Henrywood and Agarwal, Equation (13)

Average Error: 59.3 → 26.0

Time: 18.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;h \leq 1.0667044981912174 \cdot 10^{-147}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \left(\frac{D}{d} \cdot \frac{M \cdot \left(h \cdot M\right)}{d}\right)\right)\\ \mathbf{elif}\;h \leq 1.0261799473837364 \cdot 10^{+177}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot \left(\frac{D}{d} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{h \cdot \left({D}^{2} \cdot {M}^{2}\right)}{{d}^{2}}\\ \end{array} \]

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 (<= h 1.0667044981912174e-147)
   (* 0.25 (* D (* (/ D d) (/ (* M (* h M)) d))))
   (if (<= h 1.0261799473837364e+177)
     (* 0.25 (/ (* D (* (/ D d) (* h (* M M)))) d))
     (* 0.25 (/ (* h (* (pow D 2.0) (pow M 2.0))) (pow d 2.0))))))

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 (h <= 1.0667044981912174e-147) {
		tmp = 0.25 * (D * ((D / d) * ((M * (h * M)) / d)));
	} else if (h <= 1.0261799473837364e+177) {
		tmp = 0.25 * ((D * ((D / d) * (h * (M * M)))) / d);
	} else {
		tmp = 0.25 * ((h * (pow(D, 2.0) * pow(M, 2.0))) / pow(d, 2.0));
	}
	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) :: tmp
    if (h <= 1.0667044981912174d-147) then
        tmp = 0.25d0 * (d * ((d / d_1) * ((m * (h * m)) / d_1)))
    else if (h <= 1.0261799473837364d+177) then
        tmp = 0.25d0 * ((d * ((d / d_1) * (h * (m * m)))) / d_1)
    else
        tmp = 0.25d0 * ((h * ((d ** 2.0d0) * (m ** 2.0d0))) / (d_1 ** 2.0d0))
    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 tmp;
	if (h <= 1.0667044981912174e-147) {
		tmp = 0.25 * (D * ((D / d) * ((M * (h * M)) / d)));
	} else if (h <= 1.0261799473837364e+177) {
		tmp = 0.25 * ((D * ((D / d) * (h * (M * M)))) / d);
	} else {
		tmp = 0.25 * ((h * (Math.pow(D, 2.0) * Math.pow(M, 2.0))) / Math.pow(d, 2.0));
	}
	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):
	tmp = 0
	if h <= 1.0667044981912174e-147:
		tmp = 0.25 * (D * ((D / d) * ((M * (h * M)) / d)))
	elif h <= 1.0261799473837364e+177:
		tmp = 0.25 * ((D * ((D / d) * (h * (M * M)))) / d)
	else:
		tmp = 0.25 * ((h * (math.pow(D, 2.0) * math.pow(M, 2.0))) / math.pow(d, 2.0))
	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)
	tmp = 0.0
	if (h <= 1.0667044981912174e-147)
		tmp = Float64(0.25 * Float64(D * Float64(Float64(D / d) * Float64(Float64(M * Float64(h * M)) / d))));
	elseif (h <= 1.0261799473837364e+177)
		tmp = Float64(0.25 * Float64(Float64(D * Float64(Float64(D / d) * Float64(h * Float64(M * M)))) / d));
	else
		tmp = Float64(0.25 * Float64(Float64(h * Float64((D ^ 2.0) * (M ^ 2.0))) / (d ^ 2.0)));
	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)
	tmp = 0.0;
	if (h <= 1.0667044981912174e-147)
		tmp = 0.25 * (D * ((D / d) * ((M * (h * M)) / d)));
	elseif (h <= 1.0261799473837364e+177)
		tmp = 0.25 * ((D * ((D / d) * (h * (M * M)))) / d);
	else
		tmp = 0.25 * ((h * ((D ^ 2.0) * (M ^ 2.0))) / (d ^ 2.0));
	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_] := If[LessEqual[h, 1.0667044981912174e-147], N[(0.25 * N[(D * N[(N[(D / d), $MachinePrecision] * N[(N[(M * N[(h * M), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 1.0261799473837364e+177], N[(0.25 * N[(N[(D * N[(N[(D / d), $MachinePrecision] * N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(N[(h * N[(N[Power[D, 2.0], $MachinePrecision] * N[Power[M, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[d, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

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 h < 1.0667044981912174e-147

   1. Initial program 59.5

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

   2. Taylor expanded in c0 around -inf 35.3

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

   3. Applied add-sqr-sqrt_binary64 49.6

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

   4. Applied unpow-prod-down_binary64 49.6

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

   5. Applied times-frac_binary64 48.0

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

   6. Simplified 48.0

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

   7. Simplified 31.9

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

   8. Applied *-un-lft-identity_binary64 31.9

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

   9. Applied times-frac_binary64 29.1

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

   10. Applied associate-*l*_binary64 28.0

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

   11. Applied associate-*r*_binary64 24.4

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

   if 1.0667044981912174e-147 < h < 1.02617994738373639e177

   1. Initial program 58.7

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

   2. Taylor expanded in c0 around -inf 35.1

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

   3. Applied add-sqr-sqrt_binary64 49.6

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

   4. Applied unpow-prod-down_binary64 49.6

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

   5. Applied times-frac_binary64 47.9

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

   6. Simplified 47.9

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

   7. Simplified 31.8

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

   8. Applied *-un-lft-identity_binary64 31.8

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

   9. Applied times-frac_binary64 28.8

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

   10. Applied associate-*l*_binary64 27.7

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

   11. Applied associate-*r/_binary64 27.3

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

   12. Applied frac-times_binary64 27.2

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

   if 1.02617994738373639e177 < h

   1. Initial program 60.0

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

   2. Taylor expanded in c0 around -inf 39.8

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

   3. Applied associate-*r*_binary64 36.9

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

4. Final simplification 26.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq 1.0667044981912174 \cdot 10^{-147}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \left(\frac{D}{d} \cdot \frac{M \cdot \left(h \cdot M\right)}{d}\right)\right)\\ \mathbf{elif}\;h \leq 1.0261799473837364 \cdot 10^{+177}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot \left(\frac{D}{d} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{h \cdot \left({D}^{2} \cdot {M}^{2}\right)}{{d}^{2}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022131 
(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))))))