / (z * c)) + (9.0 * ((x * y) / (z * c)))) - (4.0 * ((t / (cbrt(c) * cbrt(c))) * (a / cbrt(c))));
	} else if ((x * 9.0) <= 1.2754336083427134e-101) {
		tmp = (((b / z) + (9.0 * ((x * y) / z))) + ((t * a) * -4.0)) / c;
	} else {
		tmp = ((b / (z * c)) + (9.0 * (((cbrt(x) * cbrt(x)) / (cbrt(c) * cbrt(c))) * (cbrt(x) / (z / (y / cbrt(c))))))) - (4.0 * ((t * a) / c));
	}
	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	20.1
Target	14.1
Herbie	8.4

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 x 9) < -3.2614103160154133e70 or 1.2754336083427134e-101 < (*.f64 x 9)

   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 18.1

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

   3. Taylor expanded around 0 15.8

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

   4. Simplified 15.8

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

   6. Applied associate-/l*_binary64_13322 12.2

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

   7. Simplified 11.6

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

   9. Applied add-cube-cbrt_binary64_13412 11.9

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

   10. Applied *-un-lft-identity_binary64_13377 11.9

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

   11. Applied times-frac_binary64_13383 11.9

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

   12. Applied *-un-lft-identity_binary64_13377 11.9

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

   13. Applied times-frac_binary64_13383 10.3

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

   14. Applied add-cube-cbrt_binary64_13412 10.4

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

   15. Applied times-frac_binary64_13383 9.7

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

   16. Simplified 9.7

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

   if -3.2614103160154133e70 < (*.f64 x 9) < -2.98994169938672898e-141

   1. Initial program 18.0

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

   2. Simplified 10.3

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

   3. Taylor expanded around 0 8.7

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

   4. Simplified 8.7

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

   6. Applied add-cube-cbrt_binary64_13412 9.0

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

   7. Applied times-frac_binary64_13383 7.2

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

   if -2.98994169938672898e-141 < (*.f64 x 9) < 1.2754336083427134e-101

   1. Initial program 15.4

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

   2. Simplified 7.2

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

   3. Taylor expanded around 0 7.2

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

4. Final simplification 8.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 9 \leq -3.261410316015413 \cdot 10^{+70}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{c} \cdot \sqrt[3]{c}} \cdot \frac{\sqrt[3]{x}}{\frac{z}{\frac{y}{\sqrt[3]{c}}}}\right)\right) - 4 \cdot \frac{t \cdot a}{c}\\ \mathbf{elif}\;x \cdot 9 \leq -2.989941699386729 \cdot 10^{-141}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x \cdot y}{z \cdot c}\right) - 4 \cdot \left(\frac{t}{\sqrt[3]{c} \cdot \sqrt[3]{c}} \cdot \frac{a}{\sqrt[3]{c}}\right)\\ \mathbf{elif}\;x \cdot 9 \leq 1.2754336083427134 \cdot 10^{-101}:\\ \;\;\;\;\frac{\left(\frac{b}{z} + 9 \cdot \frac{x \cdot y}{z}\right) + \left(t \cdot a\right) \cdot -4}{c}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{c} \cdot \sqrt[3]{c}} \cdot \frac{\sqrt[3]{x}}{\frac{z}{\frac{y}{\sqrt[3]{c}}}}\right)\right) - 4 \cdot \frac{t \cdot a}{c}\\ \end{array}\]

Reproduce

herbie shell --seed 2020354 
(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)))