Average Error: 59.3 → 27.7
Time: 16.3s
Precision: binary64
\[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
↓
\[\begin{array}{l} \mathbf{if}\;M \leq -4.586460145529938 \cdot 10^{+201}:\\ \;\;\;\;0.25 \cdot \frac{e^{\log h + 2 \cdot \log \left(M \cdot D\right)}}{d \cdot d}\\ \mathbf{elif}\;M \leq 5.580625602124698 \cdot 10^{+144}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot \left(D \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot e^{\left(\log \left(D \cdot D\right) + \left(\log h + \left(\log M + \log M\right)\right)\right) - \log \left(d \cdot d\right)}\\ \end{array} \]
\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 -4.586460145529938 \cdot 10^{+201}:\\
\;\;\;\;0.25 \cdot \frac{e^{\log h + 2 \cdot \log \left(M \cdot D\right)}}{d \cdot d}\\

\mathbf{elif}\;M \leq 5.580625602124698 \cdot 10^{+144}:\\
\;\;\;\;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 e^{\left(\log \left(D \cdot D\right) + \left(\log h + \left(\log M + \log M\right)\right)\right) - \log \left(d \cdot d\right)}\\


\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 -4.586460145529938e+201)
   (* 0.25 (/ (exp (+ (log h) (* 2.0 (log (* M D))))) (* d d)))
   (if (<= M 5.580625602124698e+144)
     (* 0.25 (/ (* D (* D (/ (* h (* M M)) d))) d))
     (*
      0.25
      (exp
       (- (+ (log (* D D)) (+ (log h) (+ (log M) (log M)))) (log (* 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 <= -4.586460145529938e+201) {
		tmp = 0.25 * (exp(log(h) + (2.0 * log(M * D))) / (d * d));
	} else if (M <= 5.580625602124698e+144) {
		tmp = 0.25 * ((D * (D * ((h * (M * M)) / d))) / d);
	} else {
		tmp = 0.25 * exp((log(D * D) + (log(h) + (log(M) + log(M)))) - log(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 < -4.58646014552993779e201

    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 in c0 around -inf 64.0

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

    3. Simplified64.0

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

    4. Taylor expanded in c0 around 0 64.0

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

    5. Simplified64.0

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

    7. Applied add-exp-log_binary6464.0

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

    8. Applied add-exp-log_binary6464.0

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

    9. Applied add-exp-log_binary6464.0

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

    10. Applied prod-exp_binary6464.0

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

    11. Applied prod-exp_binary6464.0

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

    12. Applied add-exp-log_binary6464.0

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

    13. Applied add-exp-log_binary6464.0

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

    14. Applied prod-exp_binary6464.0

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

    15. Applied prod-exp_binary6464.0

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

    16. Simplified53.9

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

    if -4.58646014552993779e201 < M < 5.58062560212469821e144

    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 in c0 around -inf 39.1

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

    3. Simplified39.1

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

    4. Taylor expanded in c0 around 0 31.6

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

    5. Simplified31.6

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

    7. Applied associate-/r*_binary6428.8

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

    8. Simplified28.2

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

    10. Applied associate-*l*_binary6424.5

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

    if 5.58062560212469821e144 < 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 in c0 around -inf 63.1

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

    3. Simplified63.1

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

    4. Taylor expanded in c0 around 0 62.1

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

    5. Simplified62.1

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

    7. Applied add-exp-log_binary6462.1

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

    8. Applied add-exp-log_binary6463.5

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

    9. Applied add-exp-log_binary6463.5

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

    10. Applied add-exp-log_binary6463.5

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

    11. Applied prod-exp_binary6463.5

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

    12. Applied prod-exp_binary6457.7

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

    13. Applied add-exp-log_binary6457.7

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

    14. Applied prod-exp_binary6451.3

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

    15. Applied div-exp_binary6450.3

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

  4. Final simplification27.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -4.586460145529938 \cdot 10^{+201}:\\ \;\;\;\;0.25 \cdot \frac{e^{\log h + 2 \cdot \log \left(M \cdot D\right)}}{d \cdot d}\\ \mathbf{elif}\;M \leq 5.580625602124698 \cdot 10^{+144}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot \left(D \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot e^{\left(\log \left(D \cdot D\right) + \left(\log h + \left(\log M + \log M\right)\right)\right) - \log \left(d \cdot d\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2021197 
(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))))))