Average Error: 59.5 → 27.6
Time: 20.2s
Precision: binary64
\[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
↓
\[\begin{array}{l} \mathbf{if}\;M \leq -6.359698016561584 \cdot 10^{+130}:\\ \;\;\;\;0.25 \cdot \frac{\left(D \cdot D\right) \cdot \frac{M \cdot \left(M \cdot h\right)}{d}}{d}\\ \mathbf{elif}\;M \leq 4.1923584058765576 \cdot 10^{+125}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot \left(D \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{e^{2 \cdot \log \left(M \cdot D\right) + \log \left(\frac{h}{d}\right)}}{d}\\ \end{array} \]
\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}\;M \leq -6.359698016561584 \cdot 10^{+130}:\\
\;\;\;\;0.25 \cdot \frac{\left(D \cdot D\right) \cdot \frac{M \cdot \left(M \cdot h\right)}{d}}{d}\\

\mathbf{elif}\;M \leq 4.1923584058765576 \cdot 10^{+125}:\\
\;\;\;\;0.25 \cdot \frac{D \cdot \left(D \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right)}{d}\\

\mathbf{else}:\\
\;\;\;\;0.25 \cdot \frac{e^{2 \cdot \log \left(M \cdot D\right) + \log \left(\frac{h}{d}\right)}}{d}\\


\end{array}

(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 (<= M -6.359698016561584e+130)
   (* 0.25 (/ (* (* D D) (/ (* M (* M h)) d)) d))
   (if (<= M 4.1923584058765576e+125)
     (* 0.25 (/ (* D (* D (/ (* h (* M M)) d))) d))
     (* 0.25 (/ (exp (+ (* 2.0 (log (* M D))) (log (/ h d)))) d)))))
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 (M <= -6.359698016561584e+130) {
		tmp = 0.25 * (((D * D) * ((M * (M * h)) / d)) / d);
	} else if (M <= 4.1923584058765576e+125) {
		tmp = 0.25 * ((D * (D * ((h * (M * M)) / d))) / d);
	} else {
		tmp = 0.25 * (exp((2.0 * log(M * D)) + log(h / d)) / 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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if M < -6.35969801656158433e130

    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 61.6

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

    3. Simplified61.6

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

    4. Taylor expanded around 0 61.1

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

    5. Simplified61.1

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

    7. Applied associate-/r*_binary6460.5

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

    8. Simplified60.5

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

    10. Applied associate-*r*_binary6447.2

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

    if -6.35969801656158433e130 < M < 4.19235840587655755e125

    1. Initial program 58.6

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

    2. Taylor expanded around -inf 38.4

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

    3. Simplified40.6

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

    4. Taylor expanded around 0 30.8

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

    5. Simplified30.8

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

    7. Applied associate-/r*_binary6427.8

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

    8. Simplified27.3

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

    10. Applied associate-*l*_binary6423.0

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

    if 4.19235840587655755e125 < 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 60.9

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

    3. Simplified60.9

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

    4. Taylor expanded around 0 59.2

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

    5. Simplified59.2

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

    7. Applied associate-/r*_binary6458.7

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

    8. Simplified58.4

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

    10. Applied add-exp-log_binary6460.8

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

    11. Applied add-exp-log_binary6460.8

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

    12. Applied add-exp-log_binary6460.8

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

    13. Applied prod-exp_binary6460.8

      \[\leadsto 0.25 \cdot \frac{\left(D \cdot D\right) \cdot \frac{h \cdot \color{blue}{e^{\log M + \log M}}}{e^{\log d}}}{d} \]

    14. Applied add-exp-log_binary6462.9

      \[\leadsto 0.25 \cdot \frac{\left(D \cdot D\right) \cdot \frac{\color{blue}{e^{\log h}} \cdot e^{\log M + \log M}}{e^{\log d}}}{d} \]

    15. Applied prod-exp_binary6459.1

      \[\leadsto 0.25 \cdot \frac{\left(D \cdot D\right) \cdot \frac{\color{blue}{e^{\log h + \left(\log M + \log M\right)}}}{e^{\log d}}}{d} \]

    16. Applied div-exp_binary6457.4

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

    17. Applied add-exp-log_binary6461.6

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

    18. Applied add-exp-log_binary6461.6

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

    19. Applied prod-exp_binary6461.6

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

    20. Applied prod-exp_binary6459.9

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

    21. Simplified53.6

      \[\leadsto 0.25 \cdot \frac{e^{\color{blue}{2 \cdot \log \left(D \cdot M\right) + \log \left(\frac{h}{d}\right)}}}{d} \]
  3. Recombined 3 regimes into one program.

  4. Final simplification27.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -6.359698016561584 \cdot 10^{+130}:\\ \;\;\;\;0.25 \cdot \frac{\left(D \cdot D\right) \cdot \frac{M \cdot \left(M \cdot h\right)}{d}}{d}\\ \mathbf{elif}\;M \leq 4.1923584058765576 \cdot 10^{+125}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot \left(D \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{e^{2 \cdot \log \left(M \cdot D\right) + \log \left(\frac{h}{d}\right)}}{d}\\ \end{array} \]

Reproduce

herbie shell --seed 2021202 
(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))))))