Henrywood and Agarwal, Equation (9a)

Average Error: 14.0 → 10.0

Time: 11.6s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{M \cdot D}{2 \cdot d}\\ \mathbf{if}\;t_0 \leq -1.6633635303706548 \cdot 10^{+143}:\\ \;\;\;\;w0 \cdot \left(D \cdot \sqrt{\frac{-0.25 \cdot h}{\ell} \cdot \left(\frac{M}{d} \cdot \frac{M}{d}\right)}\right)\\ \mathbf{elif}\;t_0 \leq 3.6116818720243387 \cdot 10^{+118}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{1}{\frac{\ell}{h \cdot {\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2}}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(\sqrt{-0.25 \cdot \frac{D}{\frac{\ell \cdot \left(d \cdot d\right)}{D \cdot h}}} \cdot \left(-M\right)\right)\\ \end{array} \]

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 (/ (* M D) (* 2.0 d))))
   (if (<= t_0 -1.6633635303706548e+143)
     (* w0 (* D (sqrt (* (/ (* -0.25 h) l) (* (/ M d) (/ M d))))))
     (if (<= t_0 3.6116818720243387e+118)
       (*
        w0
        (sqrt (- 1.0 (/ 1.0 (/ l (* h (pow (* 0.5 (/ (* M D) d)) 2.0)))))))
       (* w0 (* (sqrt (* -0.25 (/ D (/ (* l (* d d)) (* D h))))) (- M)))))))

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 = (M * D) / (2.0 * d);
	double tmp;
	if (t_0 <= -1.6633635303706548e+143) {
		tmp = w0 * (D * sqrt((((-0.25 * h) / l) * ((M / d) * (M / d)))));
	} else if (t_0 <= 3.6116818720243387e+118) {
		tmp = w0 * sqrt((1.0 - (1.0 / (l / (h * pow((0.5 * ((M * D) / d)), 2.0))))));
	} else {
		tmp = w0 * (sqrt((-0.25 * (D / ((l * (d * d)) / (D * h))))) * -M);
	}
	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) :: tmp
    t_0 = (m * d) / (2.0d0 * d_1)
    if (t_0 <= (-1.6633635303706548d+143)) then
        tmp = w0 * (d * sqrt(((((-0.25d0) * h) / l) * ((m / d_1) * (m / d_1)))))
    else if (t_0 <= 3.6116818720243387d+118) then
        tmp = w0 * sqrt((1.0d0 - (1.0d0 / (l / (h * ((0.5d0 * ((m * d) / d_1)) ** 2.0d0))))))
    else
        tmp = w0 * (sqrt(((-0.25d0) * (d / ((l * (d_1 * d_1)) / (d * h))))) * -m)
    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 = (M * D) / (2.0 * d);
	double tmp;
	if (t_0 <= -1.6633635303706548e+143) {
		tmp = w0 * (D * Math.sqrt((((-0.25 * h) / l) * ((M / d) * (M / d)))));
	} else if (t_0 <= 3.6116818720243387e+118) {
		tmp = w0 * Math.sqrt((1.0 - (1.0 / (l / (h * Math.pow((0.5 * ((M * D) / d)), 2.0))))));
	} else {
		tmp = w0 * (Math.sqrt((-0.25 * (D / ((l * (d * d)) / (D * h))))) * -M);
	}
	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 = (M * D) / (2.0 * d)
	tmp = 0
	if t_0 <= -1.6633635303706548e+143:
		tmp = w0 * (D * math.sqrt((((-0.25 * h) / l) * ((M / d) * (M / d)))))
	elif t_0 <= 3.6116818720243387e+118:
		tmp = w0 * math.sqrt((1.0 - (1.0 / (l / (h * math.pow((0.5 * ((M * D) / d)), 2.0))))))
	else:
		tmp = w0 * (math.sqrt((-0.25 * (D / ((l * (d * d)) / (D * h))))) * -M)
	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(M * D) / Float64(2.0 * d))
	tmp = 0.0
	if (t_0 <= -1.6633635303706548e+143)
		tmp = Float64(w0 * Float64(D * sqrt(Float64(Float64(Float64(-0.25 * h) / l) * Float64(Float64(M / d) * Float64(M / d))))));
	elseif (t_0 <= 3.6116818720243387e+118)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(1.0 / Float64(l / Float64(h * (Float64(0.5 * Float64(Float64(M * D) / d)) ^ 2.0)))))));
	else
		tmp = Float64(w0 * Float64(sqrt(Float64(-0.25 * Float64(D / Float64(Float64(l * Float64(d * d)) / Float64(D * h))))) * Float64(-M)));
	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);
	tmp = 0.0;
	if (t_0 <= -1.6633635303706548e+143)
		tmp = w0 * (D * sqrt((((-0.25 * h) / l) * ((M / d) * (M / d)))));
	elseif (t_0 <= 3.6116818720243387e+118)
		tmp = w0 * sqrt((1.0 - (1.0 / (l / (h * ((0.5 * ((M * D) / d)) ^ 2.0))))));
	else
		tmp = w0 * (sqrt((-0.25 * (D / ((l * (d * d)) / (D * h))))) * -M);
	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[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1.6633635303706548e+143], N[(w0 * N[(D * N[Sqrt[N[(N[(N[(-0.25 * h), $MachinePrecision] / l), $MachinePrecision] * N[(N[(M / d), $MachinePrecision] * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 3.6116818720243387e+118], N[(w0 * N[Sqrt[N[(1.0 - N[(1.0 / N[(l / N[(h * N[Power[N[(0.5 * N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[(N[Sqrt[N[(-0.25 * N[(D / N[(N[(l * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(D * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-M)), $MachinePrecision]), $MachinePrecision]]]]

Error

Bits error versus w0

Bits error versus M

Bits error versus D

Bits error versus h

Bits error versus l

Bits error versus d

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 M D) (*.f64 2 d)) < -1.66336353037065479e143

   1. Initial program 61.2

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

   2. Applied egg-rr 59.2

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

   3. Taylor expanded in M around 0 61.9

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

   4. Taylor expanded in D around inf 58.1

      \[\leadsto w0 \cdot \color{blue}{\left(D \cdot \sqrt{-0.25 \cdot \frac{h \cdot {M}^{2}}{\ell \cdot {d}^{2}}}\right)} \]

   5. Simplified 50.5

      \[\leadsto w0 \cdot \color{blue}{\left(D \cdot \sqrt{\frac{-0.25 \cdot h}{\ell} \cdot \left(\frac{M}{d} \cdot \frac{M}{d}\right)}\right)} \]

   if -1.66336353037065479e143 < (/.f64 (*.f64 M D) (*.f64 2 d)) < 3.6116818720243387e118

   1. Initial program 6.8

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

   2. Applied egg-rr 3.5

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

   3. Taylor expanded in M around 0 3.1

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

   if 3.6116818720243387e118 < (/.f64 (*.f64 M D) (*.f64 2 d))

   1. Initial program 57.1

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

   2. Applied egg-rr 55.8

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

   3. Taylor expanded in M around 0 57.9

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

   4. Taylor expanded in M around -inf 56.6

      \[\leadsto w0 \cdot \color{blue}{\left(-1 \cdot \left(\sqrt{-0.25 \cdot \frac{{D}^{2} \cdot h}{\ell \cdot {d}^{2}}} \cdot M\right)\right)} \]

   5. Simplified 55.1

      \[\leadsto w0 \cdot \color{blue}{\left(\sqrt{\frac{D}{\frac{\left(d \cdot d\right) \cdot \ell}{D \cdot h}} \cdot -0.25} \cdot \left(-M\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 10.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq -1.6633635303706548 \cdot 10^{+143}:\\ \;\;\;\;w0 \cdot \left(D \cdot \sqrt{\frac{-0.25 \cdot h}{\ell} \cdot \left(\frac{M}{d} \cdot \frac{M}{d}\right)}\right)\\ \mathbf{elif}\;\frac{M \cdot D}{2 \cdot d} \leq 3.6116818720243387 \cdot 10^{+118}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{1}{\frac{\ell}{h \cdot {\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2}}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(\sqrt{-0.25 \cdot \frac{D}{\frac{\ell \cdot \left(d \cdot d\right)}{D \cdot h}}} \cdot \left(-M\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022148 
(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))))))