Henrywood and Agarwal, Equation (12)

Percentage Accurate: 35.3% → 71.2%

Time: 6.7s

Alternatives: 3

Speedup: 6.2×

Specification

Math

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

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)))))

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)));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m, d_1)
use fmin_fmax_functions
    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

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)));
}

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)))

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

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

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]

Initial Program: 35.3% accurate, 1.0× speedup

Math

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

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)))))

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)));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m, d_1)
use fmin_fmax_functions
    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

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)));
}

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)))

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

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

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]

Alternative 1: 71.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ M_m = \left|M\right| \\ D_m = \left|D\right| \\ \begin{array}{l} t_0 := d\_m \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{if}\;M\_m \cdot D\_m \leq 10^{-169}:\\ \;\;\;\;t\_0 \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d\_m}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
(FPCore (d_m h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* d_m (sqrt (/ 1.0 (* h l))))))
   (if (<= (* M_m D_m) 1e-169)
     (* t_0 1.0)
     (*
      t_0
      (- 1.0 (* (* 0.5 (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0)) (/ h l)))))))

C

d_m = fabs(d);
M_m = fabs(M);
D_m = fabs(D);
double code(double d_m, double h, double l, double M_m, double D_m) {
	double t_0 = d_m * sqrt((1.0 / (h * l)));
	double tmp;
	if ((M_m * D_m) <= 1e-169) {
		tmp = t_0 * 1.0;
	} else {
		tmp = t_0 * (1.0 - ((0.5 * pow(((M_m * D_m) / (2.0 * d_m)), 2.0)) * (h / l)));
	}
	return tmp;
}

Fortran

d_m =     private
M_m =     private
D_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d_m, h, l, m_m, d_m_1)
use fmin_fmax_functions
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = d_m * sqrt((1.0d0 / (h * l)))
    if ((m_m * d_m_1) <= 1d-169) then
        tmp = t_0 * 1.0d0
    else
        tmp = t_0 * (1.0d0 - ((0.5d0 * (((m_m * d_m_1) / (2.0d0 * d_m)) ** 2.0d0)) * (h / l)))
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
M_m = Math.abs(M);
D_m = Math.abs(D);
public static double code(double d_m, double h, double l, double M_m, double D_m) {
	double t_0 = d_m * Math.sqrt((1.0 / (h * l)));
	double tmp;
	if ((M_m * D_m) <= 1e-169) {
		tmp = t_0 * 1.0;
	} else {
		tmp = t_0 * (1.0 - ((0.5 * Math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0)) * (h / l)));
	}
	return tmp;
}

Python

d_m = math.fabs(d)
M_m = math.fabs(M)
D_m = math.fabs(D)
def code(d_m, h, l, M_m, D_m):
	t_0 = d_m * math.sqrt((1.0 / (h * l)))
	tmp = 0
	if (M_m * D_m) <= 1e-169:
		tmp = t_0 * 1.0
	else:
		tmp = t_0 * (1.0 - ((0.5 * math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0)) * (h / l)))
	return tmp

Julia

d_m = abs(d)
M_m = abs(M)
D_m = abs(D)
function code(d_m, h, l, M_m, D_m)
	t_0 = Float64(d_m * sqrt(Float64(1.0 / Float64(h * l))))
	tmp = 0.0
	if (Float64(M_m * D_m) <= 1e-169)
		tmp = Float64(t_0 * 1.0);
	else
		tmp = Float64(t_0 * Float64(1.0 - Float64(Float64(0.5 * (Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) ^ 2.0)) * Float64(h / l))));
	end
	return tmp
end

MATLAB

d_m = abs(d);
M_m = abs(M);
D_m = abs(D);
function tmp_2 = code(d_m, h, l, M_m, D_m)
	t_0 = d_m * sqrt((1.0 / (h * l)));
	tmp = 0.0;
	if ((M_m * D_m) <= 1e-169)
		tmp = t_0 * 1.0;
	else
		tmp = t_0 * (1.0 - ((0.5 * (((M_m * D_m) / (2.0 * d_m)) ^ 2.0)) * (h / l)));
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
code[d$95$m_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(d$95$m * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(M$95$m * D$95$m), $MachinePrecision], 1e-169], N[(t$95$0 * 1.0), $MachinePrecision], N[(t$95$0 * N[(1.0 - N[(N[(0.5 * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M D) < 1.00000000000000002e-169

   1. Initial program 35.3%

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

   2. Taylor expanded in d around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 69.3

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

   4. Applied rewrites 69.3%

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

   5. Taylor expanded in d around inf

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

   7. Applied rewrites 42.9%

      \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \color{blue}{1} \]

   if 1.00000000000000002e-169 < (*.f64 M D)

   1. Initial program 35.3%

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

   2. Taylor expanded in d around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 69.3

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

   4. Applied rewrites 69.3%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 69.3

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

   6. Applied rewrites 69.3%

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

Alternative 2: 45.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ M_m = \left|M\right| \\ D_m = \left|D\right| \\ \begin{array}{l} \mathbf{if}\;M\_m \cdot D\_m \leq 1.6 \cdot 10^{+235}:\\ \;\;\;\;\left(d\_m \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;h \cdot \left(d\_m \cdot \sqrt{\frac{1}{{h}^{3} \cdot \ell}}\right)\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
(FPCore (d_m h l M_m D_m)
 :precision binary64
 (if (<= (* M_m D_m) 1.6e+235)
   (* (* d_m (sqrt (/ 1.0 (* h l)))) 1.0)
   (* h (* d_m (sqrt (/ 1.0 (* (pow h 3.0) l)))))))

C

d_m = fabs(d);
M_m = fabs(M);
D_m = fabs(D);
double code(double d_m, double h, double l, double M_m, double D_m) {
	double tmp;
	if ((M_m * D_m) <= 1.6e+235) {
		tmp = (d_m * sqrt((1.0 / (h * l)))) * 1.0;
	} else {
		tmp = h * (d_m * sqrt((1.0 / (pow(h, 3.0) * l))));
	}
	return tmp;
}

Fortran

d_m =     private
M_m =     private
D_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d_m, h, l, m_m, d_m_1)
use fmin_fmax_functions
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m_1
    real(8) :: tmp
    if ((m_m * d_m_1) <= 1.6d+235) then
        tmp = (d_m * sqrt((1.0d0 / (h * l)))) * 1.0d0
    else
        tmp = h * (d_m * sqrt((1.0d0 / ((h ** 3.0d0) * l))))
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
M_m = Math.abs(M);
D_m = Math.abs(D);
public static double code(double d_m, double h, double l, double M_m, double D_m) {
	double tmp;
	if ((M_m * D_m) <= 1.6e+235) {
		tmp = (d_m * Math.sqrt((1.0 / (h * l)))) * 1.0;
	} else {
		tmp = h * (d_m * Math.sqrt((1.0 / (Math.pow(h, 3.0) * l))));
	}
	return tmp;
}

Python

d_m = math.fabs(d)
M_m = math.fabs(M)
D_m = math.fabs(D)
def code(d_m, h, l, M_m, D_m):
	tmp = 0
	if (M_m * D_m) <= 1.6e+235:
		tmp = (d_m * math.sqrt((1.0 / (h * l)))) * 1.0
	else:
		tmp = h * (d_m * math.sqrt((1.0 / (math.pow(h, 3.0) * l))))
	return tmp

Julia

d_m = abs(d)
M_m = abs(M)
D_m = abs(D)
function code(d_m, h, l, M_m, D_m)
	tmp = 0.0
	if (Float64(M_m * D_m) <= 1.6e+235)
		tmp = Float64(Float64(d_m * sqrt(Float64(1.0 / Float64(h * l)))) * 1.0);
	else
		tmp = Float64(h * Float64(d_m * sqrt(Float64(1.0 / Float64((h ^ 3.0) * l)))));
	end
	return tmp
end

MATLAB

d_m = abs(d);
M_m = abs(M);
D_m = abs(D);
function tmp_2 = code(d_m, h, l, M_m, D_m)
	tmp = 0.0;
	if ((M_m * D_m) <= 1.6e+235)
		tmp = (d_m * sqrt((1.0 / (h * l)))) * 1.0;
	else
		tmp = h * (d_m * sqrt((1.0 / ((h ^ 3.0) * l))));
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
code[d$95$m_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[N[(M$95$m * D$95$m), $MachinePrecision], 1.6e+235], N[(N[(d$95$m * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(h * N[(d$95$m * N[Sqrt[N[(1.0 / N[(N[Power[h, 3.0], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M D) < 1.60000000000000003e235

   1. Initial program 35.3%

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

   2. Taylor expanded in d around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 69.3

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

   4. Applied rewrites 69.3%

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

   5. Taylor expanded in d around inf

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

   7. Applied rewrites 42.9%

      \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \color{blue}{1} \]

   if 1.60000000000000003e235 < (*.f64 M D)

   1. Initial program 35.3%

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

   2. Taylor expanded in h around inf

      \[\leadsto \color{blue}{h \cdot \left(\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{1}{h \cdot {\ell}^{3}}}\right) + d \cdot \sqrt{\frac{1}{{h}^{3} \cdot \ell}}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto h \cdot \color{blue}{\left(\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{1}{h \cdot {\ell}^{3}}}\right) + d \cdot \sqrt{\frac{1}{{h}^{3} \cdot \ell}}\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto h \cdot \mathsf{fma}\left(\frac{-1}{8}, \color{blue}{\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{1}{h \cdot {\ell}^{3}}}}, d \cdot \sqrt{\frac{1}{{h}^{3} \cdot \ell}}\right) \]

   4. Applied rewrites 13.7%

      \[\leadsto \color{blue}{h \cdot \mathsf{fma}\left(-0.125, \frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{1}{h \cdot {\ell}^{3}}}, d \cdot \sqrt{\frac{1}{{h}^{3} \cdot \ell}}\right)} \]

   5. Taylor expanded in d around 0

      \[\leadsto h \cdot \left(\frac{-1}{8} \cdot \color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{1}{h \cdot {\ell}^{3}}}\right)}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto h \cdot \left(\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \color{blue}{\sqrt{\frac{1}{h \cdot {\ell}^{3}}}}\right)\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto h \cdot \left(\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{1}{h \cdot {\ell}^{3}}}\right)\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto h \cdot \left(\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{1}{h \cdot {\ell}^{3}}}\right)\right) \]

      4. lift-*.f64 N/A

         \[\leadsto h \cdot \left(\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{1}{h \cdot {\ell}^{3}}}\right)\right) \]

      5. lift-/.f64 N/A

         \[\leadsto h \cdot \left(\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{1}{h \cdot {\ell}^{3}}}\right)\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto h \cdot \left(\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{1}{h \cdot {\ell}^{3}}}\right)\right) \]

      7. lift-*.f64 N/A

         \[\leadsto h \cdot \left(\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{1}{h \cdot {\ell}^{3}}}\right)\right) \]

      8. lift-/.f64 N/A

         \[\leadsto h \cdot \left(\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{1}{h \cdot {\ell}^{3}}}\right)\right) \]

      9. lift-sqrt.f64 N/A

         \[\leadsto h \cdot \left(\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{1}{h \cdot {\ell}^{3}}}\right)\right) \]

      10. lift-*.f64 15.3

          \[\leadsto h \cdot \left(-0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{1}{h \cdot {\ell}^{3}}}\right)\right) \]

   7. Applied rewrites 15.3%

      \[\leadsto h \cdot \left(-0.125 \cdot \color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{1}{h \cdot {\ell}^{3}}}\right)}\right) \]

   8. Taylor expanded in d around inf

      \[\leadsto h \cdot \left(d \cdot \color{blue}{\sqrt{\frac{1}{{h}^{3} \cdot \ell}}}\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto h \cdot \left(d \cdot \sqrt{\frac{1}{{h}^{3} \cdot \ell}}\right) \]

      2. lower-sqrt.f64 N/A

         \[\leadsto h \cdot \left(d \cdot \sqrt{\frac{1}{{h}^{3} \cdot \ell}}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto h \cdot \left(d \cdot \sqrt{\frac{1}{{h}^{3} \cdot \ell}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto h \cdot \left(d \cdot \sqrt{\frac{1}{{h}^{3} \cdot \ell}}\right) \]

      5. lift-pow.f64 18.4

         \[\leadsto h \cdot \left(d \cdot \sqrt{\frac{1}{{h}^{3} \cdot \ell}}\right) \]

   10. Applied rewrites 18.4%

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

Alternative 3: 42.9% accurate, 6.2× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ M_m = \left|M\right| \\ D_m = \left|D\right| \\ \left(d\_m \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot 1 \end{array} \]

FPCore

d_m = (fabs.f64 d)
M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
(FPCore (d_m h l M_m D_m)
 :precision binary64
 (* (* d_m (sqrt (/ 1.0 (* h l)))) 1.0))

C

d_m = fabs(d);
M_m = fabs(M);
D_m = fabs(D);
double code(double d_m, double h, double l, double M_m, double D_m) {
	return (d_m * sqrt((1.0 / (h * l)))) * 1.0;
}

Fortran

d_m =     private
M_m =     private
D_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d_m, h, l, m_m, d_m_1)
use fmin_fmax_functions
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m_1
    code = (d_m * sqrt((1.0d0 / (h * l)))) * 1.0d0
end function

Java

d_m = Math.abs(d);
M_m = Math.abs(M);
D_m = Math.abs(D);
public static double code(double d_m, double h, double l, double M_m, double D_m) {
	return (d_m * Math.sqrt((1.0 / (h * l)))) * 1.0;
}

Python

d_m = math.fabs(d)
M_m = math.fabs(M)
D_m = math.fabs(D)
def code(d_m, h, l, M_m, D_m):
	return (d_m * math.sqrt((1.0 / (h * l)))) * 1.0

Julia

d_m = abs(d)
M_m = abs(M)
D_m = abs(D)
function code(d_m, h, l, M_m, D_m)
	return Float64(Float64(d_m * sqrt(Float64(1.0 / Float64(h * l)))) * 1.0)
end

MATLAB

d_m = abs(d);
M_m = abs(M);
D_m = abs(D);
function tmp = code(d_m, h, l, M_m, D_m)
	tmp = (d_m * sqrt((1.0 / (h * l)))) * 1.0;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
code[d$95$m_, h_, l_, M$95$m_, D$95$m_] := N[(N[(d$95$m * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]

Derivation

1. Initial program 35.3%

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

2. Taylor expanded in d around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-*.f64 69.3

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

4. Applied rewrites 69.3%

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

5. Taylor expanded in d around inf

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

7. Applied rewrites 42.9%

   \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \color{blue}{1} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025132 
(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)))))