Henrywood and Agarwal, Equation (12)

Percentage Accurate: 65.6% → 75.0%

Time: 8.9s

Alternatives: 13

Speedup: 2.8×

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: 65.6% 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: 75.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(-{\left(\sqrt{h \cdot \ell}\right)}^{-1} \cdot d\right) \cdot \left(1 - \frac{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right)\\ \mathbf{elif}\;h \leq 5.7 \cdot 10^{+135}:\\ \;\;\;\;\left(-\left(-{\left(\ell \cdot h\right)}^{-0.5} \cdot d\right)\right) \cdot \left(1 - \frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \frac{\sqrt{d}}{\sqrt{h}}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= h -5e-310)
   (*
    (- (* (pow (sqrt (* h l)) -1.0) d))
    (- 1.0 (/ (* (* (pow (* (/ M 2.0) (/ D d)) 2.0) 0.5) h) l)))
   (if (<= h 5.7e+135)
     (*
      (- (- (* (pow (* l h) -0.5) d)))
      (- 1.0 (/ (* (* (pow (* (* M 0.5) (/ D d)) 2.0) 0.5) h) l)))
     (*
      (* (sqrt (/ d l)) (/ (sqrt d) (sqrt h)))
      (- 1.0 (* (* 0.5 (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= -5e-310) {
		tmp = -(pow(sqrt((h * l)), -1.0) * d) * (1.0 - (((pow(((M / 2.0) * (D / d)), 2.0) * 0.5) * h) / l));
	} else if (h <= 5.7e+135) {
		tmp = -(-(pow((l * h), -0.5) * d)) * (1.0 - (((pow(((M * 0.5) * (D / d)), 2.0) * 0.5) * h) / l));
	} else {
		tmp = (sqrt((d / l)) * (sqrt(d) / sqrt(h))) * (1.0 - ((0.5 * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (h <= (-5d-310)) then
        tmp = -((sqrt((h * l)) ** (-1.0d0)) * d) * (1.0d0 - ((((((m / 2.0d0) * (d_1 / d)) ** 2.0d0) * 0.5d0) * h) / l))
    else if (h <= 5.7d+135) then
        tmp = -(-(((l * h) ** (-0.5d0)) * d)) * (1.0d0 - ((((((m * 0.5d0) * (d_1 / d)) ** 2.0d0) * 0.5d0) * h) / l))
    else
        tmp = (sqrt((d / l)) * (sqrt(d) / sqrt(h))) * (1.0d0 - ((0.5d0 * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= -5e-310) {
		tmp = -(Math.pow(Math.sqrt((h * l)), -1.0) * d) * (1.0 - (((Math.pow(((M / 2.0) * (D / d)), 2.0) * 0.5) * h) / l));
	} else if (h <= 5.7e+135) {
		tmp = -(-(Math.pow((l * h), -0.5) * d)) * (1.0 - (((Math.pow(((M * 0.5) * (D / d)), 2.0) * 0.5) * h) / l));
	} else {
		tmp = (Math.sqrt((d / l)) * (Math.sqrt(d) / Math.sqrt(h))) * (1.0 - ((0.5 * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if h <= -5e-310:
		tmp = -(math.pow(math.sqrt((h * l)), -1.0) * d) * (1.0 - (((math.pow(((M / 2.0) * (D / d)), 2.0) * 0.5) * h) / l))
	elif h <= 5.7e+135:
		tmp = -(-(math.pow((l * h), -0.5) * d)) * (1.0 - (((math.pow(((M * 0.5) * (D / d)), 2.0) * 0.5) * h) / l))
	else:
		tmp = (math.sqrt((d / l)) * (math.sqrt(d) / math.sqrt(h))) * (1.0 - ((0.5 * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (h <= -5e-310)
		tmp = Float64(Float64(-Float64((sqrt(Float64(h * l)) ^ -1.0) * d)) * Float64(1.0 - Float64(Float64(Float64((Float64(Float64(M / 2.0) * Float64(D / d)) ^ 2.0) * 0.5) * h) / l)));
	elseif (h <= 5.7e+135)
		tmp = Float64(Float64(-Float64(-Float64((Float64(l * h) ^ -0.5) * d))) * Float64(1.0 - Float64(Float64(Float64((Float64(Float64(M * 0.5) * Float64(D / d)) ^ 2.0) * 0.5) * h) / l)));
	else
		tmp = Float64(Float64(sqrt(Float64(d / l)) * Float64(sqrt(d) / sqrt(h))) * Float64(1.0 - Float64(Float64(0.5 * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (h <= -5e-310)
		tmp = -((sqrt((h * l)) ^ -1.0) * d) * (1.0 - ((((((M / 2.0) * (D / d)) ^ 2.0) * 0.5) * h) / l));
	elseif (h <= 5.7e+135)
		tmp = -(-(((l * h) ^ -0.5) * d)) * (1.0 - ((((((M * 0.5) * (D / d)) ^ 2.0) * 0.5) * h) / l));
	else
		tmp = (sqrt((d / l)) * (sqrt(d) / sqrt(h))) * (1.0 - ((0.5 * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[h, -5e-310], N[((-N[(N[Power[N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision], -1.0], $MachinePrecision] * d), $MachinePrecision]) * N[(1.0 - N[(N[(N[(N[Power[N[(N[(M / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * 0.5), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 5.7e+135], N[((-(-N[(N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision] * d), $MachinePrecision])) * N[(1.0 - N[(N[(N[(N[Power[N[(N[(M * 0.5), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * 0.5), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(0.5 * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if h < -4.999999999999985e-310

   1. Initial program 65.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

   3. Applied rewrites 66.8%

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

   4. Taylor expanded in d around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sqrt-pow2 N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. inv-pow N/A

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

      13. *-commutative N/A

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

      14. sqrt-pow1 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. metadata-eval 2.9

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

   6. Applied rewrites 2.9%

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

   7. Taylor expanded in h around -inf

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

      1. sqrt-pow2 N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. sqrt-div N/A

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

      11. metadata-eval N/A

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

      12. *-commutative N/A

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

      13. inv-pow N/A

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

      14. *-commutative N/A

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

      15. lower-pow.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-sqrt.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 73.6

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

   9. Applied rewrites 73.6%

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

   if -4.999999999999985e-310 < h < 5.7000000000000002e135

   1. Initial program 71.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

   3. Applied rewrites 72.4%

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

   4. Taylor expanded in d around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sqrt-pow2 N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. inv-pow N/A

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

      13. *-commutative N/A

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

      14. sqrt-pow1 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. metadata-eval 80.2

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

   6. Applied rewrites 80.2%

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

   7. Taylor expanded in M around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 80.2

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

   9. Applied rewrites 80.2%

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

   if 5.7000000000000002e135 < h

   1. Initial program 52.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. pow1/2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. metadata-eval N/A

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

      15. pow1/2 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lift-/.f64 52.3

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

   3. Applied rewrites 52.3%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 52.3

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

   5. Applied rewrites 52.3%

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

      1. lift-/.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. sqrt-div N/A

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

      4. lower-/.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lower-sqrt.f64 66.7

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

   7. Applied rewrites 66.7%

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

Alternative 2: 69.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;h \leq -9 \cdot 10^{-143}:\\ \;\;\;\;t\_0 \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 - \frac{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)\right)\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{elif}\;h \leq 5.7 \cdot 10^{+135}:\\ \;\;\;\;\left(-\left(-{\left(\ell \cdot h\right)}^{-0.5} \cdot d\right)\right) \cdot \left(1 - \frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot \frac{\sqrt{d}}{\sqrt{h}}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l))))
   (if (<= h -9e-143)
     (*
      t_0
      (*
       (sqrt (/ d h))
       (- 1.0 (/ (* (pow (* (/ M 2.0) (/ D d)) 2.0) (* 0.5 h)) l))))
     (if (<= h -5e-310)
       (* (- d) (sqrt (pow (* l h) -1.0)))
       (if (<= h 5.7e+135)
         (*
          (- (- (* (pow (* l h) -0.5) d)))
          (- 1.0 (/ (* (* (pow (* (* M 0.5) (/ D d)) 2.0) 0.5) h) l)))
         (*
          (* t_0 (/ (sqrt d) (sqrt h)))
          (- 1.0 (* (* 0.5 (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / l));
	double tmp;
	if (h <= -9e-143) {
		tmp = t_0 * (sqrt((d / h)) * (1.0 - ((pow(((M / 2.0) * (D / d)), 2.0) * (0.5 * h)) / l)));
	} else if (h <= -5e-310) {
		tmp = -d * sqrt(pow((l * h), -1.0));
	} else if (h <= 5.7e+135) {
		tmp = -(-(pow((l * h), -0.5) * d)) * (1.0 - (((pow(((M * 0.5) * (D / d)), 2.0) * 0.5) * h) / l));
	} else {
		tmp = (t_0 * (sqrt(d) / sqrt(h))) * (1.0 - ((0.5 * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((d / l))
    if (h <= (-9d-143)) then
        tmp = t_0 * (sqrt((d / h)) * (1.0d0 - (((((m / 2.0d0) * (d_1 / d)) ** 2.0d0) * (0.5d0 * h)) / l)))
    else if (h <= (-5d-310)) then
        tmp = -d * sqrt(((l * h) ** (-1.0d0)))
    else if (h <= 5.7d+135) then
        tmp = -(-(((l * h) ** (-0.5d0)) * d)) * (1.0d0 - ((((((m * 0.5d0) * (d_1 / d)) ** 2.0d0) * 0.5d0) * h) / l))
    else
        tmp = (t_0 * (sqrt(d) / sqrt(h))) * (1.0d0 - ((0.5d0 * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((d / l));
	double tmp;
	if (h <= -9e-143) {
		tmp = t_0 * (Math.sqrt((d / h)) * (1.0 - ((Math.pow(((M / 2.0) * (D / d)), 2.0) * (0.5 * h)) / l)));
	} else if (h <= -5e-310) {
		tmp = -d * Math.sqrt(Math.pow((l * h), -1.0));
	} else if (h <= 5.7e+135) {
		tmp = -(-(Math.pow((l * h), -0.5) * d)) * (1.0 - (((Math.pow(((M * 0.5) * (D / d)), 2.0) * 0.5) * h) / l));
	} else {
		tmp = (t_0 * (Math.sqrt(d) / Math.sqrt(h))) * (1.0 - ((0.5 * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((d / l))
	tmp = 0
	if h <= -9e-143:
		tmp = t_0 * (math.sqrt((d / h)) * (1.0 - ((math.pow(((M / 2.0) * (D / d)), 2.0) * (0.5 * h)) / l)))
	elif h <= -5e-310:
		tmp = -d * math.sqrt(math.pow((l * h), -1.0))
	elif h <= 5.7e+135:
		tmp = -(-(math.pow((l * h), -0.5) * d)) * (1.0 - (((math.pow(((M * 0.5) * (D / d)), 2.0) * 0.5) * h) / l))
	else:
		tmp = (t_0 * (math.sqrt(d) / math.sqrt(h))) * (1.0 - ((0.5 * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(d / l))
	tmp = 0.0
	if (h <= -9e-143)
		tmp = Float64(t_0 * Float64(sqrt(Float64(d / h)) * Float64(1.0 - Float64(Float64((Float64(Float64(M / 2.0) * Float64(D / d)) ^ 2.0) * Float64(0.5 * h)) / l))));
	elseif (h <= -5e-310)
		tmp = Float64(Float64(-d) * sqrt((Float64(l * h) ^ -1.0)));
	elseif (h <= 5.7e+135)
		tmp = Float64(Float64(-Float64(-Float64((Float64(l * h) ^ -0.5) * d))) * Float64(1.0 - Float64(Float64(Float64((Float64(Float64(M * 0.5) * Float64(D / d)) ^ 2.0) * 0.5) * h) / l)));
	else
		tmp = Float64(Float64(t_0 * Float64(sqrt(d) / sqrt(h))) * Float64(1.0 - Float64(Float64(0.5 * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((d / l));
	tmp = 0.0;
	if (h <= -9e-143)
		tmp = t_0 * (sqrt((d / h)) * (1.0 - (((((M / 2.0) * (D / d)) ^ 2.0) * (0.5 * h)) / l)));
	elseif (h <= -5e-310)
		tmp = -d * sqrt(((l * h) ^ -1.0));
	elseif (h <= 5.7e+135)
		tmp = -(-(((l * h) ^ -0.5) * d)) * (1.0 - ((((((M * 0.5) * (D / d)) ^ 2.0) * 0.5) * h) / l));
	else
		tmp = (t_0 * (sqrt(d) / sqrt(h))) * (1.0 - ((0.5 * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[h, -9e-143], N[(t$95$0 * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(N[(N[Power[N[(N[(M / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(0.5 * h), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -5e-310], N[((-d) * N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 5.7e+135], N[((-(-N[(N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision] * d), $MachinePrecision])) * N[(1.0 - N[(N[(N[(N[Power[N[(N[(M * 0.5), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * 0.5), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(0.5 * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if h < -9.00000000000000001e-143

   1. Initial program 66.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. pow1/2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. metadata-eval N/A

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

      15. pow1/2 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lift-/.f64 66.0

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

   3. Applied rewrites 66.0%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 66.0

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

   5. Applied rewrites 66.0%

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

   7. Applied rewrites 67.6%

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

   if -9.00000000000000001e-143 < h < -4.999999999999985e-310

   1. Initial program 63.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow-prod-down N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 56.4

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

      11. lift-/.f64 N/A

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

      12. metadata-eval 56.4

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

   3. Applied rewrites 56.4%

      \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. unpow2 N/A

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

      6. *-commutative N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 46.1

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

   5. Applied rewrites 46.1%

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

   6. Taylor expanded in d around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. inv-pow N/A

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

      6. *-commutative N/A

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

      7. exp-to-pow N/A

         \[\leadsto \left(-d\right) \cdot \sqrt{e^{\log \left(\ell \cdot h\right) \cdot -1}} \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(-d\right) \cdot \sqrt{e^{\log \left(\ell \cdot h\right) \cdot -1}} \]

      9. exp-to-pow N/A

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

      10. lower-pow.f64 N/A

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

      11. lift-*.f64 52.2

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

   8. Applied rewrites 52.2%

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

   if -4.999999999999985e-310 < h < 5.7000000000000002e135

   1. Initial program 71.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

   3. Applied rewrites 72.4%

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

   4. Taylor expanded in d around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sqrt-pow2 N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. inv-pow N/A

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

      13. *-commutative N/A

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

      14. sqrt-pow1 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. metadata-eval 80.2

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

   6. Applied rewrites 80.2%

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

   7. Taylor expanded in M around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 80.2

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

   9. Applied rewrites 80.2%

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

   if 5.7000000000000002e135 < h

   1. Initial program 52.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. pow1/2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. metadata-eval N/A

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

      15. pow1/2 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lift-/.f64 52.3

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

   3. Applied rewrites 52.3%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 52.3

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

   5. Applied rewrites 52.3%

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

      1. lift-/.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. sqrt-div N/A

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

      4. lower-/.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lower-sqrt.f64 66.7

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

   7. Applied rewrites 66.7%

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

Alternative 3: 72.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\\ t_1 := -{\left(\ell \cdot h\right)}^{-0.5} \cdot d\\ \mathbf{if}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;t\_1 \cdot \left(1 - \left(\frac{1}{2} \cdot t\_0\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{elif}\;h \leq 5.7 \cdot 10^{+135}:\\ \;\;\;\;\left(-t\_1\right) \cdot \left(1 - \frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \frac{\sqrt{d}}{\sqrt{h}}\right) \cdot \left(1 - \left(0.5 \cdot t\_0\right) \cdot \frac{h}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (pow (/ (* M D) (* 2.0 d)) 2.0))
        (t_1 (- (* (pow (* l h) -0.5) d))))
   (if (<= h -5e-310)
     (* t_1 (- 1.0 (* (* (/ 1.0 2.0) t_0) (/ h l))))
     (if (<= h 5.7e+135)
       (* (- t_1) (- 1.0 (/ (* (* (pow (* (* M 0.5) (/ D d)) 2.0) 0.5) h) l)))
       (*
        (* (sqrt (/ d l)) (/ (sqrt d) (sqrt h)))
        (- 1.0 (* (* 0.5 t_0) (/ h l))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = pow(((M * D) / (2.0 * d)), 2.0);
	double t_1 = -(pow((l * h), -0.5) * d);
	double tmp;
	if (h <= -5e-310) {
		tmp = t_1 * (1.0 - (((1.0 / 2.0) * t_0) * (h / l)));
	} else if (h <= 5.7e+135) {
		tmp = -t_1 * (1.0 - (((pow(((M * 0.5) * (D / d)), 2.0) * 0.5) * h) / l));
	} else {
		tmp = (sqrt((d / l)) * (sqrt(d) / sqrt(h))) * (1.0 - ((0.5 * t_0) * (h / l)));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((m * d_1) / (2.0d0 * d)) ** 2.0d0
    t_1 = -(((l * h) ** (-0.5d0)) * d)
    if (h <= (-5d-310)) then
        tmp = t_1 * (1.0d0 - (((1.0d0 / 2.0d0) * t_0) * (h / l)))
    else if (h <= 5.7d+135) then
        tmp = -t_1 * (1.0d0 - ((((((m * 0.5d0) * (d_1 / d)) ** 2.0d0) * 0.5d0) * h) / l))
    else
        tmp = (sqrt((d / l)) * (sqrt(d) / sqrt(h))) * (1.0d0 - ((0.5d0 * t_0) * (h / l)))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.pow(((M * D) / (2.0 * d)), 2.0);
	double t_1 = -(Math.pow((l * h), -0.5) * d);
	double tmp;
	if (h <= -5e-310) {
		tmp = t_1 * (1.0 - (((1.0 / 2.0) * t_0) * (h / l)));
	} else if (h <= 5.7e+135) {
		tmp = -t_1 * (1.0 - (((Math.pow(((M * 0.5) * (D / d)), 2.0) * 0.5) * h) / l));
	} else {
		tmp = (Math.sqrt((d / l)) * (Math.sqrt(d) / Math.sqrt(h))) * (1.0 - ((0.5 * t_0) * (h / l)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.pow(((M * D) / (2.0 * d)), 2.0)
	t_1 = -(math.pow((l * h), -0.5) * d)
	tmp = 0
	if h <= -5e-310:
		tmp = t_1 * (1.0 - (((1.0 / 2.0) * t_0) * (h / l)))
	elif h <= 5.7e+135:
		tmp = -t_1 * (1.0 - (((math.pow(((M * 0.5) * (D / d)), 2.0) * 0.5) * h) / l))
	else:
		tmp = (math.sqrt((d / l)) * (math.sqrt(d) / math.sqrt(h))) * (1.0 - ((0.5 * t_0) * (h / l)))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0
	t_1 = Float64(-Float64((Float64(l * h) ^ -0.5) * d))
	tmp = 0.0
	if (h <= -5e-310)
		tmp = Float64(t_1 * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * t_0) * Float64(h / l))));
	elseif (h <= 5.7e+135)
		tmp = Float64(Float64(-t_1) * Float64(1.0 - Float64(Float64(Float64((Float64(Float64(M * 0.5) * Float64(D / d)) ^ 2.0) * 0.5) * h) / l)));
	else
		tmp = Float64(Float64(sqrt(Float64(d / l)) * Float64(sqrt(d) / sqrt(h))) * Float64(1.0 - Float64(Float64(0.5 * t_0) * Float64(h / l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = ((M * D) / (2.0 * d)) ^ 2.0;
	t_1 = -(((l * h) ^ -0.5) * d);
	tmp = 0.0;
	if (h <= -5e-310)
		tmp = t_1 * (1.0 - (((1.0 / 2.0) * t_0) * (h / l)));
	elseif (h <= 5.7e+135)
		tmp = -t_1 * (1.0 - ((((((M * 0.5) * (D / d)) ^ 2.0) * 0.5) * h) / l));
	else
		tmp = (sqrt((d / l)) * (sqrt(d) / sqrt(h))) * (1.0 - ((0.5 * t_0) * (h / l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = (-N[(N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision] * d), $MachinePrecision])}, If[LessEqual[h, -5e-310], N[(t$95$1 * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 5.7e+135], N[((-t$95$1) * N[(1.0 - N[(N[(N[(N[Power[N[(N[(M * 0.5), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * 0.5), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(0.5 * t$95$0), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if h < -4.999999999999985e-310

   1. Initial program 65.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow-prod-down N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 53.9

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

      11. lift-/.f64 N/A

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

      12. metadata-eval 53.9

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

   3. Applied rewrites 53.9%

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

   4. Taylor expanded in d around -inf

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

      1. unpow-prod-down N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. inv-pow N/A

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

      9. *-commutative N/A

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

      10. sqrt-pow1 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. metadata-eval 69.3

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

   6. Applied rewrites 69.3%

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

   if -4.999999999999985e-310 < h < 5.7000000000000002e135

   1. Initial program 71.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

   3. Applied rewrites 72.4%

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

   4. Taylor expanded in d around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sqrt-pow2 N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. inv-pow N/A

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

      13. *-commutative N/A

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

      14. sqrt-pow1 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. metadata-eval 80.2

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

   6. Applied rewrites 80.2%

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

   7. Taylor expanded in M around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 80.2

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

   9. Applied rewrites 80.2%

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

   if 5.7000000000000002e135 < h

   1. Initial program 52.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. pow1/2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. metadata-eval N/A

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

      15. pow1/2 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lift-/.f64 52.3

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

   3. Applied rewrites 52.3%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 52.3

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

   5. Applied rewrites 52.3%

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

      1. lift-/.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. sqrt-div N/A

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

      4. lower-/.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lower-sqrt.f64 66.7

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

   7. Applied rewrites 66.7%

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

Alternative 4: 71.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\\ \mathbf{if}\;d \leq 2.5 \cdot 10^{-278}:\\ \;\;\;\;t\_0 \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 - \frac{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)\right)\\ \mathbf{elif}\;d \leq 3.6 \cdot 10^{-162}:\\ \;\;\;\;\left(\frac{1}{\sqrt{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(\frac{1}{2} \cdot t\_1\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot \frac{\sqrt{d}}{\sqrt{h}}\right) \cdot \left(1 - \left(0.5 \cdot t\_1\right) \cdot \frac{h}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l))) (t_1 (pow (/ (* M D) (* 2.0 d)) 2.0)))
   (if (<= d 2.5e-278)
     (*
      t_0
      (*
       (sqrt (/ d h))
       (- 1.0 (/ (* (pow (* (/ M 2.0) (/ D d)) 2.0) (* 0.5 h)) l))))
     (if (<= d 3.6e-162)
       (* (* (/ 1.0 (sqrt (* l h))) d) (- 1.0 (* (* (/ 1.0 2.0) t_1) (/ h l))))
       (* (* t_0 (/ (sqrt d) (sqrt h))) (- 1.0 (* (* 0.5 t_1) (/ h l))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / l));
	double t_1 = pow(((M * D) / (2.0 * d)), 2.0);
	double tmp;
	if (d <= 2.5e-278) {
		tmp = t_0 * (sqrt((d / h)) * (1.0 - ((pow(((M / 2.0) * (D / d)), 2.0) * (0.5 * h)) / l)));
	} else if (d <= 3.6e-162) {
		tmp = ((1.0 / sqrt((l * h))) * d) * (1.0 - (((1.0 / 2.0) * t_1) * (h / l)));
	} else {
		tmp = (t_0 * (sqrt(d) / sqrt(h))) * (1.0 - ((0.5 * t_1) * (h / l)));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt((d / l))
    t_1 = ((m * d_1) / (2.0d0 * d)) ** 2.0d0
    if (d <= 2.5d-278) then
        tmp = t_0 * (sqrt((d / h)) * (1.0d0 - (((((m / 2.0d0) * (d_1 / d)) ** 2.0d0) * (0.5d0 * h)) / l)))
    else if (d <= 3.6d-162) then
        tmp = ((1.0d0 / sqrt((l * h))) * d) * (1.0d0 - (((1.0d0 / 2.0d0) * t_1) * (h / l)))
    else
        tmp = (t_0 * (sqrt(d) / sqrt(h))) * (1.0d0 - ((0.5d0 * t_1) * (h / l)))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((d / l));
	double t_1 = Math.pow(((M * D) / (2.0 * d)), 2.0);
	double tmp;
	if (d <= 2.5e-278) {
		tmp = t_0 * (Math.sqrt((d / h)) * (1.0 - ((Math.pow(((M / 2.0) * (D / d)), 2.0) * (0.5 * h)) / l)));
	} else if (d <= 3.6e-162) {
		tmp = ((1.0 / Math.sqrt((l * h))) * d) * (1.0 - (((1.0 / 2.0) * t_1) * (h / l)));
	} else {
		tmp = (t_0 * (Math.sqrt(d) / Math.sqrt(h))) * (1.0 - ((0.5 * t_1) * (h / l)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((d / l))
	t_1 = math.pow(((M * D) / (2.0 * d)), 2.0)
	tmp = 0
	if d <= 2.5e-278:
		tmp = t_0 * (math.sqrt((d / h)) * (1.0 - ((math.pow(((M / 2.0) * (D / d)), 2.0) * (0.5 * h)) / l)))
	elif d <= 3.6e-162:
		tmp = ((1.0 / math.sqrt((l * h))) * d) * (1.0 - (((1.0 / 2.0) * t_1) * (h / l)))
	else:
		tmp = (t_0 * (math.sqrt(d) / math.sqrt(h))) * (1.0 - ((0.5 * t_1) * (h / l)))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(d / l))
	t_1 = Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0
	tmp = 0.0
	if (d <= 2.5e-278)
		tmp = Float64(t_0 * Float64(sqrt(Float64(d / h)) * Float64(1.0 - Float64(Float64((Float64(Float64(M / 2.0) * Float64(D / d)) ^ 2.0) * Float64(0.5 * h)) / l))));
	elseif (d <= 3.6e-162)
		tmp = Float64(Float64(Float64(1.0 / sqrt(Float64(l * h))) * d) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * t_1) * Float64(h / l))));
	else
		tmp = Float64(Float64(t_0 * Float64(sqrt(d) / sqrt(h))) * Float64(1.0 - Float64(Float64(0.5 * t_1) * Float64(h / l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((d / l));
	t_1 = ((M * D) / (2.0 * d)) ^ 2.0;
	tmp = 0.0;
	if (d <= 2.5e-278)
		tmp = t_0 * (sqrt((d / h)) * (1.0 - (((((M / 2.0) * (D / d)) ^ 2.0) * (0.5 * h)) / l)));
	elseif (d <= 3.6e-162)
		tmp = ((1.0 / sqrt((l * h))) * d) * (1.0 - (((1.0 / 2.0) * t_1) * (h / l)));
	else
		tmp = (t_0 * (sqrt(d) / sqrt(h))) * (1.0 - ((0.5 * t_1) * (h / l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[d, 2.5e-278], N[(t$95$0 * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(N[(N[Power[N[(N[(M / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(0.5 * h), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 3.6e-162], N[(N[(N[(1.0 / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(0.5 * t$95$1), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if d < 2.49999999999999992e-278

   1. Initial program 63.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. pow1/2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. metadata-eval N/A

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

      15. pow1/2 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lift-/.f64 63.9

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

   3. Applied rewrites 63.9%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 63.9

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

   5. Applied rewrites 63.9%

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

   7. Applied rewrites 65.3%

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

   if 2.49999999999999992e-278 < d < 3.5999999999999998e-162

   1. Initial program 42.7%

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

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. inv-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 58.4

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

   4. Applied rewrites 58.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. *-commutative N/A

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

      5. inv-pow N/A

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

      6. sqrt-div N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lift-*.f64 58.7

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

   6. Applied rewrites 58.7%

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

   if 3.5999999999999998e-162 < d

   1. Initial program 74.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. pow1/2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. metadata-eval N/A

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

      15. pow1/2 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lift-/.f64 74.1

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

   3. Applied rewrites 74.1%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 74.1

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

   5. Applied rewrites 74.1%

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

      1. lift-/.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. sqrt-div N/A

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

      4. lower-/.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lower-sqrt.f64 82.6

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

   7. Applied rewrites 82.6%

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

Alternative 5: 68.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.3 \cdot 10^{+232}:\\ \;\;\;\;-{\left(\ell \cdot h\right)}^{-0.5} \cdot d\\ \mathbf{elif}\;\ell \leq 2.2 \cdot 10^{-304}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 - \frac{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sqrt{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -2.3e+232)
   (- (* (pow (* l h) -0.5) d))
   (if (<= l 2.2e-304)
     (*
      (sqrt (/ d l))
      (*
       (sqrt (/ d h))
       (- 1.0 (/ (* (pow (* (/ M 2.0) (/ D d)) 2.0) (* 0.5 h)) l))))
     (*
      (* (/ 1.0 (sqrt (* l h))) d)
      (- 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) {
	double tmp;
	if (l <= -2.3e+232) {
		tmp = -(pow((l * h), -0.5) * d);
	} else if (l <= 2.2e-304) {
		tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0 - ((pow(((M / 2.0) * (D / d)), 2.0) * (0.5 * h)) / l)));
	} else {
		tmp = ((1.0 / sqrt((l * h))) * d) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (l <= (-2.3d+232)) then
        tmp = -(((l * h) ** (-0.5d0)) * d)
    else if (l <= 2.2d-304) then
        tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0d0 - (((((m / 2.0d0) * (d_1 / d)) ** 2.0d0) * (0.5d0 * h)) / l)))
    else
        tmp = ((1.0d0 / sqrt((l * h))) * d) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -2.3e+232) {
		tmp = -(Math.pow((l * h), -0.5) * d);
	} else if (l <= 2.2e-304) {
		tmp = Math.sqrt((d / l)) * (Math.sqrt((d / h)) * (1.0 - ((Math.pow(((M / 2.0) * (D / d)), 2.0) * (0.5 * h)) / l)));
	} else {
		tmp = ((1.0 / Math.sqrt((l * h))) * d) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= -2.3e+232:
		tmp = -(math.pow((l * h), -0.5) * d)
	elif l <= 2.2e-304:
		tmp = math.sqrt((d / l)) * (math.sqrt((d / h)) * (1.0 - ((math.pow(((M / 2.0) * (D / d)), 2.0) * (0.5 * h)) / l)))
	else:
		tmp = ((1.0 / math.sqrt((l * h))) * d) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -2.3e+232)
		tmp = Float64(-Float64((Float64(l * h) ^ -0.5) * d));
	elseif (l <= 2.2e-304)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(sqrt(Float64(d / h)) * Float64(1.0 - Float64(Float64((Float64(Float64(M / 2.0) * Float64(D / d)) ^ 2.0) * Float64(0.5 * h)) / l))));
	else
		tmp = Float64(Float64(Float64(1.0 / sqrt(Float64(l * h))) * d) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -2.3e+232)
		tmp = -(((l * h) ^ -0.5) * d);
	elseif (l <= 2.2e-304)
		tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0 - (((((M / 2.0) * (D / d)) ^ 2.0) * (0.5 * h)) / l)));
	else
		tmp = ((1.0 / sqrt((l * h))) * d) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -2.3e+232], (-N[(N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision] * d), $MachinePrecision]), If[LessEqual[l, 2.2e-304], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(N[(N[Power[N[(N[(M / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(0.5 * h), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * d), $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]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.30000000000000006e232

   1. Initial program 44.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow-prod-down N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 36.8

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

      11. lift-/.f64 N/A

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

      12. metadata-eval 36.8

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

   3. Applied rewrites 36.8%

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

   4. Taylor expanded in d around -inf

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

      1. metadata-eval N/A

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

      2. unpow-prod-down N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. mul-1-neg N/A

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

   6. Applied rewrites 46.3%

      \[\leadsto \color{blue}{-{\left(\ell \cdot h\right)}^{-0.5} \cdot d} \]

   if -2.30000000000000006e232 < l < 2.2e-304

   1. Initial program 68.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. pow1/2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. metadata-eval N/A

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

      15. pow1/2 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lift-/.f64 68.3

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

   3. Applied rewrites 68.3%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 68.3

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

   5. Applied rewrites 68.3%

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

   7. Applied rewrites 70.1%

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

   if 2.2e-304 < l

   1. Initial program 65.8%

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

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. inv-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 68.8

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

   4. Applied rewrites 68.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. *-commutative N/A

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

      5. inv-pow N/A

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

      6. sqrt-div N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lift-*.f64 69.1

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

   6. Applied rewrites 69.1%

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

Alternative 6: 65.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\\ \mathbf{if}\;h \leq -8 \cdot 10^{-144}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(0.5 \cdot t\_0\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{elif}\;h \leq 2.2 \cdot 10^{-307}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sqrt{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(\frac{1}{2} \cdot t\_0\right) \cdot \frac{h}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (pow (/ (* M D) (* 2.0 d)) 2.0)))
   (if (<= h -8e-144)
     (* (* (sqrt (/ d l)) (sqrt (/ d h))) (- 1.0 (* (* 0.5 t_0) (/ h l))))
     (if (<= h 2.2e-307)
       (* (- d) (sqrt (pow (* l h) -1.0)))
       (*
        (* (/ 1.0 (sqrt (* l h))) d)
        (- 1.0 (* (* (/ 1.0 2.0) t_0) (/ h l))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = pow(((M * D) / (2.0 * d)), 2.0);
	double tmp;
	if (h <= -8e-144) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - ((0.5 * t_0) * (h / l)));
	} else if (h <= 2.2e-307) {
		tmp = -d * sqrt(pow((l * h), -1.0));
	} else {
		tmp = ((1.0 / sqrt((l * h))) * d) * (1.0 - (((1.0 / 2.0) * t_0) * (h / l)));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((m * d_1) / (2.0d0 * d)) ** 2.0d0
    if (h <= (-8d-144)) then
        tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0d0 - ((0.5d0 * t_0) * (h / l)))
    else if (h <= 2.2d-307) then
        tmp = -d * sqrt(((l * h) ** (-1.0d0)))
    else
        tmp = ((1.0d0 / sqrt((l * h))) * d) * (1.0d0 - (((1.0d0 / 2.0d0) * t_0) * (h / l)))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.pow(((M * D) / (2.0 * d)), 2.0);
	double tmp;
	if (h <= -8e-144) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * (1.0 - ((0.5 * t_0) * (h / l)));
	} else if (h <= 2.2e-307) {
		tmp = -d * Math.sqrt(Math.pow((l * h), -1.0));
	} else {
		tmp = ((1.0 / Math.sqrt((l * h))) * d) * (1.0 - (((1.0 / 2.0) * t_0) * (h / l)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.pow(((M * D) / (2.0 * d)), 2.0)
	tmp = 0
	if h <= -8e-144:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * (1.0 - ((0.5 * t_0) * (h / l)))
	elif h <= 2.2e-307:
		tmp = -d * math.sqrt(math.pow((l * h), -1.0))
	else:
		tmp = ((1.0 / math.sqrt((l * h))) * d) * (1.0 - (((1.0 / 2.0) * t_0) * (h / l)))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0
	tmp = 0.0
	if (h <= -8e-144)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(Float64(0.5 * t_0) * Float64(h / l))));
	elseif (h <= 2.2e-307)
		tmp = Float64(Float64(-d) * sqrt((Float64(l * h) ^ -1.0)));
	else
		tmp = Float64(Float64(Float64(1.0 / sqrt(Float64(l * h))) * d) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * t_0) * Float64(h / l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = ((M * D) / (2.0 * d)) ^ 2.0;
	tmp = 0.0;
	if (h <= -8e-144)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - ((0.5 * t_0) * (h / l)));
	elseif (h <= 2.2e-307)
		tmp = -d * sqrt(((l * h) ^ -1.0));
	else
		tmp = ((1.0 / sqrt((l * h))) * d) * (1.0 - (((1.0 / 2.0) * t_0) * (h / l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[h, -8e-144], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(0.5 * t$95$0), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 2.2e-307], N[((-d) * N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if h < -7.9999999999999996e-144

   1. Initial program 66.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. pow1/2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. metadata-eval N/A

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

      15. pow1/2 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lift-/.f64 66.0

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

   3. Applied rewrites 66.0%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 66.0

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

   5. Applied rewrites 66.0%

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

   if -7.9999999999999996e-144 < h < 2.2e-307

   1. Initial program 63.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow-prod-down N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 56.4

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

      11. lift-/.f64 N/A

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

      12. metadata-eval 56.4

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

   3. Applied rewrites 56.4%

      \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. unpow2 N/A

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

      6. *-commutative N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 46.0

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

   5. Applied rewrites 46.0%

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

   6. Taylor expanded in d around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. inv-pow N/A

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

      6. *-commutative N/A

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

      7. exp-to-pow N/A

         \[\leadsto \left(-d\right) \cdot \sqrt{e^{\log \left(\ell \cdot h\right) \cdot -1}} \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(-d\right) \cdot \sqrt{e^{\log \left(\ell \cdot h\right) \cdot -1}} \]

      9. exp-to-pow N/A

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

      10. lower-pow.f64 N/A

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

      11. lift-*.f64 52.0

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

   8. Applied rewrites 52.0%

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

   if 2.2e-307 < h

   1. Initial program 65.8%

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

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. inv-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 68.9

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

   4. Applied rewrites 68.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. *-commutative N/A

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

      5. inv-pow N/A

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

      6. sqrt-div N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lift-*.f64 69.2

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

   6. Applied rewrites 69.2%

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

Alternative 7: 65.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{M}{2} \cdot \frac{D}{d}\\ \mathbf{if}\;h \leq -9 \cdot 10^{-143}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(0.5 \cdot \left(t\_0 \cdot t\_0\right)\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{elif}\;h \leq 2.2 \cdot 10^{-307}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sqrt{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (/ M 2.0) (/ D d))))
   (if (<= h -9e-143)
     (*
      (* (sqrt (/ d l)) (sqrt (/ d h)))
      (- 1.0 (* (* 0.5 (* t_0 t_0)) (/ h l))))
     (if (<= h 2.2e-307)
       (* (- d) (sqrt (pow (* l h) -1.0)))
       (*
        (* (/ 1.0 (sqrt (* l h))) d)
        (-
         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) {
	double t_0 = (M / 2.0) * (D / d);
	double tmp;
	if (h <= -9e-143) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - ((0.5 * (t_0 * t_0)) * (h / l)));
	} else if (h <= 2.2e-307) {
		tmp = -d * sqrt(pow((l * h), -1.0));
	} else {
		tmp = ((1.0 / sqrt((l * h))) * d) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (m / 2.0d0) * (d_1 / d)
    if (h <= (-9d-143)) then
        tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0d0 - ((0.5d0 * (t_0 * t_0)) * (h / l)))
    else if (h <= 2.2d-307) then
        tmp = -d * sqrt(((l * h) ** (-1.0d0)))
    else
        tmp = ((1.0d0 / sqrt((l * h))) * d) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (M / 2.0) * (D / d);
	double tmp;
	if (h <= -9e-143) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * (1.0 - ((0.5 * (t_0 * t_0)) * (h / l)));
	} else if (h <= 2.2e-307) {
		tmp = -d * Math.sqrt(Math.pow((l * h), -1.0));
	} else {
		tmp = ((1.0 / Math.sqrt((l * h))) * d) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (M / 2.0) * (D / d)
	tmp = 0
	if h <= -9e-143:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * (1.0 - ((0.5 * (t_0 * t_0)) * (h / l)))
	elif h <= 2.2e-307:
		tmp = -d * math.sqrt(math.pow((l * h), -1.0))
	else:
		tmp = ((1.0 / math.sqrt((l * h))) * d) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(M / 2.0) * Float64(D / d))
	tmp = 0.0
	if (h <= -9e-143)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(Float64(0.5 * Float64(t_0 * t_0)) * Float64(h / l))));
	elseif (h <= 2.2e-307)
		tmp = Float64(Float64(-d) * sqrt((Float64(l * h) ^ -1.0)));
	else
		tmp = Float64(Float64(Float64(1.0 / sqrt(Float64(l * h))) * d) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (M / 2.0) * (D / d);
	tmp = 0.0;
	if (h <= -9e-143)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - ((0.5 * (t_0 * t_0)) * (h / l)));
	elseif (h <= 2.2e-307)
		tmp = -d * sqrt(((l * h) ^ -1.0));
	else
		tmp = ((1.0 / sqrt((l * h))) * d) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(M / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -9e-143], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(0.5 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 2.2e-307], N[((-d) * N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * d), $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]]]]

Derivation

1. Split input into 3 regimes

2. if h < -9.00000000000000001e-143

   1. Initial program 66.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. pow1/2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. metadata-eval N/A

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

      15. pow1/2 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lift-/.f64 66.0

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

   3. Applied rewrites 66.0%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 66.0

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

   5. Applied rewrites 66.0%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 65.6

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

   7. Applied rewrites 65.6%

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

   if -9.00000000000000001e-143 < h < 2.2e-307

   1. Initial program 63.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow-prod-down N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 56.5

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

      11. lift-/.f64 N/A

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

      12. metadata-eval 56.5

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

   3. Applied rewrites 56.5%

      \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. unpow2 N/A

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

      6. *-commutative N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 46.1

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

   5. Applied rewrites 46.1%

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

   6. Taylor expanded in d around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. inv-pow N/A

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

      6. *-commutative N/A

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

      7. exp-to-pow N/A

         \[\leadsto \left(-d\right) \cdot \sqrt{e^{\log \left(\ell \cdot h\right) \cdot -1}} \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(-d\right) \cdot \sqrt{e^{\log \left(\ell \cdot h\right) \cdot -1}} \]

      9. exp-to-pow N/A

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

      10. lower-pow.f64 N/A

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

      11. lift-*.f64 52.1

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

   8. Applied rewrites 52.1%

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

   if 2.2e-307 < h

   1. Initial program 65.8%

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

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. inv-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 68.9

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

   4. Applied rewrites 68.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. *-commutative N/A

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

      5. inv-pow N/A

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

      6. sqrt-div N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lift-*.f64 69.2

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

   6. Applied rewrites 69.2%

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

Alternative 8: 65.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{M}{2} \cdot \frac{D}{d}\\ \mathbf{if}\;h \leq -9 \cdot 10^{-143}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(0.5 \cdot \left(t\_0 \cdot t\_0\right)\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{elif}\;h \leq 2.2 \cdot 10^{-307}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{d + d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (/ M 2.0) (/ D d))))
   (if (<= h -9e-143)
     (*
      (* (sqrt (/ d l)) (sqrt (/ d h)))
      (- 1.0 (* (* 0.5 (* t_0 t_0)) (/ h l))))
     (if (<= h 2.2e-307)
       (* (- d) (sqrt (pow (* l h) -1.0)))
       (*
        (* (sqrt (/ 1.0 (* l h))) d)
        (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (+ d d)) 2.0)) (/ h l))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (M / 2.0) * (D / d);
	double tmp;
	if (h <= -9e-143) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - ((0.5 * (t_0 * t_0)) * (h / l)));
	} else if (h <= 2.2e-307) {
		tmp = -d * sqrt(pow((l * h), -1.0));
	} else {
		tmp = (sqrt((1.0 / (l * h))) * d) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (d + d)), 2.0)) * (h / l)));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (m / 2.0d0) * (d_1 / d)
    if (h <= (-9d-143)) then
        tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0d0 - ((0.5d0 * (t_0 * t_0)) * (h / l)))
    else if (h <= 2.2d-307) then
        tmp = -d * sqrt(((l * h) ** (-1.0d0)))
    else
        tmp = (sqrt((1.0d0 / (l * h))) * d) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d_1) / (d + d)) ** 2.0d0)) * (h / l)))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (M / 2.0) * (D / d);
	double tmp;
	if (h <= -9e-143) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * (1.0 - ((0.5 * (t_0 * t_0)) * (h / l)));
	} else if (h <= 2.2e-307) {
		tmp = -d * Math.sqrt(Math.pow((l * h), -1.0));
	} else {
		tmp = (Math.sqrt((1.0 / (l * h))) * d) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (d + d)), 2.0)) * (h / l)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (M / 2.0) * (D / d)
	tmp = 0
	if h <= -9e-143:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * (1.0 - ((0.5 * (t_0 * t_0)) * (h / l)))
	elif h <= 2.2e-307:
		tmp = -d * math.sqrt(math.pow((l * h), -1.0))
	else:
		tmp = (math.sqrt((1.0 / (l * h))) * d) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (d + d)), 2.0)) * (h / l)))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(M / 2.0) * Float64(D / d))
	tmp = 0.0
	if (h <= -9e-143)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(Float64(0.5 * Float64(t_0 * t_0)) * Float64(h / l))));
	elseif (h <= 2.2e-307)
		tmp = Float64(Float64(-d) * sqrt((Float64(l * h) ^ -1.0)));
	else
		tmp = Float64(Float64(sqrt(Float64(1.0 / Float64(l * h))) * d) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(d + d)) ^ 2.0)) * Float64(h / l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (M / 2.0) * (D / d);
	tmp = 0.0;
	if (h <= -9e-143)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - ((0.5 * (t_0 * t_0)) * (h / l)));
	elseif (h <= 2.2e-307)
		tmp = -d * sqrt(((l * h) ^ -1.0));
	else
		tmp = (sqrt((1.0 / (l * h))) * d) * (1.0 - (((1.0 / 2.0) * (((M * D) / (d + d)) ^ 2.0)) * (h / l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(M / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -9e-143], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(0.5 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 2.2e-307], N[((-d) * N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(d + d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if h < -9.00000000000000001e-143

   1. Initial program 66.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. pow1/2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. metadata-eval N/A

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

      15. pow1/2 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lift-/.f64 66.0

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

   3. Applied rewrites 66.0%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 66.0

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

   5. Applied rewrites 66.0%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 65.6

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

   7. Applied rewrites 65.6%

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

   if -9.00000000000000001e-143 < h < 2.2e-307

   1. Initial program 63.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow-prod-down N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 56.5

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

      11. lift-/.f64 N/A

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

      12. metadata-eval 56.5

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

   3. Applied rewrites 56.5%

      \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. unpow2 N/A

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

      6. *-commutative N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 46.1

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

   5. Applied rewrites 46.1%

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

   6. Taylor expanded in d around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. inv-pow N/A

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

      6. *-commutative N/A

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

      7. exp-to-pow N/A

         \[\leadsto \left(-d\right) \cdot \sqrt{e^{\log \left(\ell \cdot h\right) \cdot -1}} \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(-d\right) \cdot \sqrt{e^{\log \left(\ell \cdot h\right) \cdot -1}} \]

      9. exp-to-pow N/A

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

      10. lower-pow.f64 N/A

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

      11. lift-*.f64 52.1

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

   8. Applied rewrites 52.1%

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

   if 2.2e-307 < h

   1. Initial program 65.8%

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

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. inv-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 68.9

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

   4. Applied rewrites 68.9%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow-1 N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 68.9

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

   6. Applied rewrites 68.9%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 68.9

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

   8. Applied rewrites 68.9%

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

Alternative 9: 66.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{M}{2} \cdot \frac{D}{d}\\ \mathbf{if}\;\ell \leq -2.3 \cdot 10^{+232}:\\ \;\;\;\;-{\left(\ell \cdot h\right)}^{-0.5} \cdot d\\ \mathbf{elif}\;\ell \leq 2.2 \cdot 10^{+92}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(0.5 \cdot \left(t\_0 \cdot t\_0\right)\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (/ M 2.0) (/ D d))))
   (if (<= l -2.3e+232)
     (- (* (pow (* l h) -0.5) d))
     (if (<= l 2.2e+92)
       (*
        (* (sqrt (/ d l)) (sqrt (/ d h)))
        (- 1.0 (* (* 0.5 (* t_0 t_0)) (/ h l))))
       (* (/ 1.0 (* (sqrt l) (sqrt h))) d)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (M / 2.0) * (D / d);
	double tmp;
	if (l <= -2.3e+232) {
		tmp = -(pow((l * h), -0.5) * d);
	} else if (l <= 2.2e+92) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - ((0.5 * (t_0 * t_0)) * (h / l)));
	} else {
		tmp = (1.0 / (sqrt(l) * sqrt(h))) * d;
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (m / 2.0d0) * (d_1 / d)
    if (l <= (-2.3d+232)) then
        tmp = -(((l * h) ** (-0.5d0)) * d)
    else if (l <= 2.2d+92) then
        tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0d0 - ((0.5d0 * (t_0 * t_0)) * (h / l)))
    else
        tmp = (1.0d0 / (sqrt(l) * sqrt(h))) * d
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (M / 2.0) * (D / d);
	double tmp;
	if (l <= -2.3e+232) {
		tmp = -(Math.pow((l * h), -0.5) * d);
	} else if (l <= 2.2e+92) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * (1.0 - ((0.5 * (t_0 * t_0)) * (h / l)));
	} else {
		tmp = (1.0 / (Math.sqrt(l) * Math.sqrt(h))) * d;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (M / 2.0) * (D / d)
	tmp = 0
	if l <= -2.3e+232:
		tmp = -(math.pow((l * h), -0.5) * d)
	elif l <= 2.2e+92:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * (1.0 - ((0.5 * (t_0 * t_0)) * (h / l)))
	else:
		tmp = (1.0 / (math.sqrt(l) * math.sqrt(h))) * d
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(M / 2.0) * Float64(D / d))
	tmp = 0.0
	if (l <= -2.3e+232)
		tmp = Float64(-Float64((Float64(l * h) ^ -0.5) * d));
	elseif (l <= 2.2e+92)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(Float64(0.5 * Float64(t_0 * t_0)) * Float64(h / l))));
	else
		tmp = Float64(Float64(1.0 / Float64(sqrt(l) * sqrt(h))) * d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (M / 2.0) * (D / d);
	tmp = 0.0;
	if (l <= -2.3e+232)
		tmp = -(((l * h) ^ -0.5) * d);
	elseif (l <= 2.2e+92)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - ((0.5 * (t_0 * t_0)) * (h / l)));
	else
		tmp = (1.0 / (sqrt(l) * sqrt(h))) * d;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(M / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -2.3e+232], (-N[(N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision] * d), $MachinePrecision]), If[LessEqual[l, 2.2e+92], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(0.5 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.30000000000000006e232

   1. Initial program 44.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow-prod-down N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 36.8

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

      11. lift-/.f64 N/A

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

      12. metadata-eval 36.8

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

   3. Applied rewrites 36.8%

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

   4. Taylor expanded in d around -inf

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

      1. metadata-eval N/A

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

      2. unpow-prod-down N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. mul-1-neg N/A

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

   6. Applied rewrites 46.3%

      \[\leadsto \color{blue}{-{\left(\ell \cdot h\right)}^{-0.5} \cdot d} \]

   if -2.30000000000000006e232 < l < 2.19999999999999992e92

   1. Initial program 69.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. pow1/2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. metadata-eval N/A

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

      15. pow1/2 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lift-/.f64 69.8

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

   3. Applied rewrites 69.8%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 69.8

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

   5. Applied rewrites 69.8%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 69.3

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

   7. Applied rewrites 69.3%

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

   if 2.19999999999999992e92 < l

   1. Initial program 54.0%

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

   2. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

      4. inv-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 50.0

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

   4. Applied rewrites 50.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. *-commutative N/A

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

      5. inv-pow N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

      6. sqrt-div N/A

         \[\leadsto \frac{\sqrt{1}}{\sqrt{h \cdot \ell}} \cdot d \]

      7. metadata-eval N/A

         \[\leadsto \frac{1}{\sqrt{h \cdot \ell}} \cdot d \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{1}{\sqrt{h \cdot \ell}} \cdot d \]

      9. *-commutative N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      11. lift-*.f64 50.0

          \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

   6. Applied rewrites 50.0%

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      3. sqrt-prod N/A

         \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]

      6. lower-sqrt.f64 62.0

         \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]

   8. Applied rewrites 62.0%

      \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 46.7% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq 7.4 \cdot 10^{-279}:\\ \;\;\;\;-{\left(\ell \cdot h\right)}^{-0.5} \cdot d\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= d 7.4e-279)
   (- (* (pow (* l h) -0.5) d))
   (* (/ 1.0 (* (sqrt l) (sqrt h))) d)))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= 7.4e-279) {
		tmp = -(pow((l * h), -0.5) * d);
	} else {
		tmp = (1.0 / (sqrt(l) * sqrt(h))) * d;
	}
	return tmp;
}

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
    real(8) :: tmp
    if (d <= 7.4d-279) then
        tmp = -(((l * h) ** (-0.5d0)) * d)
    else
        tmp = (1.0d0 / (sqrt(l) * sqrt(h))) * d
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= 7.4e-279) {
		tmp = -(Math.pow((l * h), -0.5) * d);
	} else {
		tmp = (1.0 / (Math.sqrt(l) * Math.sqrt(h))) * d;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if d <= 7.4e-279:
		tmp = -(math.pow((l * h), -0.5) * d)
	else:
		tmp = (1.0 / (math.sqrt(l) * math.sqrt(h))) * d
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= 7.4e-279)
		tmp = Float64(-Float64((Float64(l * h) ^ -0.5) * d));
	else
		tmp = Float64(Float64(1.0 / Float64(sqrt(l) * sqrt(h))) * d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (d <= 7.4e-279)
		tmp = -(((l * h) ^ -0.5) * d);
	else
		tmp = (1.0 / (sqrt(l) * sqrt(h))) * d;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[d, 7.4e-279], (-N[(N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision] * d), $MachinePrecision]), N[(N[(1.0 / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 7.40000000000000076e-279

   1. Initial program 63.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow-prod-down N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 52.6

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

      11. lift-/.f64 N/A

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

      12. metadata-eval 52.6

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

   3. Applied rewrites 52.6%

      \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      2. unpow-prod-down N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      3. metadata-eval N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. metadata-eval N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      5. metadata-eval N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   6. Applied rewrites 41.9%

      \[\leadsto \color{blue}{-{\left(\ell \cdot h\right)}^{-0.5} \cdot d} \]

   if 7.40000000000000076e-279 < d

   1. Initial program 67.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

      4. inv-pow N/A

         \[\leadsto \sqrt{{\left(h \cdot \ell\right)}^{-1}} \cdot d \]

      5. lower-pow.f64 N/A

         \[\leadsto \sqrt{{\left(h \cdot \ell\right)}^{-1}} \cdot d \]

      6. *-commutative N/A

         \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

      7. lower-*.f64 44.2

         \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

   4. Applied rewrites 44.2%

      \[\leadsto \color{blue}{\sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

      3. lift-pow.f64 N/A

         \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

      4. *-commutative N/A

         \[\leadsto \sqrt{{\left(h \cdot \ell\right)}^{-1}} \cdot d \]

      5. inv-pow N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

      6. sqrt-div N/A

         \[\leadsto \frac{\sqrt{1}}{\sqrt{h \cdot \ell}} \cdot d \]

      7. metadata-eval N/A

         \[\leadsto \frac{1}{\sqrt{h \cdot \ell}} \cdot d \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{1}{\sqrt{h \cdot \ell}} \cdot d \]

      9. *-commutative N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      11. lift-*.f64 44.3

          \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

   6. Applied rewrites 44.3%

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      3. sqrt-prod N/A

         \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]

      6. lower-sqrt.f64 51.9

         \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]

   8. Applied rewrites 51.9%

      \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 44.5% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq 5.4 \cdot 10^{-291}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= d 5.4e-291)
   (* (* (sqrt (/ d l)) (sqrt (/ d h))) 1.0)
   (* (/ 1.0 (* (sqrt l) (sqrt h))) d)))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= 5.4e-291) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0;
	} else {
		tmp = (1.0 / (sqrt(l) * sqrt(h))) * d;
	}
	return tmp;
}

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
    real(8) :: tmp
    if (d <= 5.4d-291) then
        tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0d0
    else
        tmp = (1.0d0 / (sqrt(l) * sqrt(h))) * d
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= 5.4e-291) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * 1.0;
	} else {
		tmp = (1.0 / (Math.sqrt(l) * Math.sqrt(h))) * d;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if d <= 5.4e-291:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * 1.0
	else:
		tmp = (1.0 / (math.sqrt(l) * math.sqrt(h))) * d
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= 5.4e-291)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * 1.0);
	else
		tmp = Float64(Float64(1.0 / Float64(sqrt(l) * sqrt(h))) * d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (d <= 5.4e-291)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0;
	else
		tmp = (1.0 / (sqrt(l) * sqrt(h))) * d;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[d, 5.4e-291], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(1.0 / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 5.39999999999999983e-291

   1. Initial program 64.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. pow1/2 N/A

          \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. pow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      17. lift-/.f64 64.5

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\color{blue}{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 64.5%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. metadata-eval 64.5

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 64.5%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   6. Taylor expanded in d around inf

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 38.1%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]

   if 5.39999999999999983e-291 < d

   1. Initial program 66.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

      4. inv-pow N/A

         \[\leadsto \sqrt{{\left(h \cdot \ell\right)}^{-1}} \cdot d \]

      5. lower-pow.f64 N/A

         \[\leadsto \sqrt{{\left(h \cdot \ell\right)}^{-1}} \cdot d \]

      6. *-commutative N/A

         \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

      7. lower-*.f64 43.7

         \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

   4. Applied rewrites 43.7%

      \[\leadsto \color{blue}{\sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

      3. lift-pow.f64 N/A

         \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

      4. *-commutative N/A

         \[\leadsto \sqrt{{\left(h \cdot \ell\right)}^{-1}} \cdot d \]

      5. inv-pow N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

      6. sqrt-div N/A

         \[\leadsto \frac{\sqrt{1}}{\sqrt{h \cdot \ell}} \cdot d \]

      7. metadata-eval N/A

         \[\leadsto \frac{1}{\sqrt{h \cdot \ell}} \cdot d \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{1}{\sqrt{h \cdot \ell}} \cdot d \]

      9. *-commutative N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      11. lift-*.f64 43.8

          \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

   6. Applied rewrites 43.8%

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      3. sqrt-prod N/A

         \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]

      6. lower-sqrt.f64 51.3

         \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]

   8. Applied rewrites 51.3%

      \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 29.9% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq 4.5 \cdot 10^{-96}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot d\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= d 4.5e-96)
   (* (sqrt (/ 1.0 (* l h))) d)
   (* (/ 1.0 (* (sqrt l) (sqrt h))) d)))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= 4.5e-96) {
		tmp = sqrt((1.0 / (l * h))) * d;
	} else {
		tmp = (1.0 / (sqrt(l) * sqrt(h))) * d;
	}
	return tmp;
}

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
    real(8) :: tmp
    if (d <= 4.5d-96) then
        tmp = sqrt((1.0d0 / (l * h))) * d
    else
        tmp = (1.0d0 / (sqrt(l) * sqrt(h))) * d
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= 4.5e-96) {
		tmp = Math.sqrt((1.0 / (l * h))) * d;
	} else {
		tmp = (1.0 / (Math.sqrt(l) * Math.sqrt(h))) * d;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if d <= 4.5e-96:
		tmp = math.sqrt((1.0 / (l * h))) * d
	else:
		tmp = (1.0 / (math.sqrt(l) * math.sqrt(h))) * d
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= 4.5e-96)
		tmp = Float64(sqrt(Float64(1.0 / Float64(l * h))) * d);
	else
		tmp = Float64(Float64(1.0 / Float64(sqrt(l) * sqrt(h))) * d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (d <= 4.5e-96)
		tmp = sqrt((1.0 / (l * h))) * d;
	else
		tmp = (1.0 / (sqrt(l) * sqrt(h))) * d;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[d, 4.5e-96], N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision], N[(N[(1.0 / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 4.5e-96

   1. Initial program 60.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

      4. inv-pow N/A

         \[\leadsto \sqrt{{\left(h \cdot \ell\right)}^{-1}} \cdot d \]

      5. lower-pow.f64 N/A

         \[\leadsto \sqrt{{\left(h \cdot \ell\right)}^{-1}} \cdot d \]

      6. *-commutative N/A

         \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

      7. lower-*.f64 14.1

         \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

   4. Applied rewrites 14.1%

      \[\leadsto \color{blue}{\sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

      2. lift-pow.f64 N/A

         \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

      3. unpow-1 N/A

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

      5. lift-*.f64 14.1

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

   6. Applied rewrites 14.1%

      \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

   if 4.5e-96 < d

   1. Initial program 76.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

      4. inv-pow N/A

         \[\leadsto \sqrt{{\left(h \cdot \ell\right)}^{-1}} \cdot d \]

      5. lower-pow.f64 N/A

         \[\leadsto \sqrt{{\left(h \cdot \ell\right)}^{-1}} \cdot d \]

      6. *-commutative N/A

         \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

      7. lower-*.f64 53.7

         \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

   4. Applied rewrites 53.7%

      \[\leadsto \color{blue}{\sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

      3. lift-pow.f64 N/A

         \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

      4. *-commutative N/A

         \[\leadsto \sqrt{{\left(h \cdot \ell\right)}^{-1}} \cdot d \]

      5. inv-pow N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

      6. sqrt-div N/A

         \[\leadsto \frac{\sqrt{1}}{\sqrt{h \cdot \ell}} \cdot d \]

      7. metadata-eval N/A

         \[\leadsto \frac{1}{\sqrt{h \cdot \ell}} \cdot d \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{1}{\sqrt{h \cdot \ell}} \cdot d \]

      9. *-commutative N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      11. lift-*.f64 53.9

          \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

   6. Applied rewrites 53.9%

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      3. sqrt-prod N/A

         \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]

      6. lower-sqrt.f64 63.5

         \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]

   8. Applied rewrites 63.5%

      \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 26.8% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{1}{\ell \cdot h}} \cdot d \end{array} \]

FPCore

(FPCore (d h l M D) :precision binary64 (* (sqrt (/ 1.0 (* l h))) d))

C

double code(double d, double h, double l, double M, double D) {
	return sqrt((1.0 / (l * h))) * d;
}

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 = sqrt((1.0d0 / (l * h))) * d
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	return Math.sqrt((1.0 / (l * h))) * d;
}

Python

def code(d, h, l, M, D):
	return math.sqrt((1.0 / (l * h))) * d

Julia

function code(d, h, l, M, D)
	return Float64(sqrt(Float64(1.0 / Float64(l * h))) * d)
end

MATLAB

function tmp = code(d, h, l, M, D)
	tmp = sqrt((1.0 / (l * h))) * d;
end

Wolfram

code[d_, h_, l_, M_, D_] := N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision]

Derivation

1. Initial program 65.6%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

2. Taylor expanded in d around inf

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

   2. lower-*.f64 N/A

      \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

   3. lower-sqrt.f64 N/A

      \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

   4. inv-pow N/A

      \[\leadsto \sqrt{{\left(h \cdot \ell\right)}^{-1}} \cdot d \]

   5. lower-pow.f64 N/A

      \[\leadsto \sqrt{{\left(h \cdot \ell\right)}^{-1}} \cdot d \]

   6. *-commutative N/A

      \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

   7. lower-*.f64 26.8

      \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

4. Applied rewrites 26.8%

   \[\leadsto \color{blue}{\sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

   2. lift-pow.f64 N/A

      \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

   3. unpow-1 N/A

      \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

   4. lower-/.f64 N/A

      \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

   5. lift-*.f64 26.8

      \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

6. Applied rewrites 26.8%

   \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025092 
(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)))))