Henrywood and Agarwal, Equation (12)

Average Error: 26.9 → 18.6

Time: 14.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := 0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\\ t_1 := 1 - \frac{h \cdot t_0}{\ell}\\ \mathbf{if}\;\ell \leq -3.170040707759249 \cdot 10^{-296}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot t_1\\ \mathbf{elif}\;\ell \leq 1.9115375279058692 \cdot 10^{+23}:\\ \;\;\;\;t_1 \cdot \left({\left(\frac{d}{h}\right)}^{0.5} \cdot \frac{1}{\frac{\sqrt{\ell}}{\sqrt{d}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - t_0 \cdot \frac{h}{\ell}\right)\\ \end{array} \]

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 (* 0.5 (pow (/ (* M D) (* d 2.0)) 2.0)))
        (t_1 (- 1.0 (/ (* h t_0) l))))
   (if (<= l -3.170040707759249e-296)
     (* (* (/ (sqrt (- d)) (sqrt (- h))) (/ 1.0 (sqrt (/ l d)))) t_1)
     (if (<= l 1.9115375279058692e+23)
       (* t_1 (* (pow (/ d h) 0.5) (/ 1.0 (/ (sqrt l) (sqrt d)))))
       (*
        (* (/ (sqrt d) (sqrt h)) (pow (/ d l) 0.5))
        (- 1.0 (* t_0 (/ h l))))))))

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 = 0.5 * pow(((M * D) / (d * 2.0)), 2.0);
	double t_1 = 1.0 - ((h * t_0) / l);
	double tmp;
	if (l <= -3.170040707759249e-296) {
		tmp = ((sqrt(-d) / sqrt(-h)) * (1.0 / sqrt((l / d)))) * t_1;
	} else if (l <= 1.9115375279058692e+23) {
		tmp = t_1 * (pow((d / h), 0.5) * (1.0 / (sqrt(l) / sqrt(d))));
	} else {
		tmp = ((sqrt(d) / sqrt(h)) * pow((d / l), 0.5)) * (1.0 - (t_0 * (h / l)));
	}
	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) :: tmp
    t_0 = 0.5d0 * (((m * d_1) / (d * 2.0d0)) ** 2.0d0)
    t_1 = 1.0d0 - ((h * t_0) / l)
    if (l <= (-3.170040707759249d-296)) then
        tmp = ((sqrt(-d) / sqrt(-h)) * (1.0d0 / sqrt((l / d)))) * t_1
    else if (l <= 1.9115375279058692d+23) then
        tmp = t_1 * (((d / h) ** 0.5d0) * (1.0d0 / (sqrt(l) / sqrt(d))))
    else
        tmp = ((sqrt(d) / sqrt(h)) * ((d / l) ** 0.5d0)) * (1.0d0 - (t_0 * (h / l)))
    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 = 0.5 * Math.pow(((M * D) / (d * 2.0)), 2.0);
	double t_1 = 1.0 - ((h * t_0) / l);
	double tmp;
	if (l <= -3.170040707759249e-296) {
		tmp = ((Math.sqrt(-d) / Math.sqrt(-h)) * (1.0 / Math.sqrt((l / d)))) * t_1;
	} else if (l <= 1.9115375279058692e+23) {
		tmp = t_1 * (Math.pow((d / h), 0.5) * (1.0 / (Math.sqrt(l) / Math.sqrt(d))));
	} else {
		tmp = ((Math.sqrt(d) / Math.sqrt(h)) * Math.pow((d / l), 0.5)) * (1.0 - (t_0 * (h / l)));
	}
	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 = 0.5 * math.pow(((M * D) / (d * 2.0)), 2.0)
	t_1 = 1.0 - ((h * t_0) / l)
	tmp = 0
	if l <= -3.170040707759249e-296:
		tmp = ((math.sqrt(-d) / math.sqrt(-h)) * (1.0 / math.sqrt((l / d)))) * t_1
	elif l <= 1.9115375279058692e+23:
		tmp = t_1 * (math.pow((d / h), 0.5) * (1.0 / (math.sqrt(l) / math.sqrt(d))))
	else:
		tmp = ((math.sqrt(d) / math.sqrt(h)) * math.pow((d / l), 0.5)) * (1.0 - (t_0 * (h / l)))
	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 = Float64(0.5 * (Float64(Float64(M * D) / Float64(d * 2.0)) ^ 2.0))
	t_1 = Float64(1.0 - Float64(Float64(h * t_0) / l))
	tmp = 0.0
	if (l <= -3.170040707759249e-296)
		tmp = Float64(Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * Float64(1.0 / sqrt(Float64(l / d)))) * t_1);
	elseif (l <= 1.9115375279058692e+23)
		tmp = Float64(t_1 * Float64((Float64(d / h) ^ 0.5) * Float64(1.0 / Float64(sqrt(l) / sqrt(d)))));
	else
		tmp = Float64(Float64(Float64(sqrt(d) / sqrt(h)) * (Float64(d / l) ^ 0.5)) * Float64(1.0 - Float64(t_0 * Float64(h / l))));
	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 = 0.5 * (((M * D) / (d * 2.0)) ^ 2.0);
	t_1 = 1.0 - ((h * t_0) / l);
	tmp = 0.0;
	if (l <= -3.170040707759249e-296)
		tmp = ((sqrt(-d) / sqrt(-h)) * (1.0 / sqrt((l / d)))) * t_1;
	elseif (l <= 1.9115375279058692e+23)
		tmp = t_1 * (((d / h) ^ 0.5) * (1.0 / (sqrt(l) / sqrt(d))));
	else
		tmp = ((sqrt(d) / sqrt(h)) * ((d / l) ^ 0.5)) * (1.0 - (t_0 * (h / l)));
	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[(0.5 * N[Power[N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(N[(h * t$95$0), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -3.170040707759249e-296], N[(N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[N[(l / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[l, 1.9115375279058692e+23], N[(t$95$1 * N[(N[Power[N[(d / h), $MachinePrecision], 0.5], $MachinePrecision] * N[(1.0 / N[(N[Sqrt[l], $MachinePrecision] / N[Sqrt[d], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(t$95$0 * N[(h / l), $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 3 regimes

2. if l < -3.170040707759249e-296

   1. Initial program 26.9

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

   2. Applied egg-rr 27.0

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

   3. Applied egg-rr 26.6

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

   4. Applied egg-rr 20.1

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

   if -3.170040707759249e-296 < l < 1.9115375279058692e23

   1. Initial program 27.1

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

   2. Applied egg-rr 26.6

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

   3. Applied egg-rr 22.0

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

   4. Applied egg-rr 14.9

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

   if 1.9115375279058692e23 < l

   1. Initial program 26.7

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

   2. Applied egg-rr 18.7

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

4. Final simplification 18.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.170040707759249 \cdot 10^{-296}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{h \cdot \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)}{\ell}\right)\\ \mathbf{elif}\;\ell \leq 1.9115375279058692 \cdot 10^{+23}:\\ \;\;\;\;\left(1 - \frac{h \cdot \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{0.5} \cdot \frac{1}{\frac{\sqrt{\ell}}{\sqrt{d}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022138 
(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)))))