Henrywood and Agarwal, Equation (9a)

Average Error: 14.2 → 7.8

Time: 16.9s

Precision: binary64

Cost: 21828

Math

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

↓

\[\begin{array}{l} t_0 := \frac{D}{d} \cdot \left(M \cdot 0.5\right)\\ t_1 := 1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\\ \mathbf{if}\;t_1 \leq 2 \cdot 10^{+247}:\\ \;\;\;\;w0 \cdot \sqrt{t_1}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{\ell}{t_0} \cdot \frac{\frac{1}{h}}{t_0}\right)}^{-1}}\\ \end{array} \]

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 (* (/ D d) (* M 0.5)))
        (t_1 (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)))))
   (if (<= t_1 2e+247)
     (* w0 (sqrt t_1))
     (* w0 (sqrt (- 1.0 (pow (* (/ l t_0) (/ (/ 1.0 h) t_0)) -1.0)))))))

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 = (D / d) * (M * 0.5);
	double t_1 = 1.0 - (pow(((M * D) / (2.0 * d)), 2.0) * (h / l));
	double tmp;
	if (t_1 <= 2e+247) {
		tmp = w0 * sqrt(t_1);
	} else {
		tmp = w0 * sqrt((1.0 - pow(((l / t_0) * ((1.0 / h) / t_0)), -1.0)));
	}
	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) :: tmp
    t_0 = (d / d_1) * (m * 0.5d0)
    t_1 = 1.0d0 - ((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l))
    if (t_1 <= 2d+247) then
        tmp = w0 * sqrt(t_1)
    else
        tmp = w0 * sqrt((1.0d0 - (((l / t_0) * ((1.0d0 / h) / t_0)) ** (-1.0d0))))
    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 = (D / d) * (M * 0.5);
	double t_1 = 1.0 - (Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l));
	double tmp;
	if (t_1 <= 2e+247) {
		tmp = w0 * Math.sqrt(t_1);
	} else {
		tmp = w0 * Math.sqrt((1.0 - Math.pow(((l / t_0) * ((1.0 / h) / t_0)), -1.0)));
	}
	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 = (D / d) * (M * 0.5)
	t_1 = 1.0 - (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l))
	tmp = 0
	if t_1 <= 2e+247:
		tmp = w0 * math.sqrt(t_1)
	else:
		tmp = w0 * math.sqrt((1.0 - math.pow(((l / t_0) * ((1.0 / h) / t_0)), -1.0)))
	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(D / d) * Float64(M * 0.5))
	t_1 = Float64(1.0 - Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)))
	tmp = 0.0
	if (t_1 <= 2e+247)
		tmp = Float64(w0 * sqrt(t_1));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - (Float64(Float64(l / t_0) * Float64(Float64(1.0 / h) / t_0)) ^ -1.0))));
	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 = (D / d) * (M * 0.5);
	t_1 = 1.0 - ((((M * D) / (2.0 * d)) ^ 2.0) * (h / l));
	tmp = 0.0;
	if (t_1 <= 2e+247)
		tmp = w0 * sqrt(t_1);
	else
		tmp = w0 * sqrt((1.0 - (((l / t_0) * ((1.0 / h) / t_0)) ^ -1.0)));
	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[(D / d), $MachinePrecision] * N[(M * 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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]}, If[LessEqual[t$95$1, 2e+247], N[(w0 * N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[Power[N[(N[(l / t$95$0), $MachinePrecision] * N[(N[(1.0 / h), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))) < 1.9999999999999999e247

   1. Initial program 0.2

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

   if 1.9999999999999999e247 < (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)))

   1. Initial program 61.4

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

   2. Simplified 59.9

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

   3. Applied egg-rr 43.5

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

   4. Applied egg-rr 57.9

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

   5. Applied egg-rr 33.5

      \[\leadsto w0 \cdot \sqrt{1 - {\color{blue}{\left(\frac{\ell}{\frac{D}{d} \cdot \left(M \cdot 0.5\right)} \cdot \frac{\frac{1}{h}}{\frac{D}{d} \cdot \left(M \cdot 0.5\right)}\right)}}^{-1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 7.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 2 \cdot 10^{+247}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{\ell}{\frac{D}{d} \cdot \left(M \cdot 0.5\right)} \cdot \frac{\frac{1}{h}}{\frac{D}{d} \cdot \left(M \cdot 0.5\right)}\right)}^{-1}}\\ \end{array} \]

Alternatives

Alternative 1
Error	8.9
Cost	21444

\[\begin{array}{l} t_0 := \frac{D}{d} \cdot \left(M \cdot 0.5\right)\\ \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq 2 \cdot 10^{+52}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(\frac{D}{d \cdot \frac{2}{M}}\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{1}{t_0} \cdot \frac{\frac{\ell}{h}}{t_0}\right)}^{-1}}\\ \end{array} \]

Alternative 2
Error	10.3
Cost	21188

\[\begin{array}{l} t_0 := 1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\\ \mathbf{if}\;t_0 \leq 2 \cdot 10^{+247}:\\ \;\;\;\;w0 \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 3
Error	10.5
Cost	14084

\[\begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -5 \cdot 10^{-101}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(\frac{D}{d \cdot \frac{2}{M}}\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 4
Error	13.6
Cost	8008

\[\begin{array}{l} \mathbf{if}\;D \leq 6.416566452030646 \cdot 10^{-130}:\\ \;\;\;\;w0\\ \mathbf{elif}\;D \leq 1.35 \cdot 10^{+91}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{\frac{M \cdot \left(\left(M \cdot h\right) \cdot \left(D \cdot D\right)\right)}{d}}{d \cdot \ell} \cdot -0.25}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 5
Error	14.8
Cost	8008

\[\begin{array}{l} \mathbf{if}\;\ell \leq 2.8 \cdot 10^{-171}:\\ \;\;\;\;w0\\ \mathbf{elif}\;\ell \leq 2.5691988834742814 \cdot 10^{+159}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \left(M \cdot h\right)\right)}{\ell} \cdot -0.25}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 6
Error	17.5
Cost	8008

\[\begin{array}{l} t_0 := M \cdot \left(M \cdot h\right)\\ \mathbf{if}\;M \leq -8.5 \cdot 10^{+65}:\\ \;\;\;\;w0\\ \mathbf{elif}\;M \leq 4.782876455348571 \cdot 10^{-151}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot t_0}{\ell} \cdot -0.25}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{D \cdot t_0}{\frac{d}{\frac{D}{d \cdot \ell}}} \cdot -0.25}\\ \end{array} \]

Alternative 7
Error	13.8
Cost	64

\[w0 \]

Reproduce

herbie shell --seed 2022216 
(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))))))