Henrywood and Agarwal, Equation (13)

Average Error: 59.6 → 21.3

Time: 28.2s

Precision: binary64

Cost: 1356

Math

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

↓

\[\begin{array}{l} t_0 := \frac{0.25}{d} \cdot \left(\frac{D}{\frac{d}{M}} \cdot \left(M \cdot \left(D \cdot h\right)\right)\right)\\ \mathbf{if}\;D \leq -2 \cdot 10^{-304}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;D \leq 5 \cdot 10^{-248}:\\ \;\;\;\;\frac{M \cdot \left(D \cdot \left(M \cdot h\right)\right)}{\left(d \cdot 4\right) \cdot \frac{d}{D}}\\ \mathbf{elif}\;D \leq 4.3 \cdot 10^{-45}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left(D \cdot \frac{M}{\frac{d}{D}}\right) \cdot \frac{M}{\frac{d \cdot 4}{h}}\\ \end{array} \]

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
 (let* ((t_0 (* (/ 0.25 d) (* (/ D (/ d M)) (* M (* D h))))))
   (if (<= D -2e-304)
     t_0
     (if (<= D 5e-248)
       (/ (* M (* D (* M h))) (* (* d 4.0) (/ d D)))
       (if (<= D 4.3e-45)
         t_0
         (* (* D (/ M (/ d D))) (/ M (/ (* d 4.0) h))))))))

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 t_0 = (0.25 / d) * ((D / (d / M)) * (M * (D * h)));
	double tmp;
	if (D <= -2e-304) {
		tmp = t_0;
	} else if (D <= 5e-248) {
		tmp = (M * (D * (M * h))) / ((d * 4.0) * (d / D));
	} else if (D <= 4.3e-45) {
		tmp = t_0;
	} else {
		tmp = (D * (M / (d / D))) * (M / ((d * 4.0) / h));
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    code = (c0 / (2.0d0 * w)) * (((c0 * (d_1 * d_1)) / ((w * h) * (d * d))) + sqrt(((((c0 * (d_1 * d_1)) / ((w * h) * (d * d))) * ((c0 * (d_1 * d_1)) / ((w * h) * (d * d)))) - (m * m))))
end function

↓

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (0.25d0 / d_1) * ((d / (d_1 / m)) * (m * (d * h)))
    if (d <= (-2d-304)) then
        tmp = t_0
    else if (d <= 5d-248) then
        tmp = (m * (d * (m * h))) / ((d_1 * 4.0d0) * (d_1 / d))
    else if (d <= 4.3d-45) then
        tmp = t_0
    else
        tmp = (d * (m / (d_1 / d))) * (m / ((d_1 * 4.0d0) / h))
    end if
    code = tmp
end function

Java

public static 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))) + Math.sqrt(((((c0 * (d * d)) / ((w * h) * (D * D))) * ((c0 * (d * d)) / ((w * h) * (D * D)))) - (M * M))));
}

↓

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (0.25 / d) * ((D / (d / M)) * (M * (D * h)));
	double tmp;
	if (D <= -2e-304) {
		tmp = t_0;
	} else if (D <= 5e-248) {
		tmp = (M * (D * (M * h))) / ((d * 4.0) * (d / D));
	} else if (D <= 4.3e-45) {
		tmp = t_0;
	} else {
		tmp = (D * (M / (d / D))) * (M / ((d * 4.0) / h));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	return (c0 / (2.0 * w)) * (((c0 * (d * d)) / ((w * h) * (D * D))) + math.sqrt(((((c0 * (d * d)) / ((w * h) * (D * D))) * ((c0 * (d * d)) / ((w * h) * (D * D)))) - (M * M))))

↓

def code(c0, w, h, D, d, M):
	t_0 = (0.25 / d) * ((D / (d / M)) * (M * (D * h)))
	tmp = 0
	if D <= -2e-304:
		tmp = t_0
	elif D <= 5e-248:
		tmp = (M * (D * (M * h))) / ((d * 4.0) * (d / D))
	elif D <= 4.3e-45:
		tmp = t_0
	else:
		tmp = (D * (M / (d / D))) * (M / ((d * 4.0) / h))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	return Float64(Float64(c0 / Float64(2.0 * w)) * Float64(Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D))) + sqrt(Float64(Float64(Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D))) * Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))) - Float64(M * M)))))
end

↓

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(0.25 / d) * Float64(Float64(D / Float64(d / M)) * Float64(M * Float64(D * h))))
	tmp = 0.0
	if (D <= -2e-304)
		tmp = t_0;
	elseif (D <= 5e-248)
		tmp = Float64(Float64(M * Float64(D * Float64(M * h))) / Float64(Float64(d * 4.0) * Float64(d / D)));
	elseif (D <= 4.3e-45)
		tmp = t_0;
	else
		tmp = Float64(Float64(D * Float64(M / Float64(d / D))) * Float64(M / Float64(Float64(d * 4.0) / h)));
	end
	return tmp
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	tmp = (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))));
end

↓

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (0.25 / d) * ((D / (d / M)) * (M * (D * h)));
	tmp = 0.0;
	if (D <= -2e-304)
		tmp = t_0;
	elseif (D <= 5e-248)
		tmp = (M * (D * (M * h))) / ((d * 4.0) * (d / D));
	elseif (D <= 4.3e-45)
		tmp = t_0;
	else
		tmp = (D * (M / (d / D))) * (M / ((d * 4.0) / h));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[(N[(N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(0.25 / d), $MachinePrecision] * N[(N[(D / N[(d / M), $MachinePrecision]), $MachinePrecision] * N[(M * N[(D * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[D, -2e-304], t$95$0, If[LessEqual[D, 5e-248], N[(N[(M * N[(D * N[(M * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(d * 4.0), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[D, 4.3e-45], t$95$0, N[(N[(D * N[(M / N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(M / N[(N[(d * 4.0), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if D < -1.99999999999999994e-304 or 5.0000000000000001e-248 < D < 4.2999999999999999e-45

   1. Initial program 59.7

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

   2. Simplified 62.0

      \[\leadsto \color{blue}{\frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{c0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}, \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{c0}{w \cdot h} \cdot \left(\frac{d}{D} \cdot {\left(\frac{d}{D}\right)}^{3}\right), -M \cdot M\right)}\right)} \]
      Proof

      [Start] 59.7	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      associate-*l/ [<=] 60.2	\[ \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \left(d \cdot d\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      *-commutative [=>] 60.2	\[ \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\left(d \cdot d\right) \cdot \frac{c0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      fma-def [=>] 60.8	\[ \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(d \cdot d, \frac{c0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)} \]

   3. Taylor expanded in c0 around -inf 59.7

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(0.5 \cdot \frac{{D}^{2} \cdot \left(w \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot c0} + -1 \cdot \left(\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot c0\right)\right)} \]

   4. Simplified 36.3

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(0.5, \frac{\frac{w \cdot \left(h \cdot \left(M \cdot M\right)\right)}{{\left(\frac{d}{D}\right)}^{2}}}{c0}, c0 \cdot 0\right)} \]
      Proof

      [Start] 59.7	\[ \frac{c0}{2 \cdot w} \cdot \left(0.5 \cdot \frac{{D}^{2} \cdot \left(w \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot c0} + -1 \cdot \left(\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot c0\right)\right) \]
      fma-def [=>] 59.7	\[ \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(0.5, \frac{{D}^{2} \cdot \left(w \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot c0}, -1 \cdot \left(\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot c0\right)\right)} \]

   5. Taylor expanded in c0 around 0 34.9

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

   6. Simplified 30.1

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

      [Start] 34.9	\[ 0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2}} \]
      associate-*r/ [=>] 34.9	\[ \color{blue}{\frac{0.25 \cdot \left({D}^{2} \cdot \left(h \cdot {M}^{2}\right)\right)}{{d}^{2}}} \]
      *-commutative [=>] 34.9	\[ \frac{0.25 \cdot \left({D}^{2} \cdot \color{blue}{\left({M}^{2} \cdot h\right)}\right)}{{d}^{2}} \]
      unpow2 [=>] 34.9	\[ \frac{0.25 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{d \cdot d}} \]
      times-frac [=>] 32.3	\[ \color{blue}{\frac{0.25}{d} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d}} \]
      unpow2 [=>] 32.3	\[ \frac{0.25}{d} \cdot \frac{\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)}{d} \]
      unpow2 [=>] 32.3	\[ \frac{0.25}{d} \cdot \frac{\left(D \cdot D\right) \cdot \left(\color{blue}{\left(M \cdot M\right)} \cdot h\right)}{d} \]
      *-commutative [<=] 32.3	\[ \frac{0.25}{d} \cdot \frac{\left(D \cdot D\right) \cdot \color{blue}{\left(h \cdot \left(M \cdot M\right)\right)}}{d} \]
      *-commutative [<=] 32.3	\[ \frac{0.25}{d} \cdot \frac{\color{blue}{\left(h \cdot \left(M \cdot M\right)\right) \cdot \left(D \cdot D\right)}}{d} \]
      associate-*r* [=>] 30.1	\[ \frac{0.25}{d} \cdot \frac{\color{blue}{\left(\left(h \cdot M\right) \cdot M\right)} \cdot \left(D \cdot D\right)}{d} \]
      *-commutative [=>] 30.1	\[ \frac{0.25}{d} \cdot \frac{\color{blue}{\left(M \cdot \left(h \cdot M\right)\right)} \cdot \left(D \cdot D\right)}{d} \]

   7. Taylor expanded in M around 0 32.3

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

   8. Simplified 21.0

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

      [Start] 32.3	\[ \frac{0.25}{d} \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{d} \]
      unpow2 [=>] 32.3	\[ \frac{0.25}{d} \cdot \frac{\color{blue}{\left(D \cdot D\right)} \cdot \left(h \cdot {M}^{2}\right)}{d} \]
      *-commutative [<=] 32.3	\[ \frac{0.25}{d} \cdot \frac{\left(D \cdot D\right) \cdot \color{blue}{\left({M}^{2} \cdot h\right)}}{d} \]
      unpow2 [=>] 32.3	\[ \frac{0.25}{d} \cdot \frac{\left(D \cdot D\right) \cdot \left(\color{blue}{\left(M \cdot M\right)} \cdot h\right)}{d} \]
      associate-*r* [<=] 30.1	\[ \frac{0.25}{d} \cdot \frac{\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot \left(M \cdot h\right)\right)}}{d} \]
      associate-/l* [=>] 29.4	\[ \frac{0.25}{d} \cdot \color{blue}{\frac{D \cdot D}{\frac{d}{M \cdot \left(M \cdot h\right)}}} \]
      associate-/l* [=>] 26.2	\[ \frac{0.25}{d} \cdot \color{blue}{\frac{D}{\frac{\frac{d}{M \cdot \left(M \cdot h\right)}}{D}}} \]
      *-lft-identity [<=] 26.2	\[ \frac{0.25}{d} \cdot \frac{\color{blue}{1 \cdot D}}{\frac{\frac{d}{M \cdot \left(M \cdot h\right)}}{D}} \]
      associate-/r* [<=] 25.1	\[ \frac{0.25}{d} \cdot \frac{1 \cdot D}{\color{blue}{\frac{d}{\left(M \cdot \left(M \cdot h\right)\right) \cdot D}}} \]
      associate-*r* [<=] 22.5	\[ \frac{0.25}{d} \cdot \frac{1 \cdot D}{\frac{d}{\color{blue}{M \cdot \left(\left(M \cdot h\right) \cdot D\right)}}} \]
      *-commutative [<=] 22.5	\[ \frac{0.25}{d} \cdot \frac{\color{blue}{D \cdot 1}}{\frac{d}{M \cdot \left(\left(M \cdot h\right) \cdot D\right)}} \]
      associate-/r* [=>] 22.0	\[ \frac{0.25}{d} \cdot \frac{D \cdot 1}{\color{blue}{\frac{\frac{d}{M}}{\left(M \cdot h\right) \cdot D}}} \]
      associate-/r/ [=>] 21.8	\[ \frac{0.25}{d} \cdot \color{blue}{\left(\frac{D \cdot 1}{\frac{d}{M}} \cdot \left(\left(M \cdot h\right) \cdot D\right)\right)} \]
      *-rgt-identity [=>] 21.8	\[ \frac{0.25}{d} \cdot \left(\frac{\color{blue}{D}}{\frac{d}{M}} \cdot \left(\left(M \cdot h\right) \cdot D\right)\right) \]
      associate-*l* [=>] 21.0	\[ \frac{0.25}{d} \cdot \left(\frac{D}{\frac{d}{M}} \cdot \color{blue}{\left(M \cdot \left(h \cdot D\right)\right)}\right) \]
      *-commutative [=>] 21.0	\[ \frac{0.25}{d} \cdot \left(\frac{D}{\frac{d}{M}} \cdot \left(M \cdot \color{blue}{\left(D \cdot h\right)}\right)\right) \]

   if -1.99999999999999994e-304 < D < 5.0000000000000001e-248

   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. Simplified 64.0

      \[\leadsto \color{blue}{\frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{c0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}, \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{c0}{w \cdot h} \cdot \left(\frac{d}{D} \cdot {\left(\frac{d}{D}\right)}^{3}\right), -M \cdot M\right)}\right)} \]
      Proof

      [Start] 64.0	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      associate-*l/ [<=] 64.0	\[ \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \left(d \cdot d\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      *-commutative [=>] 64.0	\[ \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\left(d \cdot d\right) \cdot \frac{c0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      fma-def [=>] 64.0	\[ \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(d \cdot d, \frac{c0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)} \]

   3. Taylor expanded in c0 around -inf 64.0

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(0.5 \cdot \frac{{D}^{2} \cdot \left(w \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot c0} + -1 \cdot \left(\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot c0\right)\right)} \]

   4. Simplified 31.2

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(0.5, \frac{\frac{w \cdot \left(h \cdot \left(M \cdot M\right)\right)}{{\left(\frac{d}{D}\right)}^{2}}}{c0}, c0 \cdot 0\right)} \]
      Proof

      [Start] 64.0	\[ \frac{c0}{2 \cdot w} \cdot \left(0.5 \cdot \frac{{D}^{2} \cdot \left(w \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot c0} + -1 \cdot \left(\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot c0\right)\right) \]
      fma-def [=>] 64.0	\[ \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(0.5, \frac{{D}^{2} \cdot \left(w \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot c0}, -1 \cdot \left(\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot c0\right)\right)} \]

   5. Taylor expanded in c0 around 0 33.6

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

   6. Simplified 26.5

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

      [Start] 33.6	\[ 0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2}} \]
      associate-*r/ [=>] 33.6	\[ \color{blue}{\frac{0.25 \cdot \left({D}^{2} \cdot \left(h \cdot {M}^{2}\right)\right)}{{d}^{2}}} \]
      *-commutative [=>] 33.6	\[ \frac{0.25 \cdot \left({D}^{2} \cdot \color{blue}{\left({M}^{2} \cdot h\right)}\right)}{{d}^{2}} \]
      unpow2 [=>] 33.6	\[ \frac{0.25 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{d \cdot d}} \]
      times-frac [=>] 28.2	\[ \color{blue}{\frac{0.25}{d} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d}} \]
      unpow2 [=>] 28.2	\[ \frac{0.25}{d} \cdot \frac{\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)}{d} \]
      unpow2 [=>] 28.2	\[ \frac{0.25}{d} \cdot \frac{\left(D \cdot D\right) \cdot \left(\color{blue}{\left(M \cdot M\right)} \cdot h\right)}{d} \]
      *-commutative [<=] 28.2	\[ \frac{0.25}{d} \cdot \frac{\left(D \cdot D\right) \cdot \color{blue}{\left(h \cdot \left(M \cdot M\right)\right)}}{d} \]
      *-commutative [<=] 28.2	\[ \frac{0.25}{d} \cdot \frac{\color{blue}{\left(h \cdot \left(M \cdot M\right)\right) \cdot \left(D \cdot D\right)}}{d} \]
      associate-*r* [=>] 26.5	\[ \frac{0.25}{d} \cdot \frac{\color{blue}{\left(\left(h \cdot M\right) \cdot M\right)} \cdot \left(D \cdot D\right)}{d} \]
      *-commutative [=>] 26.5	\[ \frac{0.25}{d} \cdot \frac{\color{blue}{\left(M \cdot \left(h \cdot M\right)\right)} \cdot \left(D \cdot D\right)}{d} \]

   7. Applied egg-rr 18.4

      \[\leadsto \color{blue}{\frac{M \cdot \left(\left(M \cdot h\right) \cdot D\right)}{\left(d \cdot 4\right) \cdot \frac{d}{D}}} \]

   if 4.2999999999999999e-45 < D

   1. Initial program 56.9

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

   2. Simplified 60.0

      \[\leadsto \color{blue}{\frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{c0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}, \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{c0}{w \cdot h} \cdot \left(\frac{d}{D} \cdot {\left(\frac{d}{D}\right)}^{3}\right), -M \cdot M\right)}\right)} \]
      Proof

      [Start] 56.9	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      associate-*l/ [<=] 57.5	\[ \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \left(d \cdot d\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      *-commutative [=>] 57.5	\[ \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\left(d \cdot d\right) \cdot \frac{c0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      fma-def [=>] 58.3	\[ \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(d \cdot d, \frac{c0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)} \]

   3. Taylor expanded in c0 around -inf 57.7

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(0.5 \cdot \frac{{D}^{2} \cdot \left(w \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot c0} + -1 \cdot \left(\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot c0\right)\right)} \]

   4. Simplified 38.4

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(0.5, \frac{\frac{w \cdot \left(h \cdot \left(M \cdot M\right)\right)}{{\left(\frac{d}{D}\right)}^{2}}}{c0}, c0 \cdot 0\right)} \]
      Proof

      [Start] 57.7	\[ \frac{c0}{2 \cdot w} \cdot \left(0.5 \cdot \frac{{D}^{2} \cdot \left(w \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot c0} + -1 \cdot \left(\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot c0\right)\right) \]
      fma-def [=>] 57.7	\[ \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(0.5, \frac{{D}^{2} \cdot \left(w \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot c0}, -1 \cdot \left(\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot c0\right)\right)} \]

   5. Taylor expanded in c0 around 0 40.5

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

   6. Simplified 37.2

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

      [Start] 40.5	\[ 0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2}} \]
      associate-*r/ [=>] 40.5	\[ \color{blue}{\frac{0.25 \cdot \left({D}^{2} \cdot \left(h \cdot {M}^{2}\right)\right)}{{d}^{2}}} \]
      *-commutative [=>] 40.5	\[ \frac{0.25 \cdot \left({D}^{2} \cdot \color{blue}{\left({M}^{2} \cdot h\right)}\right)}{{d}^{2}} \]
      unpow2 [=>] 40.5	\[ \frac{0.25 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{d \cdot d}} \]
      times-frac [=>] 38.8	\[ \color{blue}{\frac{0.25}{d} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d}} \]
      unpow2 [=>] 38.8	\[ \frac{0.25}{d} \cdot \frac{\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)}{d} \]
      unpow2 [=>] 38.8	\[ \frac{0.25}{d} \cdot \frac{\left(D \cdot D\right) \cdot \left(\color{blue}{\left(M \cdot M\right)} \cdot h\right)}{d} \]
      *-commutative [<=] 38.8	\[ \frac{0.25}{d} \cdot \frac{\left(D \cdot D\right) \cdot \color{blue}{\left(h \cdot \left(M \cdot M\right)\right)}}{d} \]
      *-commutative [<=] 38.8	\[ \frac{0.25}{d} \cdot \frac{\color{blue}{\left(h \cdot \left(M \cdot M\right)\right) \cdot \left(D \cdot D\right)}}{d} \]
      associate-*r* [=>] 37.2	\[ \frac{0.25}{d} \cdot \frac{\color{blue}{\left(\left(h \cdot M\right) \cdot M\right)} \cdot \left(D \cdot D\right)}{d} \]
      *-commutative [=>] 37.2	\[ \frac{0.25}{d} \cdot \frac{\color{blue}{\left(M \cdot \left(h \cdot M\right)\right)} \cdot \left(D \cdot D\right)}{d} \]

   7. Applied egg-rr 35.4

      \[\leadsto \color{blue}{\frac{M}{\left(d \cdot 4\right) \cdot \frac{d}{M \cdot \left(h \cdot \left(D \cdot D\right)\right)}}} \]

   8. Simplified 24.2

      \[\leadsto \color{blue}{\left(\frac{M}{\frac{d}{D}} \cdot D\right) \cdot \frac{M}{\frac{d \cdot 4}{h}}} \]
      Proof

      [Start] 35.4	\[ \frac{M}{\left(d \cdot 4\right) \cdot \frac{d}{M \cdot \left(h \cdot \left(D \cdot D\right)\right)}} \]
      associate-*r/ [=>] 38.4	\[ \frac{M}{\color{blue}{\frac{\left(d \cdot 4\right) \cdot d}{M \cdot \left(h \cdot \left(D \cdot D\right)\right)}}} \]
      associate-*r* [=>] 37.5	\[ \frac{M}{\frac{\left(d \cdot 4\right) \cdot d}{\color{blue}{\left(M \cdot h\right) \cdot \left(D \cdot D\right)}}} \]
      associate-/l/ [<=] 38.1	\[ \frac{M}{\color{blue}{\frac{\frac{\left(d \cdot 4\right) \cdot d}{D \cdot D}}{M \cdot h}}} \]
      associate-*r/ [<=] 36.0	\[ \frac{M}{\frac{\color{blue}{\left(d \cdot 4\right) \cdot \frac{d}{D \cdot D}}}{M \cdot h}} \]
      associate-/l* [<=] 37.4	\[ \color{blue}{\frac{M \cdot \left(M \cdot h\right)}{\left(d \cdot 4\right) \cdot \frac{d}{D \cdot D}}} \]
      *-commutative [=>] 37.4	\[ \frac{M \cdot \left(M \cdot h\right)}{\color{blue}{\frac{d}{D \cdot D} \cdot \left(d \cdot 4\right)}} \]
      times-frac [=>] 32.3	\[ \color{blue}{\frac{M}{\frac{d}{D \cdot D}} \cdot \frac{M \cdot h}{d \cdot 4}} \]
      associate-/r* [=>] 26.0	\[ \frac{M}{\color{blue}{\frac{\frac{d}{D}}{D}}} \cdot \frac{M \cdot h}{d \cdot 4} \]
      associate-/r/ [=>] 24.3	\[ \color{blue}{\left(\frac{M}{\frac{d}{D}} \cdot D\right)} \cdot \frac{M \cdot h}{d \cdot 4} \]
      associate-/l* [=>] 24.2	\[ \left(\frac{M}{\frac{d}{D}} \cdot D\right) \cdot \color{blue}{\frac{M}{\frac{d \cdot 4}{h}}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 21.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq -2 \cdot 10^{-304}:\\ \;\;\;\;\frac{0.25}{d} \cdot \left(\frac{D}{\frac{d}{M}} \cdot \left(M \cdot \left(D \cdot h\right)\right)\right)\\ \mathbf{elif}\;D \leq 5 \cdot 10^{-248}:\\ \;\;\;\;\frac{M \cdot \left(D \cdot \left(M \cdot h\right)\right)}{\left(d \cdot 4\right) \cdot \frac{d}{D}}\\ \mathbf{elif}\;D \leq 4.3 \cdot 10^{-45}:\\ \;\;\;\;\frac{0.25}{d} \cdot \left(\frac{D}{\frac{d}{M}} \cdot \left(M \cdot \left(D \cdot h\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(D \cdot \frac{M}{\frac{d}{D}}\right) \cdot \frac{M}{\frac{d \cdot 4}{h}}\\ \end{array} \]

Alternatives

Alternative 1
Error	27.2
Cost	1220

\[\begin{array}{l} \mathbf{if}\;M \cdot M \leq 5 \cdot 10^{+196}:\\ \;\;\;\;0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 2
Error	21.4
Cost	1092

\[\begin{array}{l} \mathbf{if}\;M \leq -2.4 \cdot 10^{-22}:\\ \;\;\;\;\frac{0.25}{d} \cdot \left(\frac{D}{\frac{d}{M}} \cdot \left(M \cdot \left(D \cdot h\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(D \cdot \frac{M}{\frac{d}{D}}\right) \cdot \frac{M}{\frac{d \cdot 4}{h}}\\ \end{array} \]

Alternative 3
Error	22.0
Cost	960

\[\left(D \cdot \frac{M}{\frac{d}{D}}\right) \cdot \frac{M}{\frac{d \cdot 4}{h}} \]

Alternative 4
Error	31.8
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023067 
(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))))))