Henrywood and Agarwal, Equation (13)

Average Error: 60.0 → 30.7

Time: 20.2s

Precision: binary64

Math

\[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\]

↓

\[\begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq -1.364549136920434 \cdot 10^{-289}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq 0:\\ \;\;\;\;-0.25 \cdot \left(\left(\frac{\sqrt[3]{c0} \cdot \sqrt[3]{c0}}{\sqrt[3]{d} \cdot \sqrt[3]{d}} \cdot \left(\left(M \cdot M\right) \cdot \left(h \cdot \left(D \cdot D\right)\right)\right)\right) \cdot \frac{\sqrt[3]{c0}}{\frac{\sqrt[3]{d}}{\frac{\frac{-1}{c0}}{d}}}\right)\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \frac{c0}{\frac{d}{\frac{-\frac{M \cdot \left(M \cdot \left(h \cdot \left(D \cdot D\right)\right)\right)}{c0}}{d}}}\\ \end{array}\]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (*
  (/ c0 (* 2.0 w))
  (+
   (/ (* c0 (* d d)) (* (* w h) (* D D)))
   (sqrt
    (-
     (*
      (/ (* c0 (* d d)) (* (* w h) (* D D)))
      (/ (* c0 (* d d)) (* (* w h) (* D D))))
     (* M M))))))

↓

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<=
      (*
       (/ c0 (* 2.0 w))
       (+
        (/ (* c0 (* d d)) (* (* w h) (* D D)))
        (sqrt
         (-
          (*
           (/ (* c0 (* d d)) (* (* w h) (* D D)))
           (/ (* c0 (* d d)) (* (* w h) (* D D))))
          (* M M)))))
      -1.364549136920434e-289)
   (* (/ c0 (* 2.0 w)) (* 2.0 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<=
        (*
         (/ c0 (* 2.0 w))
         (+
          (/ (* c0 (* d d)) (* (* w h) (* D D)))
          (sqrt
           (-
            (*
             (/ (* c0 (* d d)) (* (* w h) (* D D)))
             (/ (* c0 (* d d)) (* (* w h) (* D D))))
            (* M M)))))
        0.0)
     (*
      -0.25
      (*
       (*
        (/ (* (cbrt c0) (cbrt c0)) (* (cbrt d) (cbrt d)))
        (* (* M M) (* h (* D D))))
       (/ (cbrt c0) (/ (cbrt d) (/ (/ -1.0 c0) d)))))
     (if (<=
          (*
           (/ c0 (* 2.0 w))
           (+
            (/ (* c0 (* d d)) (* (* w h) (* D D)))
            (sqrt
             (-
              (*
               (/ (* c0 (* d d)) (* (* w h) (* D D)))
               (/ (* c0 (* d d)) (* (* w h) (* D D))))
              (* M M)))))
          INFINITY)
       (* (/ c0 (* 2.0 w)) (* 2.0 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
       (* -0.25 (/ c0 (/ d (/ (- (/ (* M (* M (* h (* D D)))) c0)) d))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	return (c0 / (2.0 * w)) * (((c0 * (d * d)) / ((w * h) * (D * D))) + sqrt((((c0 * (d * d)) / ((w * h) * (D * D))) * ((c0 * (d * d)) / ((w * h) * (D * D)))) - (M * M)));
}

↓

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (((c0 / (2.0 * w)) * (((c0 * (d * d)) / ((w * h) * (D * D))) + sqrt((((c0 * (d * d)) / ((w * h) * (D * D))) * ((c0 * (d * d)) / ((w * h) * (D * D)))) - (M * M)))) <= -1.364549136920434e-289) {
		tmp = (c0 / (2.0 * w)) * (2.0 * ((c0 * (d * d)) / ((w * h) * (D * D))));
	} else if (((c0 / (2.0 * w)) * (((c0 * (d * d)) / ((w * h) * (D * D))) + sqrt((((c0 * (d * d)) / ((w * h) * (D * D))) * ((c0 * (d * d)) / ((w * h) * (D * D)))) - (M * M)))) <= 0.0) {
		tmp = -0.25 * ((((cbrt(c0) * cbrt(c0)) / (cbrt(d) * cbrt(d))) * ((M * M) * (h * (D * D)))) * (cbrt(c0) / (cbrt(d) / ((-1.0 / c0) / d))));
	} else if (((c0 / (2.0 * w)) * (((c0 * (d * d)) / ((w * h) * (D * D))) + sqrt((((c0 * (d * d)) / ((w * h) * (D * D))) * ((c0 * (d * d)) / ((w * h) * (D * D)))) - (M * M)))) <= ((double) INFINITY)) {
		tmp = (c0 / (2.0 * w)) * (2.0 * ((c0 * (d * d)) / ((w * h) * (D * D))));
	} else {
		tmp = -0.25 * (c0 / (d / (-((M * (M * (h * (D * D)))) / c0) / d)));
	}
	return tmp;
}

Error

Bits error versus c0

Bits error versus w

Bits error versus h

Bits error versus D

Bits error versus d

Bits error versus M

Derivation

1. Split input into 3 regimes

2. if (*.f64 (/.f64 c0 (*.f64 2 w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < -1.364549136920434e-289 or -0.0 < (*.f64 (/.f64 c0 (*.f64 2 w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < +inf.0

   1. Initial program 49.3

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\]

   2. Taylor expanded around 0 43.4

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(2 \cdot \frac{{d}^{2} \cdot c0}{w \cdot \left({D}^{2} \cdot h\right)}\right)}\]

   3. Simplified 43.2

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot 2\right)}\]

   if -1.364549136920434e-289 < (*.f64 (/.f64 c0 (*.f64 2 w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < -0.0

   1. Initial program 30.2

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\]

   2. Taylor expanded around -inf 35.3

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(-0.5 \cdot \frac{w \cdot \left(c0 \cdot \left({M}^{2} \cdot \left({D}^{2} \cdot h\right)\right)\right)}{{d}^{2}}\right)}\]

   3. Simplified 35.3

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(-0.5 \cdot \frac{w \cdot \left(c0 \cdot \left(\left(M \cdot M\right) \cdot \left(h \cdot \left(D \cdot D\right)\right)\right)\right)}{d \cdot d}\right)}\]

   4. Taylor expanded around 0 37.3

      \[\leadsto \color{blue}{-0.25 \cdot \frac{{c0}^{2} \cdot \left({M}^{2} \cdot \left({D}^{2} \cdot h\right)\right)}{{d}^{2}}}\]

   5. Simplified 33.3

      \[\leadsto \color{blue}{-0.25 \cdot \frac{c0}{\frac{d}{\frac{c0 \cdot \left(\left(h \cdot \left(D \cdot D\right)\right) \cdot \left(M \cdot M\right)\right)}{d}}}}\]

   6. Taylor expanded around -inf 24.8

      \[\leadsto -0.25 \cdot \frac{c0}{\frac{d}{\frac{\color{blue}{-1 \cdot \frac{{M}^{2} \cdot \left({D}^{2} \cdot h\right)}{c0}}}{d}}}\]

   7. Simplified 24.8

      \[\leadsto -0.25 \cdot \frac{c0}{\frac{d}{\frac{\color{blue}{-\frac{\left(h \cdot \left(D \cdot D\right)\right) \cdot \left(M \cdot M\right)}{c0}}}{d}}}\]
   8. Using strategy rm

   9. Applied *-un-lft-identity_binary64_1101 24.8

      \[\leadsto -0.25 \cdot \frac{c0}{\frac{d}{\frac{-\frac{\left(h \cdot \left(D \cdot D\right)\right) \cdot \left(M \cdot M\right)}{c0}}{\color{blue}{1 \cdot d}}}}\]

   10. Applied div-inv_binary64_1098 24.8

       \[\leadsto -0.25 \cdot \frac{c0}{\frac{d}{\frac{-\color{blue}{\left(\left(h \cdot \left(D \cdot D\right)\right) \cdot \left(M \cdot M\right)\right) \cdot \frac{1}{c0}}}{1 \cdot d}}}\]

   11. Applied distribute-rgt-neg-in_binary64_1059 24.8

       \[\leadsto -0.25 \cdot \frac{c0}{\frac{d}{\frac{\color{blue}{\left(\left(h \cdot \left(D \cdot D\right)\right) \cdot \left(M \cdot M\right)\right) \cdot \left(-\frac{1}{c0}\right)}}{1 \cdot d}}}\]

   12. Applied times-frac_binary64_1107 24.4

       \[\leadsto -0.25 \cdot \frac{c0}{\frac{d}{\color{blue}{\frac{\left(h \cdot \left(D \cdot D\right)\right) \cdot \left(M \cdot M\right)}{1} \cdot \frac{-\frac{1}{c0}}{d}}}}\]

   13. Applied add-cube-cbrt_binary64_1136 24.5

       \[\leadsto -0.25 \cdot \frac{c0}{\frac{\color{blue}{\left(\sqrt[3]{d} \cdot \sqrt[3]{d}\right) \cdot \sqrt[3]{d}}}{\frac{\left(h \cdot \left(D \cdot D\right)\right) \cdot \left(M \cdot M\right)}{1} \cdot \frac{-\frac{1}{c0}}{d}}}\]

   14. Applied times-frac_binary64_1107 23.9

       \[\leadsto -0.25 \cdot \frac{c0}{\color{blue}{\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\frac{\left(h \cdot \left(D \cdot D\right)\right) \cdot \left(M \cdot M\right)}{1}} \cdot \frac{\sqrt[3]{d}}{\frac{-\frac{1}{c0}}{d}}}}\]

   15. Applied add-cube-cbrt_binary64_1136 24.0

       \[\leadsto -0.25 \cdot \frac{\color{blue}{\left(\sqrt[3]{c0} \cdot \sqrt[3]{c0}\right) \cdot \sqrt[3]{c0}}}{\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\frac{\left(h \cdot \left(D \cdot D\right)\right) \cdot \left(M \cdot M\right)}{1}} \cdot \frac{\sqrt[3]{d}}{\frac{-\frac{1}{c0}}{d}}}\]

   16. Applied times-frac_binary64_1107 23.8

       \[\leadsto -0.25 \cdot \color{blue}{\left(\frac{\sqrt[3]{c0} \cdot \sqrt[3]{c0}}{\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\frac{\left(h \cdot \left(D \cdot D\right)\right) \cdot \left(M \cdot M\right)}{1}}} \cdot \frac{\sqrt[3]{c0}}{\frac{\sqrt[3]{d}}{\frac{-\frac{1}{c0}}{d}}}\right)}\]

   17. Simplified 24.0

       \[\leadsto -0.25 \cdot \left(\color{blue}{\left(\frac{\sqrt[3]{c0} \cdot \sqrt[3]{c0}}{\sqrt[3]{d} \cdot \sqrt[3]{d}} \cdot \left(\left(h \cdot \left(D \cdot D\right)\right) \cdot \left(M \cdot M\right)\right)\right)} \cdot \frac{\sqrt[3]{c0}}{\frac{\sqrt[3]{d}}{\frac{-\frac{1}{c0}}{d}}}\right)\]

   18. Simplified 24.0

       \[\leadsto -0.25 \cdot \left(\left(\frac{\sqrt[3]{c0} \cdot \sqrt[3]{c0}}{\sqrt[3]{d} \cdot \sqrt[3]{d}} \cdot \left(\left(h \cdot \left(D \cdot D\right)\right) \cdot \left(M \cdot M\right)\right)\right) \cdot \color{blue}{\frac{\sqrt[3]{c0}}{\frac{\sqrt[3]{d}}{\frac{\frac{-1}{c0}}{d}}}}\right)\]

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 2 w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M)))))

   1. Initial program 64.0

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\]

   2. Taylor expanded around -inf 40.6

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(-0.5 \cdot \frac{w \cdot \left(c0 \cdot \left({M}^{2} \cdot \left({D}^{2} \cdot h\right)\right)\right)}{{d}^{2}}\right)}\]

   3. Simplified 40.6

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(-0.5 \cdot \frac{w \cdot \left(c0 \cdot \left(\left(M \cdot M\right) \cdot \left(h \cdot \left(D \cdot D\right)\right)\right)\right)}{d \cdot d}\right)}\]

   4. Taylor expanded around 0 43.7

      \[\leadsto \color{blue}{-0.25 \cdot \frac{{c0}^{2} \cdot \left({M}^{2} \cdot \left({D}^{2} \cdot h\right)\right)}{{d}^{2}}}\]

   5. Simplified 34.7

      \[\leadsto \color{blue}{-0.25 \cdot \frac{c0}{\frac{d}{\frac{c0 \cdot \left(\left(h \cdot \left(D \cdot D\right)\right) \cdot \left(M \cdot M\right)\right)}{d}}}}\]

   6. Taylor expanded around -inf 32.2

      \[\leadsto -0.25 \cdot \frac{c0}{\frac{d}{\frac{\color{blue}{-1 \cdot \frac{{M}^{2} \cdot \left({D}^{2} \cdot h\right)}{c0}}}{d}}}\]

   7. Simplified 32.2

      \[\leadsto -0.25 \cdot \frac{c0}{\frac{d}{\frac{\color{blue}{-\frac{\left(h \cdot \left(D \cdot D\right)\right) \cdot \left(M \cdot M\right)}{c0}}}{d}}}\]
   8. Using strategy rm

   9. Applied associate-*r*_binary64_1041 29.3

      \[\leadsto -0.25 \cdot \frac{c0}{\frac{d}{\frac{-\frac{\color{blue}{\left(\left(h \cdot \left(D \cdot D\right)\right) \cdot M\right) \cdot M}}{c0}}{d}}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 30.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq -1.364549136920434 \cdot 10^{-289}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq 0:\\ \;\;\;\;-0.25 \cdot \left(\left(\frac{\sqrt[3]{c0} \cdot \sqrt[3]{c0}}{\sqrt[3]{d} \cdot \sqrt[3]{d}} \cdot \left(\left(M \cdot M\right) \cdot \left(h \cdot \left(D \cdot D\right)\right)\right)\right) \cdot \frac{\sqrt[3]{c0}}{\frac{\sqrt[3]{d}}{\frac{\frac{-1}{c0}}{d}}}\right)\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \frac{c0}{\frac{d}{\frac{-\frac{M \cdot \left(M \cdot \left(h \cdot \left(D \cdot D\right)\right)\right)}{c0}}{d}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020344 
(FPCore (c0 w h D d M)
  :name "Henrywood and Agarwal, Equation (13)"
  :precision binary64
  (* (/ c0 (* 2.0 w)) (+ (/ (* c0 (* d d)) (* (* w h) (* D D))) (sqrt (- (* (/ (* c0 (* d d)) (* (* w h) (* D D))) (/ (* c0 (* d d)) (* (* w h) (* D D)))) (* M M))))))