Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.9% → 87.7%

Time: 17.0s

Alternatives: 3

Speedup: 1.8×

Specification

Math

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

FPCore

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

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))));
}

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

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))));
}

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))))

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

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

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]

Initial Program: 80.9% accurate, 1.0× speedup

Math

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

FPCore

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

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))));
}

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

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))));
}

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))))

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

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

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]

Alternative 1: 87.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ w0 \cdot \sqrt{1 - \left(\frac{D\_m \cdot M\_m}{d \cdot 2} \cdot \frac{D\_m \cdot \frac{M\_m}{d \cdot 2}}{\ell}\right) \cdot h} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (*
  w0
  (sqrt
   (-
    1.0
    (* (* (/ (* D_m M_m) (* d 2.0)) (/ (* D_m (/ M_m (* d 2.0))) l)) h)))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	return w0 * sqrt((1.0 - ((((D_m * M_m) / (d * 2.0)) * ((D_m * (M_m / (d * 2.0))) / l)) * h)));
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    code = w0 * sqrt((1.0d0 - ((((d_m * m_m) / (d * 2.0d0)) * ((d_m * (m_m / (d * 2.0d0))) / l)) * h)))
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	return w0 * Math.sqrt((1.0 - ((((D_m * M_m) / (d * 2.0)) * ((D_m * (M_m / (d * 2.0))) / l)) * h)));
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	return w0 * math.sqrt((1.0 - ((((D_m * M_m) / (d * 2.0)) * ((D_m * (M_m / (d * 2.0))) / l)) * h)))

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	return Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(Float64(D_m * M_m) / Float64(d * 2.0)) * Float64(Float64(D_m * Float64(M_m / Float64(d * 2.0))) / l)) * h))))
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp = code(w0, M_m, D_m, h, l, d)
	tmp = w0 * sqrt((1.0 - ((((D_m * M_m) / (d * 2.0)) * ((D_m * (M_m / (d * 2.0))) / l)) * h)));
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(D$95$m * N[(M$95$m / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.6%

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

3. Simplified 83.1%

   \[\leadsto \color{blue}{w0 \cdot \sqrt{1 - {\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot \frac{h}{\ell}}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. clear-num 83.1%

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

   2. un-div-inv 83.9%

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

   3. associate-/r* 83.9%

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

6. Applied egg-rr 83.9%

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

   1. associate-/r/ 89.5%

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

   2. associate-/r* 89.5%

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

8. Simplified 89.5%

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

   1. unpow2 89.5%

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

   2. associate-/l/ 89.5%

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

   3. associate-/l/ 89.5%

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

10. Applied egg-rr 89.5%

    \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{\left(D \cdot \frac{M}{d \cdot 2}\right) \cdot \left(D \cdot \frac{M}{d \cdot 2}\right)}}{\ell} \cdot h} \]
11. Step-by-step derivation

    1. associate-/l* 91.0%

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

    2. associate-*r/ 90.6%

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

    3. associate-*r/ 91.8%

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

12. Applied egg-rr 91.8%

    \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(\frac{D \cdot M}{d \cdot 2} \cdot \frac{\frac{D \cdot M}{d \cdot 2}}{\ell}\right)} \cdot h} \]
13. Step-by-step derivation

    1. associate-/l* 90.6%

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

14. Applied egg-rr 90.6%

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

Alternative 2: 68.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;D\_m \leq 5.5 \cdot 10^{+265}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\left(\left(D\_m \cdot M\_m\right) \cdot \left(D\_m \cdot M\_m\right)\right) \cdot \left(h \cdot \frac{\frac{w0}{\ell}}{{d}^{2}}\right)\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= D_m 5.5e+265)
   w0
   (* -0.125 (* (* (* D_m M_m) (* D_m M_m)) (* h (/ (/ w0 l) (pow d 2.0)))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (D_m <= 5.5e+265) {
		tmp = w0;
	} else {
		tmp = -0.125 * (((D_m * M_m) * (D_m * M_m)) * (h * ((w0 / l) / pow(d, 2.0))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if (d_m <= 5.5d+265) then
        tmp = w0
    else
        tmp = (-0.125d0) * (((d_m * m_m) * (d_m * m_m)) * (h * ((w0 / l) / (d ** 2.0d0))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (D_m <= 5.5e+265) {
		tmp = w0;
	} else {
		tmp = -0.125 * (((D_m * M_m) * (D_m * M_m)) * (h * ((w0 / l) / Math.pow(d, 2.0))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if D_m <= 5.5e+265:
		tmp = w0
	else:
		tmp = -0.125 * (((D_m * M_m) * (D_m * M_m)) * (h * ((w0 / l) / math.pow(d, 2.0))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (D_m <= 5.5e+265)
		tmp = w0;
	else
		tmp = Float64(-0.125 * Float64(Float64(Float64(D_m * M_m) * Float64(D_m * M_m)) * Float64(h * Float64(Float64(w0 / l) / (d ^ 2.0)))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (D_m <= 5.5e+265)
		tmp = w0;
	else
		tmp = -0.125 * (((D_m * M_m) * (D_m * M_m)) * (h * ((w0 / l) / (d ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[D$95$m, 5.5e+265], w0, N[(-0.125 * N[(N[(N[(D$95$m * M$95$m), $MachinePrecision] * N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] * N[(h * N[(N[(w0 / l), $MachinePrecision] / N[Power[d, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 5.4999999999999997e265

   1. Initial program 83.7%

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

   3. Simplified 83.3%

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

   5. Taylor expanded in D around 0 73.6%

      \[\leadsto \color{blue}{w0} \]

   if 5.4999999999999997e265 < D

   1. Initial program 67.9%

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

   3. Simplified 67.9%

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

   5. Taylor expanded in D around 0 33.3%

      \[\leadsto \color{blue}{w0 + -0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}} \]
   6. Step-by-step derivation

      1. +-commutative 33.3%

         \[\leadsto \color{blue}{-0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell} + w0} \]

      2. fma-define 33.3%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-0.125, \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}, w0\right)} \]

      3. associate-*r* 33.3%

         \[\leadsto \mathsf{fma}\left(-0.125, \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot \left(h \cdot w0\right)}}{{d}^{2} \cdot \ell}, w0\right) \]

      4. unpow2 33.3%

         \[\leadsto \mathsf{fma}\left(-0.125, \frac{\left(\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}\right) \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}, w0\right) \]

      5. unpow2 33.3%

         \[\leadsto \mathsf{fma}\left(-0.125, \frac{\left(\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}\right) \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}, w0\right) \]

      6. swap-sqr 66.7%

         \[\leadsto \mathsf{fma}\left(-0.125, \frac{\color{blue}{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}, w0\right) \]

      7. unpow2 66.7%

         \[\leadsto \mathsf{fma}\left(-0.125, \frac{\color{blue}{{\left(D \cdot M\right)}^{2}} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}, w0\right) \]

      8. *-commutative 66.7%

         \[\leadsto \mathsf{fma}\left(-0.125, \frac{{\left(D \cdot M\right)}^{2} \cdot \color{blue}{\left(w0 \cdot h\right)}}{{d}^{2} \cdot \ell}, w0\right) \]

   7. Simplified 66.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.125, \frac{{\left(D \cdot M\right)}^{2} \cdot \left(w0 \cdot h\right)}{{d}^{2} \cdot \ell}, w0\right)} \]
   8. Step-by-step derivation

      1. expm1-log1p-u 0.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(-0.125, \frac{{\left(D \cdot M\right)}^{2} \cdot \left(w0 \cdot h\right)}{{d}^{2} \cdot \ell}, w0\right)\right)\right)} \]

      2. expm1-undefine 0.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(-0.125, \frac{{\left(D \cdot M\right)}^{2} \cdot \left(w0 \cdot h\right)}{{d}^{2} \cdot \ell}, w0\right)\right)} - 1} \]

      3. times-frac 0.0%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(-0.125, \color{blue}{\frac{{\left(D \cdot M\right)}^{2}}{{d}^{2}} \cdot \frac{w0 \cdot h}{\ell}}, w0\right)\right)} - 1 \]

      4. *-commutative 0.0%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(-0.125, \frac{{\left(D \cdot M\right)}^{2}}{{d}^{2}} \cdot \frac{\color{blue}{h \cdot w0}}{\ell}, w0\right)\right)} - 1 \]

   9. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(-0.125, \frac{{\left(D \cdot M\right)}^{2}}{{d}^{2}} \cdot \frac{h \cdot w0}{\ell}, w0\right)\right)} - 1} \]
   10. Step-by-step derivation

       1. expm1-define 0.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(-0.125, \frac{{\left(D \cdot M\right)}^{2}}{{d}^{2}} \cdot \frac{h \cdot w0}{\ell}, w0\right)\right)\right)} \]

       2. associate-*l/ 0.0%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(-0.125, \color{blue}{\frac{{\left(D \cdot M\right)}^{2} \cdot \frac{h \cdot w0}{\ell}}{{d}^{2}}}, w0\right)\right)\right) \]

       3. associate-*r/ 0.0%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(-0.125, \color{blue}{{\left(D \cdot M\right)}^{2} \cdot \frac{\frac{h \cdot w0}{\ell}}{{d}^{2}}}, w0\right)\right)\right) \]

       4. associate-/r* 0.0%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(-0.125, {\left(D \cdot M\right)}^{2} \cdot \color{blue}{\frac{h \cdot w0}{\ell \cdot {d}^{2}}}, w0\right)\right)\right) \]

       5. associate-/l* 0.0%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(-0.125, {\left(D \cdot M\right)}^{2} \cdot \color{blue}{\left(h \cdot \frac{w0}{\ell \cdot {d}^{2}}\right)}, w0\right)\right)\right) \]

       6. *-commutative 0.0%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(-0.125, {\left(D \cdot M\right)}^{2} \cdot \left(h \cdot \frac{w0}{\color{blue}{{d}^{2} \cdot \ell}}\right), w0\right)\right)\right) \]

   11. Simplified 0.0%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(-0.125, {\left(D \cdot M\right)}^{2} \cdot \left(h \cdot \frac{w0}{{d}^{2} \cdot \ell}\right), w0\right)\right)\right)} \]

   12. Taylor expanded in D around inf 33.3%

       \[\leadsto \color{blue}{-0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}} \]
   13. Step-by-step derivation

       1. associate-*r* 33.3%

          \[\leadsto -0.125 \cdot \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot \left(h \cdot w0\right)}}{{d}^{2} \cdot \ell} \]

       2. unpow2 33.3%

          \[\leadsto -0.125 \cdot \frac{\left(\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}\right) \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell} \]

       3. unpow2 33.3%

          \[\leadsto -0.125 \cdot \frac{\left(\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}\right) \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell} \]

       4. swap-sqr 33.9%

          \[\leadsto -0.125 \cdot \frac{\color{blue}{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell} \]

       5. unpow2 33.9%

          \[\leadsto -0.125 \cdot \frac{\color{blue}{{\left(D \cdot M\right)}^{2}} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell} \]

       6. *-commutative 33.9%

          \[\leadsto -0.125 \cdot \frac{{\left(D \cdot M\right)}^{2} \cdot \left(h \cdot w0\right)}{\color{blue}{\ell \cdot {d}^{2}}} \]

       7. associate-*r/ 33.9%

          \[\leadsto -0.125 \cdot \color{blue}{\left({\left(D \cdot M\right)}^{2} \cdot \frac{h \cdot w0}{\ell \cdot {d}^{2}}\right)} \]

       8. associate-/l* 33.9%

          \[\leadsto -0.125 \cdot \left({\left(D \cdot M\right)}^{2} \cdot \color{blue}{\left(h \cdot \frac{w0}{\ell \cdot {d}^{2}}\right)}\right) \]

       9. associate-/r* 33.3%

          \[\leadsto -0.125 \cdot \left({\left(D \cdot M\right)}^{2} \cdot \left(h \cdot \color{blue}{\frac{\frac{w0}{\ell}}{{d}^{2}}}\right)\right) \]

   14. Simplified 33.3%

       \[\leadsto \color{blue}{-0.125 \cdot \left({\left(D \cdot M\right)}^{2} \cdot \left(h \cdot \frac{\frac{w0}{\ell}}{{d}^{2}}\right)\right)} \]
   15. Step-by-step derivation

       1. unpow2 33.3%

          \[\leadsto -0.125 \cdot \left(\color{blue}{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)} \cdot \left(h \cdot \frac{\frac{w0}{\ell}}{{d}^{2}}\right)\right) \]

   16. Applied egg-rr 33.3%

       \[\leadsto -0.125 \cdot \left(\color{blue}{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)} \cdot \left(h \cdot \frac{\frac{w0}{\ell}}{{d}^{2}}\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 67.9% accurate, 216.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ w0 \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d) :precision binary64 w0)

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	return w0;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    code = w0
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	return w0;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	return w0

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	return w0
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp = code(w0, M_m, D_m, h, l, d)
	tmp = w0;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := w0

Derivation

1. Initial program 83.6%

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

3. Simplified 83.1%

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

5. Taylor expanded in D around 0 73.2%

   \[\leadsto \color{blue}{w0} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024125 
(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))))))