Average Error: 59.5 → 26.9
Time: 18.7s
Precision: binary64
\[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
↓
\[\begin{array}{l} \mathbf{if}\;M \leq -2.5521439251508326 \cdot 10^{+127}:\\ \;\;\;\;0.25 \cdot \frac{\frac{e^{\mathsf{fma}\left(2, \log \left(M \cdot D\right), \log h\right)}}{d}}{d}\\ \mathbf{elif}\;M \leq 3.4252903958142705 \cdot 10^{+75}:\\ \;\;\;\;0.25 \cdot \left(\frac{D}{d} \cdot \frac{D \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{\frac{D \cdot \left(D \cdot \left(M \cdot \left(M \cdot h\right)\right)\right)}{d}}{d}\\ \end{array} \]
\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 -2.5521439251508326 \cdot 10^{+127}:\\
\;\;\;\;0.25 \cdot \frac{\frac{e^{\mathsf{fma}\left(2, \log \left(M \cdot D\right), \log h\right)}}{d}}{d}\\

\mathbf{elif}\;M \leq 3.4252903958142705 \cdot 10^{+75}:\\
\;\;\;\;0.25 \cdot \left(\frac{D}{d} \cdot \frac{D \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d}\right)\\

\mathbf{else}:\\
\;\;\;\;0.25 \cdot \frac{\frac{D \cdot \left(D \cdot \left(M \cdot \left(M \cdot h\right)\right)\right)}{d}}{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 -2.5521439251508326e+127)
   (* 0.25 (/ (/ (exp (fma 2.0 (log (* M D)) (log h))) d) d))
   (if (<= M 3.4252903958142705e+75)
     (* 0.25 (* (/ D d) (/ (* D (* h (* M M))) d)))
     (* 0.25 (/ (/ (* D (* D (* M (* M 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 <= -2.5521439251508326e+127) {
		tmp = 0.25 * ((exp(fma(2.0, log(M * D), log(h))) / d) / d);
	} else if (M <= 3.4252903958142705e+75) {
		tmp = 0.25 * ((D / d) * ((D * (h * (M * M))) / d));
	} else {
		tmp = 0.25 * (((D * (D * (M * (M * 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

Derivation

  1. Split input into 3 regimes

  2. if M < -2.5521439251508326e127

    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 59.9

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

    3. Applied unpow2_binary6459.9

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

    4. Applied associate-/r*_binary6459.5

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

    5. Applied add-exp-log_binary6461.7

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

    6. Applied pow-to-exp_binary6464.0

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

    7. Applied prod-exp_binary6464.0

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

    8. Applied pow-to-exp_binary6464.0

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

    9. Applied prod-exp_binary6464.0

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

    10. Simplified55.2

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

    if -2.5521439251508326e127 < M < 3.42529039581427048e75

    1. Initial program 58.4

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

    2. Taylor expanded in c0 around -inf 30.5

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

    3. Applied unpow2_binary6430.5

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

    4. Applied associate-/r*_binary6427.5

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

    5. Applied unpow2_binary6427.5

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

    6. Applied associate-*l*_binary6423.7

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

    7. Simplified23.7

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

    8. Applied add-sqr-sqrt_binary6443.5

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

    9. Applied add-sqr-sqrt_binary6443.6

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

    10. Applied times-frac_binary6442.6

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

    11. Applied times-frac_binary6442.5

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

    12. Simplified42.5

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

    13. Simplified21.6

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

    if 3.42529039581427048e75 < M

    1. Initial program 63.5

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

    2. Taylor expanded in c0 around -inf 53.5

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

    3. Applied unpow2_binary6453.5

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

    4. Applied associate-/r*_binary6452.5

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

    5. Applied unpow2_binary6452.5

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

    6. Applied associate-*l*_binary6451.3

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

    7. Simplified51.3

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

    8. Applied associate-*r*_binary6441.7

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

  4. Final simplification26.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -2.5521439251508326 \cdot 10^{+127}:\\ \;\;\;\;0.25 \cdot \frac{\frac{e^{\mathsf{fma}\left(2, \log \left(M \cdot D\right), \log h\right)}}{d}}{d}\\ \mathbf{elif}\;M \leq 3.4252903958142705 \cdot 10^{+75}:\\ \;\;\;\;0.25 \cdot \left(\frac{D}{d} \cdot \frac{D \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{\frac{D \cdot \left(D \cdot \left(M \cdot \left(M \cdot h\right)\right)\right)}{d}}{d}\\ \end{array} \]

Reproduce

herbie shell --seed 2022063 
(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))))))