Henrywood and Agarwal, Equation (13)

Average Error: 59.5 → 23.2

Time: 16.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \frac{0.25}{d} \cdot \left(D \cdot \left(\left(h \cdot \frac{D}{d}\right) \cdot \left(M \cdot M\right)\right)\right)\\ t_1 := \left(M \cdot D\right) \cdot \sqrt{h}\\ \mathbf{if}\;M \leq -2.6 \cdot 10^{+153}:\\ \;\;\;\;\frac{0.25}{d} \cdot \left(t_1 \cdot \left(t_1 \cdot {d}^{-1}\right)\right)\\ \mathbf{elif}\;M \leq -2.1 \cdot 10^{-261}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;M \leq -1.15 \cdot 10^{-280}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{h \cdot w}\right)\right)\\ \mathbf{elif}\;M \leq 4.4 \cdot 10^{+149}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{d} \cdot {\left(\frac{d}{h \cdot {\left(M \cdot D\right)}^{2}}\right)}^{-1}\\ \end{array} \]

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 (* (* h (/ D d)) (* M M)))))
        (t_1 (* (* M D) (sqrt h))))
   (if (<= M -2.6e+153)
     (* (/ 0.25 d) (* t_1 (* t_1 (pow d -1.0))))
     (if (<= M -2.1e-261)
       t_0
       (if (<= M -1.15e-280)
         (* (/ c0 (* 2.0 w)) (* 2.0 (* (* (/ d D) (/ d D)) (/ c0 (* h w)))))
         (if (<= M 4.4e+149)
           t_0
           (* (/ 0.25 d) (pow (/ d (* h (pow (* M D) 2.0))) -1.0))))))))

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 t_0 = (0.25 / d) * (D * ((h * (D / d)) * (M * M)));
	double t_1 = (M * D) * sqrt(h);
	double tmp;
	if (M <= -2.6e+153) {
		tmp = (0.25 / d) * (t_1 * (t_1 * pow(d, -1.0)));
	} else if (M <= -2.1e-261) {
		tmp = t_0;
	} else if (M <= -1.15e-280) {
		tmp = (c0 / (2.0 * w)) * (2.0 * (((d / D) * (d / D)) * (c0 / (h * w))));
	} else if (M <= 4.4e+149) {
		tmp = t_0;
	} else {
		tmp = (0.25 / d) * pow((d / (h * pow((M * D), 2.0))), -1.0);
	}
	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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (0.25d0 / d_1) * (d * ((h * (d / d_1)) * (m * m)))
    t_1 = (m * d) * sqrt(h)
    if (m <= (-2.6d+153)) then
        tmp = (0.25d0 / d_1) * (t_1 * (t_1 * (d_1 ** (-1.0d0))))
    else if (m <= (-2.1d-261)) then
        tmp = t_0
    else if (m <= (-1.15d-280)) then
        tmp = (c0 / (2.0d0 * w)) * (2.0d0 * (((d_1 / d) * (d_1 / d)) * (c0 / (h * w))))
    else if (m <= 4.4d+149) then
        tmp = t_0
    else
        tmp = (0.25d0 / d_1) * ((d_1 / (h * ((m * d) ** 2.0d0))) ** (-1.0d0))
    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 t_0 = (0.25 / d) * (D * ((h * (D / d)) * (M * M)));
	double t_1 = (M * D) * Math.sqrt(h);
	double tmp;
	if (M <= -2.6e+153) {
		tmp = (0.25 / d) * (t_1 * (t_1 * Math.pow(d, -1.0)));
	} else if (M <= -2.1e-261) {
		tmp = t_0;
	} else if (M <= -1.15e-280) {
		tmp = (c0 / (2.0 * w)) * (2.0 * (((d / D) * (d / D)) * (c0 / (h * w))));
	} else if (M <= 4.4e+149) {
		tmp = t_0;
	} else {
		tmp = (0.25 / d) * Math.pow((d / (h * Math.pow((M * D), 2.0))), -1.0);
	}
	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):
	t_0 = (0.25 / d) * (D * ((h * (D / d)) * (M * M)))
	t_1 = (M * D) * math.sqrt(h)
	tmp = 0
	if M <= -2.6e+153:
		tmp = (0.25 / d) * (t_1 * (t_1 * math.pow(d, -1.0)))
	elif M <= -2.1e-261:
		tmp = t_0
	elif M <= -1.15e-280:
		tmp = (c0 / (2.0 * w)) * (2.0 * (((d / D) * (d / D)) * (c0 / (h * w))))
	elif M <= 4.4e+149:
		tmp = t_0
	else:
		tmp = (0.25 / d) * math.pow((d / (h * math.pow((M * D), 2.0))), -1.0)
	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)
	t_0 = Float64(Float64(0.25 / d) * Float64(D * Float64(Float64(h * Float64(D / d)) * Float64(M * M))))
	t_1 = Float64(Float64(M * D) * sqrt(h))
	tmp = 0.0
	if (M <= -2.6e+153)
		tmp = Float64(Float64(0.25 / d) * Float64(t_1 * Float64(t_1 * (d ^ -1.0))));
	elseif (M <= -2.1e-261)
		tmp = t_0;
	elseif (M <= -1.15e-280)
		tmp = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(2.0 * Float64(Float64(Float64(d / D) * Float64(d / D)) * Float64(c0 / Float64(h * w)))));
	elseif (M <= 4.4e+149)
		tmp = t_0;
	else
		tmp = Float64(Float64(0.25 / d) * (Float64(d / Float64(h * (Float64(M * D) ^ 2.0))) ^ -1.0));
	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)
	t_0 = (0.25 / d) * (D * ((h * (D / d)) * (M * M)));
	t_1 = (M * D) * sqrt(h);
	tmp = 0.0;
	if (M <= -2.6e+153)
		tmp = (0.25 / d) * (t_1 * (t_1 * (d ^ -1.0)));
	elseif (M <= -2.1e-261)
		tmp = t_0;
	elseif (M <= -1.15e-280)
		tmp = (c0 / (2.0 * w)) * (2.0 * (((d / D) * (d / D)) * (c0 / (h * w))));
	elseif (M <= 4.4e+149)
		tmp = t_0;
	else
		tmp = (0.25 / d) * ((d / (h * ((M * D) ^ 2.0))) ^ -1.0);
	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_] := Block[{t$95$0 = N[(N[(0.25 / d), $MachinePrecision] * N[(D * N[(N[(h * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(M * D), $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[M, -2.6e+153], N[(N[(0.25 / d), $MachinePrecision] * N[(t$95$1 * N[(t$95$1 * N[Power[d, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[M, -2.1e-261], t$95$0, If[LessEqual[M, -1.15e-280], N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(N[(N[(d / D), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision] * N[(c0 / N[(h * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[M, 4.4e+149], t$95$0, N[(N[(0.25 / d), $MachinePrecision] * N[Power[N[(d / N[(h * N[Power[N[(M * D), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $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 4 regimes

2. if M < -2.5999999999999999e153

   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 63.9

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

   3. Simplified 63.8

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

   4. Taylor expanded in c0 around 0 63.8

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

   5. Simplified 63.9

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

   6. Applied egg-rr 31.2

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

   7. Applied egg-rr 40.8

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

   if -2.5999999999999999e153 < M < -2.09999999999999996e-261 or -1.15e-280 < M < 4.4e149

   1. Initial program 58.9

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

   2. Taylor expanded in c0 around -inf 38.5

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

   3. Simplified 34.6

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

   4. Taylor expanded in c0 around 0 31.3

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

   5. Simplified 27.3

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

   6. Applied egg-rr 23.5

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

   7. Taylor expanded in h around 0 28.4

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

   8. Simplified 20.4

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

   if -2.09999999999999996e-261 < M < -1.15e-280

   1. Initial program 54.9

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

   2. Taylor expanded in c0 around inf 55.0

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

   3. Simplified 46.4

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

   if 4.4e149 < 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 63.9

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

   3. Simplified 63.5

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

   4. Taylor expanded in c0 around 0 63.5

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

   5. Simplified 63.3

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

   6. Applied egg-rr 29.7

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

   7. Applied egg-rr 29.8

      \[\leadsto \frac{0.25}{d} \cdot \color{blue}{{\left(\frac{d}{h \cdot {\left(D \cdot M\right)}^{2}}\right)}^{-1}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 23.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -2.6 \cdot 10^{+153}:\\ \;\;\;\;\frac{0.25}{d} \cdot \left(\left(\left(M \cdot D\right) \cdot \sqrt{h}\right) \cdot \left(\left(\left(M \cdot D\right) \cdot \sqrt{h}\right) \cdot {d}^{-1}\right)\right)\\ \mathbf{elif}\;M \leq -2.1 \cdot 10^{-261}:\\ \;\;\;\;\frac{0.25}{d} \cdot \left(D \cdot \left(\left(h \cdot \frac{D}{d}\right) \cdot \left(M \cdot M\right)\right)\right)\\ \mathbf{elif}\;M \leq -1.15 \cdot 10^{-280}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{h \cdot w}\right)\right)\\ \mathbf{elif}\;M \leq 4.4 \cdot 10^{+149}:\\ \;\;\;\;\frac{0.25}{d} \cdot \left(D \cdot \left(\left(h \cdot \frac{D}{d}\right) \cdot \left(M \cdot M\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{d} \cdot {\left(\frac{d}{h \cdot {\left(M \cdot D\right)}^{2}}\right)}^{-1}\\ \end{array} \]

Reproduce

herbie shell --seed 2022162 
(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))))))