Henrywood and Agarwal, Equation (12)

Percentage Accurate: 67.1% → 77.6%

Time: 8.8s

Alternatives: 15

Speedup: 1.7×

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: 67.1% 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: 77.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;h \leq -2.6 \cdot 10^{+67}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \frac{h}{\ell} \cdot \left(\left(\left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right) \cdot \left(M \cdot \frac{D}{2 \cdot d}\right)\right) \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= h -2.6e+67)
   (*
    (sqrt (/ d h))
    (*
     (sqrt (/ d l))
     (-
      1.0
      (* (/ h l) (* (* (* (/ D d) (* 0.5 M)) (* M (/ D (* 2.0 d)))) 0.5)))))
   (if (<= h -5e-310)
     (*
      (* d (- (pow (* l h) -0.5)))
      (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l))))
     (*
      (/ (sqrt d) (sqrt h))
      (*
       (/ (sqrt d) (sqrt l))
       (- 1.0 (* (/ h l) (* (pow (* (/ M 2.0) (/ D d)) 2.0) 0.5))))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= -2.6e+67) {
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 - ((h / l) * ((((D / d) * (0.5 * M)) * (M * (D / (2.0 * d)))) * 0.5))));
	} else if (h <= -5e-310) {
		tmp = (d * -pow((l * h), -0.5)) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	} else {
		tmp = (sqrt(d) / sqrt(h)) * ((sqrt(d) / sqrt(l)) * (1.0 - ((h / l) * (pow(((M / 2.0) * (D / d)), 2.0) * 0.5))));
	}
	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 <= (-2.6d+67)) then
        tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0d0 - ((h / l) * ((((d_1 / d) * (0.5d0 * m)) * (m * (d_1 / (2.0d0 * d)))) * 0.5d0))))
    else if (h <= (-5d-310)) then
        tmp = (d * -((l * h) ** (-0.5d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    else
        tmp = (sqrt(d) / sqrt(h)) * ((sqrt(d) / sqrt(l)) * (1.0d0 - ((h / l) * ((((m / 2.0d0) * (d_1 / d)) ** 2.0d0) * 0.5d0))))
    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 <= -2.6e+67) {
		tmp = Math.sqrt((d / h)) * (Math.sqrt((d / l)) * (1.0 - ((h / l) * ((((D / d) * (0.5 * M)) * (M * (D / (2.0 * d)))) * 0.5))));
	} else if (h <= -5e-310) {
		tmp = (d * -Math.pow((l * h), -0.5)) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	} else {
		tmp = (Math.sqrt(d) / Math.sqrt(h)) * ((Math.sqrt(d) / Math.sqrt(l)) * (1.0 - ((h / l) * (Math.pow(((M / 2.0) * (D / d)), 2.0) * 0.5))));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if h <= -2.6e+67:
		tmp = math.sqrt((d / h)) * (math.sqrt((d / l)) * (1.0 - ((h / l) * ((((D / d) * (0.5 * M)) * (M * (D / (2.0 * d)))) * 0.5))))
	elif h <= -5e-310:
		tmp = (d * -math.pow((l * h), -0.5)) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))
	else:
		tmp = (math.sqrt(d) / math.sqrt(h)) * ((math.sqrt(d) / math.sqrt(l)) * (1.0 - ((h / l) * (math.pow(((M / 2.0) * (D / d)), 2.0) * 0.5))))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (h <= -2.6e+67)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * Float64(1.0 - Float64(Float64(h / l) * Float64(Float64(Float64(Float64(D / d) * Float64(0.5 * M)) * Float64(M * Float64(D / Float64(2.0 * d)))) * 0.5)))));
	elseif (h <= -5e-310)
		tmp = Float64(Float64(d * Float64(-(Float64(l * h) ^ -0.5))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))));
	else
		tmp = Float64(Float64(sqrt(d) / sqrt(h)) * Float64(Float64(sqrt(d) / sqrt(l)) * Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(M / 2.0) * Float64(D / d)) ^ 2.0) * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (h <= -2.6e+67)
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 - ((h / l) * ((((D / d) * (0.5 * M)) * (M * (D / (2.0 * d)))) * 0.5))));
	elseif (h <= -5e-310)
		tmp = (d * -((l * h) ^ -0.5)) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	else
		tmp = (sqrt(d) / sqrt(h)) * ((sqrt(d) / sqrt(l)) * (1.0 - ((h / l) * ((((M / 2.0) * (D / d)) ^ 2.0) * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[h, -2.6e+67], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[(N[(N[(D / d), $MachinePrecision] * N[(0.5 * M), $MachinePrecision]), $MachinePrecision] * N[(M * N[(D / N[(2.0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -5e-310], N[(N[(d * (-N[Power[N[(l * h), $MachinePrecision], -0.5], $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], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(M / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if h < -2.6e67

   1. Initial program 58.9%

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

   3. Applied rewrites 58.5%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 58.5

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 58.5

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-/.f64 58.5

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

   5. Applied rewrites 58.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 58.5

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

   7. Applied rewrites 58.5%

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

   8. Taylor expanded in M around 0

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

      1. lower-*.f64 58.5

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

   10. Applied rewrites 58.5%

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

   if -2.6e67 < h < -4.999999999999985e-310

   1. Initial program 71.6%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow1/2 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 0.0

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

   3. Applied rewrites 0.0%

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

   4. Taylor expanded in h around -inf

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

      1. sqrt-pow2 N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. inv-pow N/A

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

      9. *-commutative N/A

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

      10. sqrt-pow1 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. metadata-eval 79.4

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

   6. Applied rewrites 79.4%

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

   if -4.999999999999985e-310 < h

   1. Initial program 67.5%

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

   3. Applied rewrites 66.9%

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

      1. lift-/.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. sqrt-div N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-/.f64 71.2

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

   5. Applied rewrites 71.2%

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

      1. lift-/.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. sqrt-div N/A

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

      4. lower-/.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lower-sqrt.f64 83.7

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

   7. Applied rewrites 83.7%

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

Alternative 2: 54.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{h}{\ell}}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+51}:\\ \;\;\;\;\left(\frac{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot -1}{d} \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}}\right) \cdot -0.125\\ \mathbf{elif}\;t\_1 \leq 10^{+220}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{t\_0 \cdot d}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{d \cdot \left(-t\_0\right)}{h}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ h l)))
        (t_1
         (*
          (* (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))))))
   (if (<= t_1 -5e+51)
     (*
      (* (/ (* (* (* M D) (* M D)) -1.0) d) (sqrt (/ h (* (* l l) l))))
      -0.125)
     (if (<= t_1 1e+220)
       (* (sqrt (/ d h)) (sqrt (/ d l)))
       (if (<= t_1 INFINITY) (/ (* t_0 d) h) (/ (* d (- t_0)) h))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((h / l));
	double t_1 = (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)));
	double tmp;
	if (t_1 <= -5e+51) {
		tmp = (((((M * D) * (M * D)) * -1.0) / d) * sqrt((h / ((l * l) * l)))) * -0.125;
	} else if (t_1 <= 1e+220) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (t_0 * d) / h;
	} else {
		tmp = (d * -t_0) / h;
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((h / l));
	double t_1 = (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)));
	double tmp;
	if (t_1 <= -5e+51) {
		tmp = (((((M * D) * (M * D)) * -1.0) / d) * Math.sqrt((h / ((l * l) * l)))) * -0.125;
	} else if (t_1 <= 1e+220) {
		tmp = Math.sqrt((d / h)) * Math.sqrt((d / l));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (t_0 * d) / h;
	} else {
		tmp = (d * -t_0) / h;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((h / l))
	t_1 = (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)))
	tmp = 0
	if t_1 <= -5e+51:
		tmp = (((((M * D) * (M * D)) * -1.0) / d) * math.sqrt((h / ((l * l) * l)))) * -0.125
	elif t_1 <= 1e+220:
		tmp = math.sqrt((d / h)) * math.sqrt((d / l))
	elif t_1 <= math.inf:
		tmp = (t_0 * d) / h
	else:
		tmp = (d * -t_0) / h
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(h / l))
	t_1 = 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))))
	tmp = 0.0
	if (t_1 <= -5e+51)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(M * D) * Float64(M * D)) * -1.0) / d) * sqrt(Float64(h / Float64(Float64(l * l) * l)))) * -0.125);
	elseif (t_1 <= 1e+220)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(t_0 * d) / h);
	else
		tmp = Float64(Float64(d * Float64(-t_0)) / h);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((h / l));
	t_1 = (((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)));
	tmp = 0.0;
	if (t_1 <= -5e+51)
		tmp = (((((M * D) * (M * D)) * -1.0) / d) * sqrt((h / ((l * l) * l)))) * -0.125;
	elseif (t_1 <= 1e+220)
		tmp = sqrt((d / h)) * sqrt((d / l));
	elseif (t_1 <= Inf)
		tmp = (t_0 * d) / h;
	else
		tmp = (d * -t_0) / h;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = 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]}, If[LessEqual[t$95$1, -5e+51], N[(N[(N[(N[(N[(N[(M * D), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] * -1.0), $MachinePrecision] / d), $MachinePrecision] * N[Sqrt[N[(h / N[(N[(l * l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision], If[LessEqual[t$95$1, 1e+220], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(t$95$0 * d), $MachinePrecision] / h), $MachinePrecision], N[(N[(d * (-t$95$0)), $MachinePrecision] / h), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -5e51

   1. Initial program 85.9%

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

   2. Taylor expanded in h around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 37.0%

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

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 37.0

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

   6. Applied rewrites 37.0%

      \[\leadsto \left(\frac{{\left(D \cdot M\right)}^{2} \cdot -1}{d} \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}}\right) \cdot -0.125 \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 37.0

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

   8. Applied rewrites 37.0%

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

   if -5e51 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 1e220

   1. Initial program 89.1%

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

   3. Applied rewrites 87.9%

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

   4. Taylor expanded in d around inf

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

      1. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

      2. lift-/.f64 81.5

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

   6. Applied rewrites 81.5%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

   if 1e220 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 56.3%

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

   2. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 49.4%

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

   5. Taylor expanded in d around inf

      \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

      4. lift-sqrt.f64 70.8

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

   7. Applied rewrites 70.8%

      \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

   if +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 0.0%

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

   2. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 19.6%

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

   5. Taylor expanded in l around -inf

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

      1. sqrt-pow2 N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-sqrt.f64 17.9

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

   7. Applied rewrites 17.9%

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

Alternative 3: 53.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{h}{\ell}}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+51}:\\ \;\;\;\;\left(\left(\left(D \cdot D\right) \cdot \frac{\left(M \cdot M\right) \cdot -1}{d}\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}}\right) \cdot -0.125\\ \mathbf{elif}\;t\_1 \leq 10^{+220}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{t\_0 \cdot d}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{d \cdot \left(-t\_0\right)}{h}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ h l)))
        (t_1
         (*
          (* (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))))))
   (if (<= t_1 -5e+51)
     (*
      (* (* (* D D) (/ (* (* M M) -1.0) d)) (sqrt (/ h (* (* l l) l))))
      -0.125)
     (if (<= t_1 1e+220)
       (* (sqrt (/ d h)) (sqrt (/ d l)))
       (if (<= t_1 INFINITY) (/ (* t_0 d) h) (/ (* d (- t_0)) h))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((h / l));
	double t_1 = (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)));
	double tmp;
	if (t_1 <= -5e+51) {
		tmp = (((D * D) * (((M * M) * -1.0) / d)) * sqrt((h / ((l * l) * l)))) * -0.125;
	} else if (t_1 <= 1e+220) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (t_0 * d) / h;
	} else {
		tmp = (d * -t_0) / h;
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((h / l));
	double t_1 = (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)));
	double tmp;
	if (t_1 <= -5e+51) {
		tmp = (((D * D) * (((M * M) * -1.0) / d)) * Math.sqrt((h / ((l * l) * l)))) * -0.125;
	} else if (t_1 <= 1e+220) {
		tmp = Math.sqrt((d / h)) * Math.sqrt((d / l));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (t_0 * d) / h;
	} else {
		tmp = (d * -t_0) / h;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((h / l))
	t_1 = (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)))
	tmp = 0
	if t_1 <= -5e+51:
		tmp = (((D * D) * (((M * M) * -1.0) / d)) * math.sqrt((h / ((l * l) * l)))) * -0.125
	elif t_1 <= 1e+220:
		tmp = math.sqrt((d / h)) * math.sqrt((d / l))
	elif t_1 <= math.inf:
		tmp = (t_0 * d) / h
	else:
		tmp = (d * -t_0) / h
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(h / l))
	t_1 = 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))))
	tmp = 0.0
	if (t_1 <= -5e+51)
		tmp = Float64(Float64(Float64(Float64(D * D) * Float64(Float64(Float64(M * M) * -1.0) / d)) * sqrt(Float64(h / Float64(Float64(l * l) * l)))) * -0.125);
	elseif (t_1 <= 1e+220)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(t_0 * d) / h);
	else
		tmp = Float64(Float64(d * Float64(-t_0)) / h);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((h / l));
	t_1 = (((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)));
	tmp = 0.0;
	if (t_1 <= -5e+51)
		tmp = (((D * D) * (((M * M) * -1.0) / d)) * sqrt((h / ((l * l) * l)))) * -0.125;
	elseif (t_1 <= 1e+220)
		tmp = sqrt((d / h)) * sqrt((d / l));
	elseif (t_1 <= Inf)
		tmp = (t_0 * d) / h;
	else
		tmp = (d * -t_0) / h;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = 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]}, If[LessEqual[t$95$1, -5e+51], N[(N[(N[(N[(D * D), $MachinePrecision] * N[(N[(N[(M * M), $MachinePrecision] * -1.0), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(h / N[(N[(l * l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision], If[LessEqual[t$95$1, 1e+220], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(t$95$0 * d), $MachinePrecision] / h), $MachinePrecision], N[(N[(d * (-t$95$0)), $MachinePrecision] / h), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -5e51

   1. Initial program 85.9%

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

   2. Taylor expanded in h around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 37.0%

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

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 37.0

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

   6. Applied rewrites 37.0%

      \[\leadsto \left(\frac{{\left(D \cdot M\right)}^{2} \cdot -1}{d} \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}}\right) \cdot -0.125 \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow-prod-down N/A

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

      6. associate-*r* N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. sqrt-pow2 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. sqrt-pow2 N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. lower-*.f64 N/A

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

      19. unpow2 N/A

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

      20. lower-*.f64 31.3

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

   8. Applied rewrites 31.3%

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

   if -5e51 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 1e220

   1. Initial program 89.1%

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

   3. Applied rewrites 87.9%

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

   4. Taylor expanded in d around inf

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

      1. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

      2. lift-/.f64 81.5

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

   6. Applied rewrites 81.5%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

   if 1e220 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 56.3%

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

   2. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 49.4%

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

   5. Taylor expanded in d around inf

      \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

      4. lift-sqrt.f64 70.8

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

   7. Applied rewrites 70.8%

      \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

   if +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 0.0%

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

   2. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 19.6%

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

   5. Taylor expanded in l around -inf

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

      1. sqrt-pow2 N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-sqrt.f64 17.9

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

   7. Applied rewrites 17.9%

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

Alternative 4: 52.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{h}{\ell}}\\ t_1 := -t\_0\\ t_2 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{-78}:\\ \;\;\;\;\frac{\frac{\left(d \cdot d\right) \cdot t\_1}{d}}{h}\\ \mathbf{elif}\;t\_2 \leq 10^{+220}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\frac{t\_0 \cdot d}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{d \cdot t\_1}{h}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ h l)))
        (t_1 (- t_0))
        (t_2
         (*
          (* (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))))))
   (if (<= t_2 -2e-78)
     (/ (/ (* (* d d) t_1) d) h)
     (if (<= t_2 1e+220)
       (* (sqrt (/ d h)) (sqrt (/ d l)))
       (if (<= t_2 INFINITY) (/ (* t_0 d) h) (/ (* d t_1) h))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((h / l));
	double t_1 = -t_0;
	double t_2 = (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)));
	double tmp;
	if (t_2 <= -2e-78) {
		tmp = (((d * d) * t_1) / d) / h;
	} else if (t_2 <= 1e+220) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = (t_0 * d) / h;
	} else {
		tmp = (d * t_1) / h;
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((h / l));
	double t_1 = -t_0;
	double t_2 = (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)));
	double tmp;
	if (t_2 <= -2e-78) {
		tmp = (((d * d) * t_1) / d) / h;
	} else if (t_2 <= 1e+220) {
		tmp = Math.sqrt((d / h)) * Math.sqrt((d / l));
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = (t_0 * d) / h;
	} else {
		tmp = (d * t_1) / h;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((h / l))
	t_1 = -t_0
	t_2 = (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)))
	tmp = 0
	if t_2 <= -2e-78:
		tmp = (((d * d) * t_1) / d) / h
	elif t_2 <= 1e+220:
		tmp = math.sqrt((d / h)) * math.sqrt((d / l))
	elif t_2 <= math.inf:
		tmp = (t_0 * d) / h
	else:
		tmp = (d * t_1) / h
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(h / l))
	t_1 = Float64(-t_0)
	t_2 = 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))))
	tmp = 0.0
	if (t_2 <= -2e-78)
		tmp = Float64(Float64(Float64(Float64(d * d) * t_1) / d) / h);
	elseif (t_2 <= 1e+220)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	elseif (t_2 <= Inf)
		tmp = Float64(Float64(t_0 * d) / h);
	else
		tmp = Float64(Float64(d * t_1) / h);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((h / l));
	t_1 = -t_0;
	t_2 = (((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)));
	tmp = 0.0;
	if (t_2 <= -2e-78)
		tmp = (((d * d) * t_1) / d) / h;
	elseif (t_2 <= 1e+220)
		tmp = sqrt((d / h)) * sqrt((d / l));
	elseif (t_2 <= Inf)
		tmp = (t_0 * d) / h;
	else
		tmp = (d * t_1) / h;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = (-t$95$0)}, Block[{t$95$2 = 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]}, If[LessEqual[t$95$2, -2e-78], N[(N[(N[(N[(d * d), $MachinePrecision] * t$95$1), $MachinePrecision] / d), $MachinePrecision] / h), $MachinePrecision], If[LessEqual[t$95$2, 1e+220], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[(t$95$0 * d), $MachinePrecision] / h), $MachinePrecision], N[(N[(d * t$95$1), $MachinePrecision] / h), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -2e-78

   1. Initial program 86.6%

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

   2. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 55.3%

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

   5. Taylor expanded in d around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 54.7%

      \[\leadsto \frac{\frac{\mathsf{fma}\left({\left(\frac{h}{\ell}\right)}^{1.5} \cdot {\left(D \cdot M\right)}^{2}, -0.125, \sqrt{\frac{h}{\ell}} \cdot \left(d \cdot d\right)\right)}{d}}{h} \]

   8. Taylor expanded in l around -inf

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

      1. sqrt-pow2 N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-/.f64 27.5

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

   10. Applied rewrites 27.5%

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

   if -2e-78 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 1e220

   1. Initial program 88.6%

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

   3. Applied rewrites 87.9%

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

   4. Taylor expanded in d around inf

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

      1. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

      2. lift-/.f64 85.7

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

   6. Applied rewrites 85.7%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

   if 1e220 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 56.3%

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

   2. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 49.4%

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

   5. Taylor expanded in d around inf

      \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

      4. lift-sqrt.f64 70.8

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

   7. Applied rewrites 70.8%

      \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

   if +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 0.0%

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

   2. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 19.6%

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

   5. Taylor expanded in l around -inf

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

      1. sqrt-pow2 N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-sqrt.f64 17.9

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

   7. Applied rewrites 17.9%

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

Alternative 5: 50.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{h}{\ell}}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_2 := \frac{d \cdot \left(-t\_0\right)}{h}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-245}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{+220}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{t\_0 \cdot d}{h}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ h l)))
        (t_1
         (*
          (* (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)))))
        (t_2 (/ (* d (- t_0)) h)))
   (if (<= t_1 2e-245)
     t_2
     (if (<= t_1 1e+220)
       (* (sqrt (/ d h)) (sqrt (/ d l)))
       (if (<= t_1 INFINITY) (/ (* t_0 d) h) t_2)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((h / l));
	double t_1 = (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)));
	double t_2 = (d * -t_0) / h;
	double tmp;
	if (t_1 <= 2e-245) {
		tmp = t_2;
	} else if (t_1 <= 1e+220) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (t_0 * d) / h;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((h / l));
	double t_1 = (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)));
	double t_2 = (d * -t_0) / h;
	double tmp;
	if (t_1 <= 2e-245) {
		tmp = t_2;
	} else if (t_1 <= 1e+220) {
		tmp = Math.sqrt((d / h)) * Math.sqrt((d / l));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (t_0 * d) / h;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((h / l))
	t_1 = (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)))
	t_2 = (d * -t_0) / h
	tmp = 0
	if t_1 <= 2e-245:
		tmp = t_2
	elif t_1 <= 1e+220:
		tmp = math.sqrt((d / h)) * math.sqrt((d / l))
	elif t_1 <= math.inf:
		tmp = (t_0 * d) / h
	else:
		tmp = t_2
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(h / l))
	t_1 = 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))))
	t_2 = Float64(Float64(d * Float64(-t_0)) / h)
	tmp = 0.0
	if (t_1 <= 2e-245)
		tmp = t_2;
	elseif (t_1 <= 1e+220)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(t_0 * d) / h);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((h / l));
	t_1 = (((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)));
	t_2 = (d * -t_0) / h;
	tmp = 0.0;
	if (t_1 <= 2e-245)
		tmp = t_2;
	elseif (t_1 <= 1e+220)
		tmp = sqrt((d / h)) * sqrt((d / l));
	elseif (t_1 <= Inf)
		tmp = (t_0 * d) / h;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = 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]}, Block[{t$95$2 = N[(N[(d * (-t$95$0)), $MachinePrecision] / h), $MachinePrecision]}, If[LessEqual[t$95$1, 2e-245], t$95$2, If[LessEqual[t$95$1, 1e+220], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(t$95$0 * d), $MachinePrecision] / h), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 1.9999999999999999e-245 or +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 55.0%

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

   2. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 43.8%

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

   5. Taylor expanded in l around -inf

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

      1. sqrt-pow2 N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-sqrt.f64 22.2

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

   7. Applied rewrites 22.2%

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

   if 1.9999999999999999e-245 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 1e220

   1. Initial program 98.8%

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

   3. Applied rewrites 98.3%

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

   4. Taylor expanded in d around inf

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

      1. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

      2. lift-/.f64 98.1

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

   6. Applied rewrites 98.1%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

   if 1e220 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 56.3%

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

   2. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 49.4%

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

   5. Taylor expanded in d around inf

      \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

      4. lift-sqrt.f64 70.8

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

   7. Applied rewrites 70.8%

      \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 73.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{h}{\ell}}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_1 \leq 10^{+220}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \frac{h}{\ell} \cdot \left(\left(\left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right) \cdot \left(M \cdot \frac{D}{2 \cdot d}\right)\right) \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{t\_0 \cdot d}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left({\left(\frac{h}{\ell}\right)}^{1.5} \cdot \left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right), -0.125, t\_0 \cdot \left(d \cdot d\right)\right)}{d}}{h}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ h l)))
        (t_1
         (*
          (* (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))))))
   (if (<= t_1 1e+220)
     (*
      (sqrt (/ d h))
      (*
       (sqrt (/ d l))
       (-
        1.0
        (* (/ h l) (* (* (* (/ D d) (* 0.5 M)) (* M (/ D (* 2.0 d)))) 0.5)))))
     (if (<= t_1 INFINITY)
       (/ (* t_0 d) h)
       (/
        (/
         (fma (* (pow (/ h l) 1.5) (* (* M D) (* M D))) -0.125 (* t_0 (* d d)))
         d)
        h)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((h / l));
	double t_1 = (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)));
	double tmp;
	if (t_1 <= 1e+220) {
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 - ((h / l) * ((((D / d) * (0.5 * M)) * (M * (D / (2.0 * d)))) * 0.5))));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (t_0 * d) / h;
	} else {
		tmp = (fma((pow((h / l), 1.5) * ((M * D) * (M * D))), -0.125, (t_0 * (d * d))) / d) / h;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(h / l))
	t_1 = 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))))
	tmp = 0.0
	if (t_1 <= 1e+220)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * Float64(1.0 - Float64(Float64(h / l) * Float64(Float64(Float64(Float64(D / d) * Float64(0.5 * M)) * Float64(M * Float64(D / Float64(2.0 * d)))) * 0.5)))));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(t_0 * d) / h);
	else
		tmp = Float64(Float64(fma(Float64((Float64(h / l) ^ 1.5) * Float64(Float64(M * D) * Float64(M * D))), -0.125, Float64(t_0 * Float64(d * d))) / d) / h);
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = 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]}, If[LessEqual[t$95$1, 1e+220], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[(N[(N[(D / d), $MachinePrecision] * N[(0.5 * M), $MachinePrecision]), $MachinePrecision] * N[(M * N[(D / N[(2.0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(t$95$0 * d), $MachinePrecision] / h), $MachinePrecision], N[(N[(N[(N[(N[Power[N[(h / l), $MachinePrecision], 1.5], $MachinePrecision] * N[(N[(M * D), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125 + N[(t$95$0 * N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] / h), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 1e220

   1. Initial program 87.6%

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

   3. Applied rewrites 86.7%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 86.7

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 86.7

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-/.f64 86.7

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

   5. Applied rewrites 86.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 86.7

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

   7. Applied rewrites 86.7%

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

   8. Taylor expanded in M around 0

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

      1. lower-*.f64 86.7

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

   10. Applied rewrites 86.7%

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

   if 1e220 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 56.3%

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

   2. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 49.4%

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

   5. Taylor expanded in d around inf

      \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

      4. lift-sqrt.f64 70.8

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

   7. Applied rewrites 70.8%

      \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

   if +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 0.0%

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

   2. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 19.6%

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

   5. Taylor expanded in d around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 26.6%

      \[\leadsto \frac{\frac{\mathsf{fma}\left({\left(\frac{h}{\ell}\right)}^{1.5} \cdot {\left(D \cdot M\right)}^{2}, -0.125, \sqrt{\frac{h}{\ell}} \cdot \left(d \cdot d\right)\right)}{d}}{h} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 26.6

         \[\leadsto \frac{\frac{\mathsf{fma}\left({\left(\frac{h}{\ell}\right)}^{1.5} \cdot \left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right), -0.125, \sqrt{\frac{h}{\ell}} \cdot \left(d \cdot d\right)\right)}{d}}{h} \]

   9. Applied rewrites 26.6%

      \[\leadsto \frac{\frac{\mathsf{fma}\left({\left(\frac{h}{\ell}\right)}^{1.5} \cdot \left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right), -0.125, \sqrt{\frac{h}{\ell}} \cdot \left(d \cdot d\right)\right)}{d}}{h} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 71.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{h}{\ell}}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_1 \leq 10^{+220}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \frac{h}{\ell} \cdot \left(\left(\left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right) \cdot \left(M \cdot \frac{D}{2 \cdot d}\right)\right) \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{t\_0 \cdot d}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{d \cdot \left(-t\_0\right)}{h}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ h l)))
        (t_1
         (*
          (* (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))))))
   (if (<= t_1 1e+220)
     (*
      (sqrt (/ d h))
      (*
       (sqrt (/ d l))
       (-
        1.0
        (* (/ h l) (* (* (* (/ D d) (* 0.5 M)) (* M (/ D (* 2.0 d)))) 0.5)))))
     (if (<= t_1 INFINITY) (/ (* t_0 d) h) (/ (* d (- t_0)) h)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((h / l));
	double t_1 = (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)));
	double tmp;
	if (t_1 <= 1e+220) {
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 - ((h / l) * ((((D / d) * (0.5 * M)) * (M * (D / (2.0 * d)))) * 0.5))));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (t_0 * d) / h;
	} else {
		tmp = (d * -t_0) / h;
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((h / l));
	double t_1 = (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)));
	double tmp;
	if (t_1 <= 1e+220) {
		tmp = Math.sqrt((d / h)) * (Math.sqrt((d / l)) * (1.0 - ((h / l) * ((((D / d) * (0.5 * M)) * (M * (D / (2.0 * d)))) * 0.5))));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (t_0 * d) / h;
	} else {
		tmp = (d * -t_0) / h;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((h / l))
	t_1 = (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)))
	tmp = 0
	if t_1 <= 1e+220:
		tmp = math.sqrt((d / h)) * (math.sqrt((d / l)) * (1.0 - ((h / l) * ((((D / d) * (0.5 * M)) * (M * (D / (2.0 * d)))) * 0.5))))
	elif t_1 <= math.inf:
		tmp = (t_0 * d) / h
	else:
		tmp = (d * -t_0) / h
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(h / l))
	t_1 = 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))))
	tmp = 0.0
	if (t_1 <= 1e+220)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * Float64(1.0 - Float64(Float64(h / l) * Float64(Float64(Float64(Float64(D / d) * Float64(0.5 * M)) * Float64(M * Float64(D / Float64(2.0 * d)))) * 0.5)))));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(t_0 * d) / h);
	else
		tmp = Float64(Float64(d * Float64(-t_0)) / h);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((h / l));
	t_1 = (((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)));
	tmp = 0.0;
	if (t_1 <= 1e+220)
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 - ((h / l) * ((((D / d) * (0.5 * M)) * (M * (D / (2.0 * d)))) * 0.5))));
	elseif (t_1 <= Inf)
		tmp = (t_0 * d) / h;
	else
		tmp = (d * -t_0) / h;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = 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]}, If[LessEqual[t$95$1, 1e+220], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[(N[(N[(D / d), $MachinePrecision] * N[(0.5 * M), $MachinePrecision]), $MachinePrecision] * N[(M * N[(D / N[(2.0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(t$95$0 * d), $MachinePrecision] / h), $MachinePrecision], N[(N[(d * (-t$95$0)), $MachinePrecision] / h), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 1e220

   1. Initial program 87.6%

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

   3. Applied rewrites 86.7%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 86.7

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 86.7

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-/.f64 86.7

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

   5. Applied rewrites 86.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 86.7

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

   7. Applied rewrites 86.7%

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

   8. Taylor expanded in M around 0

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

      1. lower-*.f64 86.7

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

   10. Applied rewrites 86.7%

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

   if 1e220 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 56.3%

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

   2. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 49.4%

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

   5. Taylor expanded in d around inf

      \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

      4. lift-sqrt.f64 70.8

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

   7. Applied rewrites 70.8%

      \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

   if +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 0.0%

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

   2. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 19.6%

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

   5. Taylor expanded in l around -inf

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

      1. sqrt-pow2 N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-sqrt.f64 17.9

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

   7. Applied rewrites 17.9%

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

Alternative 8: 70.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \frac{D}{2 \cdot d}\\ t_2 := \sqrt{\frac{h}{\ell}}\\ \mathbf{if}\;t\_0 \leq 10^{+220}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \frac{h}{\ell} \cdot \left(\left(\left(\left(t\_1 \cdot M\right) \cdot M\right) \cdot t\_1\right) \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{t\_2 \cdot d}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{d \cdot \left(-t\_2\right)}{h}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (* (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)))))
        (t_1 (/ D (* 2.0 d)))
        (t_2 (sqrt (/ h l))))
   (if (<= t_0 1e+220)
     (*
      (sqrt (/ d h))
      (* (sqrt (/ d l)) (- 1.0 (* (/ h l) (* (* (* (* t_1 M) M) t_1) 0.5)))))
     (if (<= t_0 INFINITY) (/ (* t_2 d) h) (/ (* d (- t_2)) h)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (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)));
	double t_1 = D / (2.0 * d);
	double t_2 = sqrt((h / l));
	double tmp;
	if (t_0 <= 1e+220) {
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 - ((h / l) * ((((t_1 * M) * M) * t_1) * 0.5))));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (t_2 * d) / h;
	} else {
		tmp = (d * -t_2) / h;
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (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)));
	double t_1 = D / (2.0 * d);
	double t_2 = Math.sqrt((h / l));
	double tmp;
	if (t_0 <= 1e+220) {
		tmp = Math.sqrt((d / h)) * (Math.sqrt((d / l)) * (1.0 - ((h / l) * ((((t_1 * M) * M) * t_1) * 0.5))));
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = (t_2 * d) / h;
	} else {
		tmp = (d * -t_2) / h;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (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)))
	t_1 = D / (2.0 * d)
	t_2 = math.sqrt((h / l))
	tmp = 0
	if t_0 <= 1e+220:
		tmp = math.sqrt((d / h)) * (math.sqrt((d / l)) * (1.0 - ((h / l) * ((((t_1 * M) * M) * t_1) * 0.5))))
	elif t_0 <= math.inf:
		tmp = (t_2 * d) / h
	else:
		tmp = (d * -t_2) / h
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = 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))))
	t_1 = Float64(D / Float64(2.0 * d))
	t_2 = sqrt(Float64(h / l))
	tmp = 0.0
	if (t_0 <= 1e+220)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * Float64(1.0 - Float64(Float64(h / l) * Float64(Float64(Float64(Float64(t_1 * M) * M) * t_1) * 0.5)))));
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(t_2 * d) / h);
	else
		tmp = Float64(Float64(d * Float64(-t_2)) / h);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (((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)));
	t_1 = D / (2.0 * d);
	t_2 = sqrt((h / l));
	tmp = 0.0;
	if (t_0 <= 1e+220)
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 - ((h / l) * ((((t_1 * M) * M) * t_1) * 0.5))));
	elseif (t_0 <= Inf)
		tmp = (t_2 * d) / h;
	else
		tmp = (d * -t_2) / h;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = 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]}, Block[{t$95$1 = N[(D / N[(2.0 * d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 1e+220], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[(N[(N[(t$95$1 * M), $MachinePrecision] * M), $MachinePrecision] * t$95$1), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(t$95$2 * d), $MachinePrecision] / h), $MachinePrecision], N[(N[(d * (-t$95$2)), $MachinePrecision] / h), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 1e220

   1. Initial program 87.6%

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

   3. Applied rewrites 86.7%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 86.7

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 86.7

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-/.f64 86.7

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

   5. Applied rewrites 86.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 86.7

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

   7. Applied rewrites 86.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-/.f64 N/A

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

      11. times-frac N/A

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

      12. associate-*r/ N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. lift-*.f64 84.9

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

   9. Applied rewrites 84.9%

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

   if 1e220 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 56.3%

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

   2. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 49.4%

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

   5. Taylor expanded in d around inf

      \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

      4. lift-sqrt.f64 70.8

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

   7. Applied rewrites 70.8%

      \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

   if +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 0.0%

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

   2. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 19.6%

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

   5. Taylor expanded in l around -inf

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

      1. sqrt-pow2 N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-sqrt.f64 17.9

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

   7. Applied rewrites 17.9%

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

Alternative 9: 47.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{h}{\ell}}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_2 := \frac{d \cdot \left(-t\_0\right)}{h}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-78}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{t\_0 \cdot d}{h}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ h l)))
        (t_1
         (*
          (* (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)))))
        (t_2 (/ (* d (- t_0)) h)))
   (if (<= t_1 -2e-78) t_2 (if (<= t_1 INFINITY) (/ (* t_0 d) h) t_2))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((h / l));
	double t_1 = (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)));
	double t_2 = (d * -t_0) / h;
	double tmp;
	if (t_1 <= -2e-78) {
		tmp = t_2;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (t_0 * d) / h;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((h / l));
	double t_1 = (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)));
	double t_2 = (d * -t_0) / h;
	double tmp;
	if (t_1 <= -2e-78) {
		tmp = t_2;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (t_0 * d) / h;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((h / l))
	t_1 = (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)))
	t_2 = (d * -t_0) / h
	tmp = 0
	if t_1 <= -2e-78:
		tmp = t_2
	elif t_1 <= math.inf:
		tmp = (t_0 * d) / h
	else:
		tmp = t_2
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(h / l))
	t_1 = 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))))
	t_2 = Float64(Float64(d * Float64(-t_0)) / h)
	tmp = 0.0
	if (t_1 <= -2e-78)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(t_0 * d) / h);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((h / l));
	t_1 = (((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)));
	t_2 = (d * -t_0) / h;
	tmp = 0.0;
	if (t_1 <= -2e-78)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = (t_0 * d) / h;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = 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]}, Block[{t$95$2 = N[(N[(d * (-t$95$0)), $MachinePrecision] / h), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-78], t$95$2, If[LessEqual[t$95$1, Infinity], N[(N[(t$95$0 * d), $MachinePrecision] / h), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -2e-78 or +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 55.8%

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

   2. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 42.6%

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

   5. Taylor expanded in l around -inf

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

      1. sqrt-pow2 N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-sqrt.f64 20.8

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

   7. Applied rewrites 20.8%

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

   if -2e-78 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 78.7%

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

   2. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 57.8%

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

   5. Taylor expanded in d around inf

      \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

      4. lift-sqrt.f64 74.8

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

   7. Applied rewrites 74.8%

      \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 41.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 2 \cdot 10^{-245}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot d\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot d}{h}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<=
      (*
       (* (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))))
      2e-245)
   (* (sqrt (/ (/ 1.0 l) h)) d)
   (/ (* (sqrt (/ h l)) d) h)))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (((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)))) <= 2e-245) {
		tmp = sqrt(((1.0 / l) / h)) * d;
	} else {
		tmp = (sqrt((h / l)) * d) / h;
	}
	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 / 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)))) <= 2d-245) then
        tmp = sqrt(((1.0d0 / l) / h)) * d
    else
        tmp = (sqrt((h / l)) * d) / h
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (((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)))) <= 2e-245) {
		tmp = Math.sqrt(((1.0 / l) / h)) * d;
	} else {
		tmp = (Math.sqrt((h / l)) * d) / h;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if ((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)))) <= 2e-245:
		tmp = math.sqrt(((1.0 / l) / h)) * d
	else:
		tmp = (math.sqrt((h / l)) * d) / h
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (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)))) <= 2e-245)
		tmp = Float64(sqrt(Float64(Float64(1.0 / l) / h)) * d);
	else
		tmp = Float64(Float64(sqrt(Float64(h / l)) * d) / h);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (((((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)))) <= 2e-245)
		tmp = sqrt(((1.0 / l) / h)) * d;
	else
		tmp = (sqrt((h / l)) * d) / h;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[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], 2e-245], N[(N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision], N[(N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision] / h), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 1.9999999999999999e-245

   1. Initial program 80.0%

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

   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 17.3

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

   4. Applied rewrites 17.3%

      \[\leadsto \color{blue}{\sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d} \]
   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 17.3

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

   6. Applied rewrites 17.3%

      \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

      3. associate-/r* N/A

         \[\leadsto \sqrt{\frac{\frac{1}{\ell}}{h}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\frac{\frac{1}{\ell}}{h}} \cdot d \]

      5. lower-/.f64 17.3

         \[\leadsto \sqrt{\frac{\frac{1}{\ell}}{h}} \cdot d \]

   8. Applied rewrites 17.3%

      \[\leadsto \sqrt{\frac{\frac{1}{\ell}}{h}} \cdot d \]

   if 1.9999999999999999e-245 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 58.6%

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

   2. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 46.9%

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

   5. Taylor expanded in d around inf

      \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

      4. lift-sqrt.f64 56.6

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

   7. Applied rewrites 56.6%

      \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 75.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -7.5 \cdot 10^{-119}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \frac{h}{\ell} \cdot \left(\left(\left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right) \cdot \left(M \cdot \frac{D}{2 \cdot d}\right)\right) \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;d \leq 1.9 \cdot 10^{-160}:\\ \;\;\;\;\frac{\left({\left(\frac{h}{\ell}\right)}^{1.5} \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}\right) \cdot -0.125}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= d -7.5e-119)
   (*
    (sqrt (/ d h))
    (*
     (sqrt (/ d l))
     (-
      1.0
      (* (/ h l) (* (* (* (/ D d) (* 0.5 M)) (* M (/ D (* 2.0 d)))) 0.5)))))
   (if (<= d 1.9e-160)
     (/ (* (* (pow (/ h l) 1.5) (/ (pow (* D M) 2.0) d)) -0.125) h)
     (*
      (/ (sqrt d) (sqrt h))
      (*
       (/ (sqrt d) (sqrt l))
       (- 1.0 (* (/ h l) (* (pow (* (/ M 2.0) (/ D d)) 2.0) 0.5))))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -7.5e-119) {
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 - ((h / l) * ((((D / d) * (0.5 * M)) * (M * (D / (2.0 * d)))) * 0.5))));
	} else if (d <= 1.9e-160) {
		tmp = ((pow((h / l), 1.5) * (pow((D * M), 2.0) / d)) * -0.125) / h;
	} else {
		tmp = (sqrt(d) / sqrt(h)) * ((sqrt(d) / sqrt(l)) * (1.0 - ((h / l) * (pow(((M / 2.0) * (D / d)), 2.0) * 0.5))));
	}
	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 <= (-7.5d-119)) then
        tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0d0 - ((h / l) * ((((d_1 / d) * (0.5d0 * m)) * (m * (d_1 / (2.0d0 * d)))) * 0.5d0))))
    else if (d <= 1.9d-160) then
        tmp = ((((h / l) ** 1.5d0) * (((d_1 * m) ** 2.0d0) / d)) * (-0.125d0)) / h
    else
        tmp = (sqrt(d) / sqrt(h)) * ((sqrt(d) / sqrt(l)) * (1.0d0 - ((h / l) * ((((m / 2.0d0) * (d_1 / d)) ** 2.0d0) * 0.5d0))))
    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 <= -7.5e-119) {
		tmp = Math.sqrt((d / h)) * (Math.sqrt((d / l)) * (1.0 - ((h / l) * ((((D / d) * (0.5 * M)) * (M * (D / (2.0 * d)))) * 0.5))));
	} else if (d <= 1.9e-160) {
		tmp = ((Math.pow((h / l), 1.5) * (Math.pow((D * M), 2.0) / d)) * -0.125) / h;
	} else {
		tmp = (Math.sqrt(d) / Math.sqrt(h)) * ((Math.sqrt(d) / Math.sqrt(l)) * (1.0 - ((h / l) * (Math.pow(((M / 2.0) * (D / d)), 2.0) * 0.5))));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if d <= -7.5e-119:
		tmp = math.sqrt((d / h)) * (math.sqrt((d / l)) * (1.0 - ((h / l) * ((((D / d) * (0.5 * M)) * (M * (D / (2.0 * d)))) * 0.5))))
	elif d <= 1.9e-160:
		tmp = ((math.pow((h / l), 1.5) * (math.pow((D * M), 2.0) / d)) * -0.125) / h
	else:
		tmp = (math.sqrt(d) / math.sqrt(h)) * ((math.sqrt(d) / math.sqrt(l)) * (1.0 - ((h / l) * (math.pow(((M / 2.0) * (D / d)), 2.0) * 0.5))))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -7.5e-119)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * Float64(1.0 - Float64(Float64(h / l) * Float64(Float64(Float64(Float64(D / d) * Float64(0.5 * M)) * Float64(M * Float64(D / Float64(2.0 * d)))) * 0.5)))));
	elseif (d <= 1.9e-160)
		tmp = Float64(Float64(Float64((Float64(h / l) ^ 1.5) * Float64((Float64(D * M) ^ 2.0) / d)) * -0.125) / h);
	else
		tmp = Float64(Float64(sqrt(d) / sqrt(h)) * Float64(Float64(sqrt(d) / sqrt(l)) * Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(M / 2.0) * Float64(D / d)) ^ 2.0) * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (d <= -7.5e-119)
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 - ((h / l) * ((((D / d) * (0.5 * M)) * (M * (D / (2.0 * d)))) * 0.5))));
	elseif (d <= 1.9e-160)
		tmp = ((((h / l) ^ 1.5) * (((D * M) ^ 2.0) / d)) * -0.125) / h;
	else
		tmp = (sqrt(d) / sqrt(h)) * ((sqrt(d) / sqrt(l)) * (1.0 - ((h / l) * ((((M / 2.0) * (D / d)) ^ 2.0) * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[d, -7.5e-119], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[(N[(N[(D / d), $MachinePrecision] * N[(0.5 * M), $MachinePrecision]), $MachinePrecision] * N[(M * N[(D / N[(2.0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.9e-160], N[(N[(N[(N[Power[N[(h / l), $MachinePrecision], 1.5], $MachinePrecision] * N[(N[Power[N[(D * M), $MachinePrecision], 2.0], $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] / h), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(M / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -7.50000000000000044e-119

   1. Initial program 76.2%

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

   3. Applied rewrites 75.8%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 75.8

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 75.8

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-/.f64 75.8

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

   5. Applied rewrites 75.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 75.8

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

   7. Applied rewrites 75.8%

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

   8. Taylor expanded in M around 0

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

      1. lower-*.f64 75.8

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

   10. Applied rewrites 75.8%

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

   if -7.50000000000000044e-119 < d < 1.8999999999999999e-160

   1. Initial program 45.6%

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

   2. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 47.4%

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

   5. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. cube-div N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. unpow-prod-down N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 48.7

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

   7. Applied rewrites 57.8%

      \[\leadsto \frac{\left({\left(\frac{h}{\ell}\right)}^{1.5} \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}\right) \cdot -0.125}{h} \]

   if 1.8999999999999999e-160 < d

   1. Initial program 74.4%

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

   3. Applied rewrites 74.1%

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

      1. lift-/.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. sqrt-div N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-/.f64 77.2

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

   5. Applied rewrites 77.2%

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

      1. lift-/.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. sqrt-div N/A

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

      4. lower-/.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lower-sqrt.f64 86.5

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

   7. Applied rewrites 86.5%

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

Alternative 12: 75.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -2.5 \cdot 10^{-27}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \frac{h}{\ell} \cdot \left(\left(\left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right) \cdot \left(M \cdot \frac{D}{2 \cdot d}\right)\right) \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left({\left(\frac{h}{\ell}\right)}^{1.5} \cdot \left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right), -0.125, \sqrt{\frac{h}{\ell}} \cdot \left(d \cdot d\right)\right)}{d}}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= d -2.5e-27)
   (*
    (sqrt (/ d h))
    (*
     (sqrt (/ d l))
     (-
      1.0
      (* (/ h l) (* (* (* (/ D d) (* 0.5 M)) (* M (/ D (* 2.0 d)))) 0.5)))))
   (if (<= d -1e-309)
     (/
      (/
       (fma
        (* (pow (/ h l) 1.5) (* (* M D) (* M D)))
        -0.125
        (* (sqrt (/ h l)) (* d d)))
       d)
      h)
     (*
      (/ (sqrt d) (sqrt h))
      (*
       (/ (sqrt d) (sqrt l))
       (- 1.0 (* (/ h l) (* (pow (* (/ M 2.0) (/ D d)) 2.0) 0.5))))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -2.5e-27) {
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 - ((h / l) * ((((D / d) * (0.5 * M)) * (M * (D / (2.0 * d)))) * 0.5))));
	} else if (d <= -1e-309) {
		tmp = (fma((pow((h / l), 1.5) * ((M * D) * (M * D))), -0.125, (sqrt((h / l)) * (d * d))) / d) / h;
	} else {
		tmp = (sqrt(d) / sqrt(h)) * ((sqrt(d) / sqrt(l)) * (1.0 - ((h / l) * (pow(((M / 2.0) * (D / d)), 2.0) * 0.5))));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -2.5e-27)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * Float64(1.0 - Float64(Float64(h / l) * Float64(Float64(Float64(Float64(D / d) * Float64(0.5 * M)) * Float64(M * Float64(D / Float64(2.0 * d)))) * 0.5)))));
	elseif (d <= -1e-309)
		tmp = Float64(Float64(fma(Float64((Float64(h / l) ^ 1.5) * Float64(Float64(M * D) * Float64(M * D))), -0.125, Float64(sqrt(Float64(h / l)) * Float64(d * d))) / d) / h);
	else
		tmp = Float64(Float64(sqrt(d) / sqrt(h)) * Float64(Float64(sqrt(d) / sqrt(l)) * Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(M / 2.0) * Float64(D / d)) ^ 2.0) * 0.5)))));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[d, -2.5e-27], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[(N[(N[(D / d), $MachinePrecision] * N[(0.5 * M), $MachinePrecision]), $MachinePrecision] * N[(M * N[(D / N[(2.0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -1e-309], N[(N[(N[(N[(N[Power[N[(h / l), $MachinePrecision], 1.5], $MachinePrecision] * N[(N[(M * D), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125 + N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] / h), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(M / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.5000000000000001e-27

   1. Initial program 76.8%

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

   3. Applied rewrites 76.6%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 76.6

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 76.6

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-/.f64 76.6

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

   5. Applied rewrites 76.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 76.6

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

   7. Applied rewrites 76.6%

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

   8. Taylor expanded in M around 0

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

      1. lower-*.f64 76.6

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

   10. Applied rewrites 76.6%

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

   if -2.5000000000000001e-27 < d < -1.000000000000002e-309

   1. Initial program 54.5%

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

   2. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 48.6%

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

   5. Taylor expanded in d around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 54.1%

      \[\leadsto \frac{\frac{\mathsf{fma}\left({\left(\frac{h}{\ell}\right)}^{1.5} \cdot {\left(D \cdot M\right)}^{2}, -0.125, \sqrt{\frac{h}{\ell}} \cdot \left(d \cdot d\right)\right)}{d}}{h} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 54.1

         \[\leadsto \frac{\frac{\mathsf{fma}\left({\left(\frac{h}{\ell}\right)}^{1.5} \cdot \left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right), -0.125, \sqrt{\frac{h}{\ell}} \cdot \left(d \cdot d\right)\right)}{d}}{h} \]

   9. Applied rewrites 54.1%

      \[\leadsto \frac{\frac{\mathsf{fma}\left({\left(\frac{h}{\ell}\right)}^{1.5} \cdot \left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right), -0.125, \sqrt{\frac{h}{\ell}} \cdot \left(d \cdot d\right)\right)}{d}}{h} \]

   if -1.000000000000002e-309 < d

   1. Initial program 67.5%

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

   3. Applied rewrites 66.9%

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

      1. lift-/.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. sqrt-div N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-/.f64 71.2

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

   5. Applied rewrites 71.2%

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

      1. lift-/.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. sqrt-div N/A

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

      4. lower-/.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lower-sqrt.f64 83.7

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

   7. Applied rewrites 83.7%

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

Alternative 13: 52.3% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.05 \cdot 10^{+126}:\\ \;\;\;\;\left(-{\left(\ell \cdot h\right)}^{-0.5}\right) \cdot d\\ \mathbf{elif}\;d \leq -1.45 \cdot 10^{-31}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(\frac{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot -1}{d} \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}}\right) \cdot -0.125\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= d -1.05e+126)
   (* (- (pow (* l h) -0.5)) d)
   (if (<= d -1.45e-31)
     (* (sqrt (/ d h)) (sqrt (/ d l)))
     (if (<= d -1e-309)
       (*
        (* (/ (* (* (* M D) (* M D)) -1.0) d) (sqrt (/ h (* (* l l) l))))
        -0.125)
       (* (/ 1.0 (* (sqrt l) (sqrt h))) d)))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -1.05e+126) {
		tmp = -pow((l * h), -0.5) * d;
	} else if (d <= -1.45e-31) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else if (d <= -1e-309) {
		tmp = (((((M * D) * (M * D)) * -1.0) / d) * sqrt((h / ((l * l) * l)))) * -0.125;
	} 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 <= (-1.05d+126)) then
        tmp = -((l * h) ** (-0.5d0)) * d
    else if (d <= (-1.45d-31)) then
        tmp = sqrt((d / h)) * sqrt((d / l))
    else if (d <= (-1d-309)) then
        tmp = (((((m * d_1) * (m * d_1)) * (-1.0d0)) / d) * sqrt((h / ((l * l) * l)))) * (-0.125d0)
    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 <= -1.05e+126) {
		tmp = -Math.pow((l * h), -0.5) * d;
	} else if (d <= -1.45e-31) {
		tmp = Math.sqrt((d / h)) * Math.sqrt((d / l));
	} else if (d <= -1e-309) {
		tmp = (((((M * D) * (M * D)) * -1.0) / d) * Math.sqrt((h / ((l * l) * l)))) * -0.125;
	} 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 <= -1.05e+126:
		tmp = -math.pow((l * h), -0.5) * d
	elif d <= -1.45e-31:
		tmp = math.sqrt((d / h)) * math.sqrt((d / l))
	elif d <= -1e-309:
		tmp = (((((M * D) * (M * D)) * -1.0) / d) * math.sqrt((h / ((l * l) * l)))) * -0.125
	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 <= -1.05e+126)
		tmp = Float64(Float64(-(Float64(l * h) ^ -0.5)) * d);
	elseif (d <= -1.45e-31)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	elseif (d <= -1e-309)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(M * D) * Float64(M * D)) * -1.0) / d) * sqrt(Float64(h / Float64(Float64(l * l) * l)))) * -0.125);
	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 <= -1.05e+126)
		tmp = -((l * h) ^ -0.5) * d;
	elseif (d <= -1.45e-31)
		tmp = sqrt((d / h)) * sqrt((d / l));
	elseif (d <= -1e-309)
		tmp = (((((M * D) * (M * D)) * -1.0) / d) * sqrt((h / ((l * l) * l)))) * -0.125;
	else
		tmp = (1.0 / (sqrt(l) * sqrt(h))) * d;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[d, -1.05e+126], N[((-N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]) * d), $MachinePrecision], If[LessEqual[d, -1.45e-31], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -1e-309], N[(N[(N[(N[(N[(N[(M * D), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] * -1.0), $MachinePrecision] / d), $MachinePrecision] * N[Sqrt[N[(h / N[(N[(l * l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision], N[(N[(1.0 / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.05e126

   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. 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 6.5

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

   4. Applied rewrites 6.5%

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

   5. Taylor expanded in h around -inf

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

      1. sqrt-pow2 N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. inv-pow N/A

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

      8. *-commutative N/A

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

      9. sqrt-pow1 N/A

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

      10. lower-pow.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. metadata-eval 66.2

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

   7. Applied rewrites 66.2%

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

   if -1.05e126 < d < -1.45e-31

   1. Initial program 79.9%

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

   3. Applied rewrites 79.2%

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

   4. Taylor expanded in d around inf

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

      1. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

      2. lift-/.f64 48.4

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

   6. Applied rewrites 48.4%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

   if -1.45e-31 < d < -1.000000000000002e-309

   1. Initial program 54.1%

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

   2. Taylor expanded in h around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 45.2%

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

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 45.1

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

   6. Applied rewrites 45.1%

      \[\leadsto \left(\frac{{\left(D \cdot M\right)}^{2} \cdot -1}{d} \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}}\right) \cdot -0.125 \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 45.1

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

   8. Applied rewrites 45.1%

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

   if -1.000000000000002e-309 < d

   1. Initial program 67.5%

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

   2. 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 44.4

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

   4. Applied rewrites 44.4%

      \[\leadsto \color{blue}{\sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d} \]
   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 44.5

          \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

   6. Applied rewrites 44.5%

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]
   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. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]

      6. lower-sqrt.f64 52.4

         \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]

   8. Applied rewrites 52.4%

      \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 14: 43.4% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{h}{\ell}}\\ \mathbf{if}\;d \leq -5.6 \cdot 10^{-108}:\\ \;\;\;\;\frac{t\_0 \cdot d}{h}\\ \mathbf{elif}\;d \leq 7 \cdot 10^{-122}:\\ \;\;\;\;\frac{d \cdot \left(-t\_0\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ h l))))
   (if (<= d -5.6e-108)
     (/ (* t_0 d) h)
     (if (<= d 7e-122)
       (/ (* d (- t_0)) h)
       (* (/ 1.0 (* (sqrt l) (sqrt h))) d)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((h / l));
	double tmp;
	if (d <= -5.6e-108) {
		tmp = (t_0 * d) / h;
	} else if (d <= 7e-122) {
		tmp = (d * -t_0) / h;
	} 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 = sqrt((h / l))
    if (d <= (-5.6d-108)) then
        tmp = (t_0 * d) / h
    else if (d <= 7d-122) then
        tmp = (d * -t_0) / h
    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 = Math.sqrt((h / l));
	double tmp;
	if (d <= -5.6e-108) {
		tmp = (t_0 * d) / h;
	} else if (d <= 7e-122) {
		tmp = (d * -t_0) / h;
	} else {
		tmp = (1.0 / (Math.sqrt(l) * Math.sqrt(h))) * d;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((h / l))
	tmp = 0
	if d <= -5.6e-108:
		tmp = (t_0 * d) / h
	elif d <= 7e-122:
		tmp = (d * -t_0) / h
	else:
		tmp = (1.0 / (math.sqrt(l) * math.sqrt(h))) * d
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(h / l))
	tmp = 0.0
	if (d <= -5.6e-108)
		tmp = Float64(Float64(t_0 * d) / h);
	elseif (d <= 7e-122)
		tmp = Float64(Float64(d * Float64(-t_0)) / h);
	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 = sqrt((h / l));
	tmp = 0.0;
	if (d <= -5.6e-108)
		tmp = (t_0 * d) / h;
	elseif (d <= 7e-122)
		tmp = (d * -t_0) / h;
	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[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -5.6e-108], N[(N[(t$95$0 * d), $MachinePrecision] / h), $MachinePrecision], If[LessEqual[d, 7e-122], N[(N[(d * (-t$95$0)), $MachinePrecision] / h), $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 d < -5.6e-108

   1. Initial program 76.5%

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

   2. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 51.6%

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

   5. Taylor expanded in d around inf

      \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

      4. lift-sqrt.f64 46.9

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

   7. Applied rewrites 46.9%

      \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

   if -5.6e-108 < d < 7.0000000000000003e-122

   1. Initial program 46.9%

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

   2. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 47.7%

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

   5. Taylor expanded in l around -inf

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

      1. sqrt-pow2 N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-sqrt.f64 17.7

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

   7. Applied rewrites 17.7%

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

   if 7.0000000000000003e-122 < d

   1. Initial program 76.0%

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

   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 52.7

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

   4. Applied rewrites 52.7%

      \[\leadsto \color{blue}{\sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d} \]
   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 52.9

          \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

   6. Applied rewrites 52.9%

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]
   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. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]

      6. lower-sqrt.f64 62.7

         \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]

   8. Applied rewrites 62.7%

      \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 27.4% 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 67.1%

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

2. 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 27.4

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

4. Applied rewrites 27.4%

   \[\leadsto \color{blue}{\sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d} \]
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 27.4

      \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

6. Applied rewrites 27.4%

   \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025106 
(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)))))