Henrywood and Agarwal, Equation (9a)

Average Error: 14.7 → 8.7

Time: 15.7s

Precision: binary64

Cost: 28296

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

Math

\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]

↓

\[\begin{array}{l} t_0 := {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\\ t_1 := \frac{d}{D \cdot 0.5}\\ t_2 := h \cdot \frac{M}{\ell \cdot t_1}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{+211}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{t_2}{2 \cdot \frac{\frac{d}{D}}{M}}}\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{-83}:\\ \;\;\;\;w0 \cdot \sqrt{1 - t_0}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{M \cdot t_2}{t_1}}\\ \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (* w0 (sqrt (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l))))))

↓

(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)))
        (t_1 (/ d (* D 0.5)))
        (t_2 (* h (/ M (* l t_1)))))
   (if (<= t_0 -5e+211)
     (* w0 (sqrt (- 1.0 (/ t_2 (* 2.0 (/ (/ d D) M))))))
     (if (<= t_0 2e-83)
       (* w0 (sqrt (- 1.0 t_0)))
       (* w0 (sqrt (- 1.0 (/ (* M t_2) t_1))))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * sqrt((1.0 - (pow(((M * D) / (2.0 * d)), 2.0) * (h / l))));
}

↓

double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = pow(((M * D) / (2.0 * d)), 2.0) * (h / l);
	double t_1 = d / (D * 0.5);
	double t_2 = h * (M / (l * t_1));
	double tmp;
	if (t_0 <= -5e+211) {
		tmp = w0 * sqrt((1.0 - (t_2 / (2.0 * ((d / D) / M)))));
	} else if (t_0 <= 2e-83) {
		tmp = w0 * sqrt((1.0 - t_0));
	} else {
		tmp = w0 * sqrt((1.0 - ((M * t_2) / t_1)));
	}
	return tmp;
}

Fortran

real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    code = w0 * sqrt((1.0d0 - ((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l))))
end function

↓

real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l)
    t_1 = d_1 / (d * 0.5d0)
    t_2 = h * (m / (l * t_1))
    if (t_0 <= (-5d+211)) then
        tmp = w0 * sqrt((1.0d0 - (t_2 / (2.0d0 * ((d_1 / d) / m)))))
    else if (t_0 <= 2d-83) then
        tmp = w0 * sqrt((1.0d0 - t_0))
    else
        tmp = w0 * sqrt((1.0d0 - ((m * t_2) / t_1)))
    end if
    code = tmp
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * Math.sqrt((1.0 - (Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l))));
}

↓

public static double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l);
	double t_1 = d / (D * 0.5);
	double t_2 = h * (M / (l * t_1));
	double tmp;
	if (t_0 <= -5e+211) {
		tmp = w0 * Math.sqrt((1.0 - (t_2 / (2.0 * ((d / D) / M)))));
	} else if (t_0 <= 2e-83) {
		tmp = w0 * Math.sqrt((1.0 - t_0));
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((M * t_2) / t_1)));
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	return w0 * math.sqrt((1.0 - (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l))))

↓

def code(w0, M, D, h, l, d):
	t_0 = math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)
	t_1 = d / (D * 0.5)
	t_2 = h * (M / (l * t_1))
	tmp = 0
	if t_0 <= -5e+211:
		tmp = w0 * math.sqrt((1.0 - (t_2 / (2.0 * ((d / D) / M)))))
	elif t_0 <= 2e-83:
		tmp = w0 * math.sqrt((1.0 - t_0))
	else:
		tmp = w0 * math.sqrt((1.0 - ((M * t_2) / t_1)))
	return tmp

Julia

function code(w0, M, D, h, l, d)
	return Float64(w0 * sqrt(Float64(1.0 - Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)))))
end

↓

function code(w0, M, D, h, l, d)
	t_0 = Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l))
	t_1 = Float64(d / Float64(D * 0.5))
	t_2 = Float64(h * Float64(M / Float64(l * t_1)))
	tmp = 0.0
	if (t_0 <= -5e+211)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(t_2 / Float64(2.0 * Float64(Float64(d / D) / M))))));
	elseif (t_0 <= 2e-83)
		tmp = Float64(w0 * sqrt(Float64(1.0 - t_0)));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(M * t_2) / t_1))));
	end
	return tmp
end

MATLAB

function tmp = code(w0, M, D, h, l, d)
	tmp = w0 * sqrt((1.0 - ((((M * D) / (2.0 * d)) ^ 2.0) * (h / l))));
end

↓

function tmp_2 = code(w0, M, D, h, l, d)
	t_0 = (((M * D) / (2.0 * d)) ^ 2.0) * (h / l);
	t_1 = d / (D * 0.5);
	t_2 = h * (M / (l * t_1));
	tmp = 0.0;
	if (t_0 <= -5e+211)
		tmp = w0 * sqrt((1.0 - (t_2 / (2.0 * ((d / D) / M)))));
	elseif (t_0 <= 2e-83)
		tmp = w0 * sqrt((1.0 - t_0));
	else
		tmp = w0 * sqrt((1.0 - ((M * t_2) / t_1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := N[(w0 * N[Sqrt[N[(1.0 - N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

↓

code[w0_, M_, D_, h_, l_, d_] := Block[{t$95$0 = N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(d / N[(D * 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(h * N[(M / N[(l * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+211], N[(w0 * N[Sqrt[N[(1.0 - N[(t$95$2 / N[(2.0 * N[(N[(d / D), $MachinePrecision] / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-83], N[(w0 * N[Sqrt[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(M * t$95$2), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)) < -4.9999999999999995e211

   1. Initial program 57.9

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]

   2. Applied egg-rr 54.9

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot h}{\ell}}} \]

   3. Applied egg-rr 54.3

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell} \cdot h}} \]

   4. Applied egg-rr 51.8

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(\frac{M \cdot 0.5}{\frac{d}{D}} \cdot \left(\frac{M \cdot 0.5}{\frac{d}{D}} \cdot \frac{1}{\ell}\right)\right)} \cdot h} \]

   5. Applied egg-rr 50.8

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{\frac{M}{\ell \cdot \frac{d}{0.5 \cdot D}} \cdot h}{2 \cdot \frac{\frac{d}{D}}{M}}}} \]

   if -4.9999999999999995e211 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)) < 2.0000000000000001e-83

   1. Initial program 0.1

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]

   if 2.0000000000000001e-83 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))

   1. Initial program 49.7

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]

   2. Applied egg-rr 17.7

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot h}{\ell}}} \]

   3. Applied egg-rr 17.7

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell} \cdot h}} \]

   4. Applied egg-rr 9.1

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(\frac{M \cdot 0.5}{\frac{d}{D}} \cdot \left(\frac{M \cdot 0.5}{\frac{d}{D}} \cdot \frac{1}{\ell}\right)\right)} \cdot h} \]

   5. Applied egg-rr 8.7

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{M \cdot \left(\frac{M}{\ell \cdot \frac{d}{0.5 \cdot D}} \cdot h\right)}{\frac{d}{0.5 \cdot D}}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 8.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+211}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot \frac{M}{\ell \cdot \frac{d}{D \cdot 0.5}}}{2 \cdot \frac{\frac{d}{D}}{M}}}\\ \mathbf{elif}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 2 \cdot 10^{-83}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{M \cdot \left(h \cdot \frac{M}{\ell \cdot \frac{d}{D \cdot 0.5}}\right)}{\frac{d}{D \cdot 0.5}}}\\ \end{array} \]

Alternatives

Alternative 1
Error	12.8
Cost	8264

\[\begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -4 \cdot 10^{+220}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \left(h \cdot \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d \cdot d\right)}\right) \cdot -0.25}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -1 \cdot 10^{-269}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{\frac{D}{d} \cdot \left(M \cdot \frac{M}{\frac{\ell}{h}}\right)}{\frac{d}{D}} \cdot -0.25}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 2
Error	9.5
Cost	8132

\[\begin{array}{l} t_0 := \frac{M \cdot 0.5}{\frac{d}{D}}\\ t_1 := \frac{d}{D \cdot 0.5}\\ \mathbf{if}\;\ell \leq 10^{-80}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \left(t_0 \cdot \left(t_0 \cdot \frac{1}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{M \cdot \left(h \cdot \frac{M}{\ell \cdot t_1}\right)}{t_1}}\\ \end{array} \]

Alternative 3
Error	9.9
Cost	8004

\[\begin{array}{l} t_0 := \frac{d}{D \cdot 0.5}\\ \mathbf{if}\;\ell \leq 4 \cdot 10^{-73}:\\ \;\;\;\;w0 \cdot \sqrt{1 + h \cdot \left(\frac{M \cdot 0.5}{\frac{d}{D}} \cdot \left(\frac{D}{\frac{d}{M}} \cdot \frac{-0.5}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{M \cdot \left(h \cdot \frac{M}{\ell \cdot t_0}\right)}{t_0}}\\ \end{array} \]

Alternative 4
Error	10.4
Cost	7872

\[w0 \cdot \sqrt{1 + h \cdot \left(\frac{M \cdot 0.5}{\frac{d}{D}} \cdot \left(\frac{D}{\frac{d}{M}} \cdot \frac{-0.5}{\ell}\right)\right)} \]

Alternative 5
Error	14.0
Cost	64

\[w0 \]

Reproduce

herbie shell --seed 2022217 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  :precision binary64
  (* w0 (sqrt (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l))))))