Henrywood and Agarwal, Equation (12)

Average Error: 27.0 → 19.0

Time: 13.0s

Precision: binary64

\[ \begin{array}{c}[M, D] = \mathsf{sort}([M, D])\\ \end{array} \]

Math

\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

↓

\[\begin{array}{l} t_0 := \sqrt{-d}\\ t_1 := \frac{D}{2} \cdot \frac{M}{d}\\ t_2 := 1 - {\left(\sqrt{\frac{h}{\ell}} \cdot \left(t_1 \cdot \sqrt{0.5}\right)\right)}^{2}\\ t_3 := {\left(\frac{d}{h}\right)}^{0.5}\\ t_4 := \left(t_3 \cdot \frac{t_0}{\sqrt{-\ell}}\right) \cdot t_2\\ t_5 := {\left(\frac{d}{\ell}\right)}^{0.5}\\ \mathbf{if}\;\ell \leq -2.65 \cdot 10^{+156}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;\ell \leq -3.8 \cdot 10^{-39}:\\ \;\;\;\;\left(\frac{t_0}{\sqrt{-h}} \cdot t_5\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right)\right)\\ \mathbf{elif}\;\ell \leq -2.5 \cdot 10^{-124}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;\ell \leq 5.2 \cdot 10^{-31}:\\ \;\;\;\;\left(t_3 \cdot t_5\right) \cdot \left(1 - \frac{h \cdot \left(0.5 \cdot {t_1}^{2}\right)}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \left(t_5 \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (*
  (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
  (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))

↓

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (- d)))
        (t_1 (* (/ D 2.0) (/ M d)))
        (t_2 (- 1.0 (pow (* (sqrt (/ h l)) (* t_1 (sqrt 0.5))) 2.0)))
        (t_3 (pow (/ d h) 0.5))
        (t_4 (* (* t_3 (/ t_0 (sqrt (- l)))) t_2))
        (t_5 (pow (/ d l) 0.5)))
   (if (<= l -2.65e+156)
     t_4
     (if (<= l -3.8e-39)
       (*
        (* (/ t_0 (sqrt (- h))) t_5)
        (- 1.0 (* (/ h l) (* 0.5 (pow (/ (* D M) (* d 2.0)) 2.0)))))
       (if (<= l -2.5e-124)
         t_4
         (if (<= l 5.2e-31)
           (* (* t_3 t_5) (- 1.0 (/ (* h (* 0.5 (pow t_1 2.0))) l)))
           (* t_2 (* t_5 (/ (sqrt d) (sqrt h))))))))))

C

double code(double d, double h, double l, double M, double D) {
	return (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
}

↓

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt(-d);
	double t_1 = (D / 2.0) * (M / d);
	double t_2 = 1.0 - pow((sqrt((h / l)) * (t_1 * sqrt(0.5))), 2.0);
	double t_3 = pow((d / h), 0.5);
	double t_4 = (t_3 * (t_0 / sqrt(-l))) * t_2;
	double t_5 = pow((d / l), 0.5);
	double tmp;
	if (l <= -2.65e+156) {
		tmp = t_4;
	} else if (l <= -3.8e-39) {
		tmp = ((t_0 / sqrt(-h)) * t_5) * (1.0 - ((h / l) * (0.5 * pow(((D * M) / (d * 2.0)), 2.0))));
	} else if (l <= -2.5e-124) {
		tmp = t_4;
	} else if (l <= 5.2e-31) {
		tmp = (t_3 * t_5) * (1.0 - ((h * (0.5 * pow(t_1, 2.0))) / l));
	} else {
		tmp = t_2 * (t_5 * (sqrt(d) / sqrt(h)));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    code = (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
end function

↓

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: tmp
    t_0 = sqrt(-d)
    t_1 = (d_1 / 2.0d0) * (m / d)
    t_2 = 1.0d0 - ((sqrt((h / l)) * (t_1 * sqrt(0.5d0))) ** 2.0d0)
    t_3 = (d / h) ** 0.5d0
    t_4 = (t_3 * (t_0 / sqrt(-l))) * t_2
    t_5 = (d / l) ** 0.5d0
    if (l <= (-2.65d+156)) then
        tmp = t_4
    else if (l <= (-3.8d-39)) then
        tmp = ((t_0 / sqrt(-h)) * t_5) * (1.0d0 - ((h / l) * (0.5d0 * (((d_1 * m) / (d * 2.0d0)) ** 2.0d0))))
    else if (l <= (-2.5d-124)) then
        tmp = t_4
    else if (l <= 5.2d-31) then
        tmp = (t_3 * t_5) * (1.0d0 - ((h * (0.5d0 * (t_1 ** 2.0d0))) / l))
    else
        tmp = t_2 * (t_5 * (sqrt(d) / sqrt(h)))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	return (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
}

↓

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt(-d);
	double t_1 = (D / 2.0) * (M / d);
	double t_2 = 1.0 - Math.pow((Math.sqrt((h / l)) * (t_1 * Math.sqrt(0.5))), 2.0);
	double t_3 = Math.pow((d / h), 0.5);
	double t_4 = (t_3 * (t_0 / Math.sqrt(-l))) * t_2;
	double t_5 = Math.pow((d / l), 0.5);
	double tmp;
	if (l <= -2.65e+156) {
		tmp = t_4;
	} else if (l <= -3.8e-39) {
		tmp = ((t_0 / Math.sqrt(-h)) * t_5) * (1.0 - ((h / l) * (0.5 * Math.pow(((D * M) / (d * 2.0)), 2.0))));
	} else if (l <= -2.5e-124) {
		tmp = t_4;
	} else if (l <= 5.2e-31) {
		tmp = (t_3 * t_5) * (1.0 - ((h * (0.5 * Math.pow(t_1, 2.0))) / l));
	} else {
		tmp = t_2 * (t_5 * (Math.sqrt(d) / Math.sqrt(h)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	return (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))

↓

def code(d, h, l, M, D):
	t_0 = math.sqrt(-d)
	t_1 = (D / 2.0) * (M / d)
	t_2 = 1.0 - math.pow((math.sqrt((h / l)) * (t_1 * math.sqrt(0.5))), 2.0)
	t_3 = math.pow((d / h), 0.5)
	t_4 = (t_3 * (t_0 / math.sqrt(-l))) * t_2
	t_5 = math.pow((d / l), 0.5)
	tmp = 0
	if l <= -2.65e+156:
		tmp = t_4
	elif l <= -3.8e-39:
		tmp = ((t_0 / math.sqrt(-h)) * t_5) * (1.0 - ((h / l) * (0.5 * math.pow(((D * M) / (d * 2.0)), 2.0))))
	elif l <= -2.5e-124:
		tmp = t_4
	elif l <= 5.2e-31:
		tmp = (t_3 * t_5) * (1.0 - ((h * (0.5 * math.pow(t_1, 2.0))) / l))
	else:
		tmp = t_2 * (t_5 * (math.sqrt(d) / math.sqrt(h)))
	return tmp

Julia

function code(d, h, l, M, D)
	return Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
end

↓

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(-d))
	t_1 = Float64(Float64(D / 2.0) * Float64(M / d))
	t_2 = Float64(1.0 - (Float64(sqrt(Float64(h / l)) * Float64(t_1 * sqrt(0.5))) ^ 2.0))
	t_3 = Float64(d / h) ^ 0.5
	t_4 = Float64(Float64(t_3 * Float64(t_0 / sqrt(Float64(-l)))) * t_2)
	t_5 = Float64(d / l) ^ 0.5
	tmp = 0.0
	if (l <= -2.65e+156)
		tmp = t_4;
	elseif (l <= -3.8e-39)
		tmp = Float64(Float64(Float64(t_0 / sqrt(Float64(-h))) * t_5) * Float64(1.0 - Float64(Float64(h / l) * Float64(0.5 * (Float64(Float64(D * M) / Float64(d * 2.0)) ^ 2.0)))));
	elseif (l <= -2.5e-124)
		tmp = t_4;
	elseif (l <= 5.2e-31)
		tmp = Float64(Float64(t_3 * t_5) * Float64(1.0 - Float64(Float64(h * Float64(0.5 * (t_1 ^ 2.0))) / l)));
	else
		tmp = Float64(t_2 * Float64(t_5 * Float64(sqrt(d) / sqrt(h))));
	end
	return tmp
end

MATLAB

function tmp = code(d, h, l, M, D)
	tmp = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
end

↓

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt(-d);
	t_1 = (D / 2.0) * (M / d);
	t_2 = 1.0 - ((sqrt((h / l)) * (t_1 * sqrt(0.5))) ^ 2.0);
	t_3 = (d / h) ^ 0.5;
	t_4 = (t_3 * (t_0 / sqrt(-l))) * t_2;
	t_5 = (d / l) ^ 0.5;
	tmp = 0.0;
	if (l <= -2.65e+156)
		tmp = t_4;
	elseif (l <= -3.8e-39)
		tmp = ((t_0 / sqrt(-h)) * t_5) * (1.0 - ((h / l) * (0.5 * (((D * M) / (d * 2.0)) ^ 2.0))));
	elseif (l <= -2.5e-124)
		tmp = t_4;
	elseif (l <= 5.2e-31)
		tmp = (t_3 * t_5) * (1.0 - ((h * (0.5 * (t_1 ^ 2.0))) / l));
	else
		tmp = t_2 * (t_5 * (sqrt(d) / sqrt(h)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[(-d)], $MachinePrecision]}, Block[{t$95$1 = N[(N[(D / 2.0), $MachinePrecision] * N[(M / d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - N[Power[N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * N[(t$95$1 * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[(d / h), $MachinePrecision], 0.5], $MachinePrecision]}, Block[{t$95$4 = N[(N[(t$95$3 * N[(t$95$0 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]}, Block[{t$95$5 = N[Power[N[(d / l), $MachinePrecision], 0.5], $MachinePrecision]}, If[LessEqual[l, -2.65e+156], t$95$4, If[LessEqual[l, -3.8e-39], N[(N[(N[(t$95$0 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * t$95$5), $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(0.5 * N[Power[N[(N[(D * M), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -2.5e-124], t$95$4, If[LessEqual[l, 5.2e-31], N[(N[(t$95$3 * t$95$5), $MachinePrecision] * N[(1.0 - N[(N[(h * N[(0.5 * N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 * N[(t$95$5 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Error

Bits error versus d

Bits error versus h

Bits error versus l

Bits error versus M

Bits error versus D

Derivation

1. Split input into 4 regimes

2. if l < -2.6499999999999999e156 or -3.8000000000000002e-39 < l < -2.5000000000000001e-124

   1. Initial program 29.8

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Applied egg-rr 27.9

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

   3. Applied egg-rr 21.5

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

   if -2.6499999999999999e156 < l < -3.8000000000000002e-39

   1. Initial program 20.9

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Applied egg-rr 13.5

      \[\leadsto \left(\color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   if -2.5000000000000001e-124 < l < 5.19999999999999991e-31

   1. Initial program 31.1

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Applied egg-rr 25.4

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

   if 5.19999999999999991e-31 < l

   1. Initial program 25.5

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Applied egg-rr 23.7

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

   3. Applied egg-rr 15.8

      \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\sqrt{\frac{h}{\ell}} \cdot \left(\left(\frac{D}{2} \cdot \frac{M}{d}\right) \cdot \sqrt{0.5}\right)\right)}^{2}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 19.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.65 \cdot 10^{+156}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right) \cdot \left(1 - {\left(\sqrt{\frac{h}{\ell}} \cdot \left(\left(\frac{D}{2} \cdot \frac{M}{d}\right) \cdot \sqrt{0.5}\right)\right)}^{2}\right)\\ \mathbf{elif}\;\ell \leq -3.8 \cdot 10^{-39}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right)\right)\\ \mathbf{elif}\;\ell \leq -2.5 \cdot 10^{-124}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right) \cdot \left(1 - {\left(\sqrt{\frac{h}{\ell}} \cdot \left(\left(\frac{D}{2} \cdot \frac{M}{d}\right) \cdot \sqrt{0.5}\right)\right)}^{2}\right)\\ \mathbf{elif}\;\ell \leq 5.2 \cdot 10^{-31}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h \cdot \left(0.5 \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}\right)}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - {\left(\sqrt{\frac{h}{\ell}} \cdot \left(\left(\frac{D}{2} \cdot \frac{M}{d}\right) \cdot \sqrt{0.5}\right)\right)}^{2}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{0.5} \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022166 
(FPCore (d h l M D)
  :name "Henrywood and Agarwal, Equation (12)"
  :precision binary64
  (* (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0))) (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))