Henrywood and Agarwal, Equation (12)

Percentage Accurate: 36.2% → 77.2%

Time: 8.4s

Alternatives: 9

Speedup: 2.7×

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: 36.2% 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: 77.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} t_0 := 1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d\_m}\right)}^{2}\right) \cdot \frac{h}{\ell}\\ t_1 := {\left(\frac{d\_m}{h}\right)}^{\left(\frac{1}{2}\right)}\\ t_2 := \frac{D}{d\_m + d\_m}\\ \mathbf{if}\;\left(t\_1 \cdot {\left(\frac{d\_m}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot t\_0 \leq 2 \cdot 10^{+221}:\\ \;\;\;\;\left(t\_1 \cdot \frac{\sqrt{d\_m}}{\sqrt{\ell}}\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(d\_m \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - \frac{\left(\left(\left(\left(M \cdot t\_2\right) \cdot t\_2\right) \cdot M\right) \cdot 0.5\right) \cdot h}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
(FPCore (d_m h l M D)
 :precision binary64
 (let* ((t_0
         (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d_m)) 2.0)) (/ h l))))
        (t_1 (pow (/ d_m h) (/ 1.0 2.0)))
        (t_2 (/ D (+ d_m d_m))))
   (if (<= (* (* t_1 (pow (/ d_m l) (/ 1.0 2.0))) t_0) 2e+221)
     (* (* t_1 (/ (sqrt d_m) (sqrt l))) t_0)
     (*
      (* d_m (sqrt (/ 1.0 (* h l))))
      (- 1.0 (/ (* (* (* (* (* M t_2) t_2) M) 0.5) h) l))))))

C

d_m = fabs(d);
double code(double d_m, double h, double l, double M, double D) {
	double t_0 = 1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d_m)), 2.0)) * (h / l));
	double t_1 = pow((d_m / h), (1.0 / 2.0));
	double t_2 = D / (d_m + d_m);
	double tmp;
	if (((t_1 * pow((d_m / l), (1.0 / 2.0))) * t_0) <= 2e+221) {
		tmp = (t_1 * (sqrt(d_m) / sqrt(l))) * t_0;
	} else {
		tmp = (d_m * sqrt((1.0 / (h * l)))) * (1.0 - ((((((M * t_2) * t_2) * M) * 0.5) * h) / l));
	}
	return tmp;
}

Fortran

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

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

real(8) function code(d_m, h, l, m, d)
use fmin_fmax_functions
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 1.0d0 - (((1.0d0 / 2.0d0) * (((m * d) / (2.0d0 * d_m)) ** 2.0d0)) * (h / l))
    t_1 = (d_m / h) ** (1.0d0 / 2.0d0)
    t_2 = d / (d_m + d_m)
    if (((t_1 * ((d_m / l) ** (1.0d0 / 2.0d0))) * t_0) <= 2d+221) then
        tmp = (t_1 * (sqrt(d_m) / sqrt(l))) * t_0
    else
        tmp = (d_m * sqrt((1.0d0 / (h * l)))) * (1.0d0 - ((((((m * t_2) * t_2) * m) * 0.5d0) * h) / l))
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
public static double code(double d_m, double h, double l, double M, double D) {
	double t_0 = 1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d_m)), 2.0)) * (h / l));
	double t_1 = Math.pow((d_m / h), (1.0 / 2.0));
	double t_2 = D / (d_m + d_m);
	double tmp;
	if (((t_1 * Math.pow((d_m / l), (1.0 / 2.0))) * t_0) <= 2e+221) {
		tmp = (t_1 * (Math.sqrt(d_m) / Math.sqrt(l))) * t_0;
	} else {
		tmp = (d_m * Math.sqrt((1.0 / (h * l)))) * (1.0 - ((((((M * t_2) * t_2) * M) * 0.5) * h) / l));
	}
	return tmp;
}

Python

d_m = math.fabs(d)
def code(d_m, h, l, M, D):
	t_0 = 1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d_m)), 2.0)) * (h / l))
	t_1 = math.pow((d_m / h), (1.0 / 2.0))
	t_2 = D / (d_m + d_m)
	tmp = 0
	if ((t_1 * math.pow((d_m / l), (1.0 / 2.0))) * t_0) <= 2e+221:
		tmp = (t_1 * (math.sqrt(d_m) / math.sqrt(l))) * t_0
	else:
		tmp = (d_m * math.sqrt((1.0 / (h * l)))) * (1.0 - ((((((M * t_2) * t_2) * M) * 0.5) * h) / l))
	return tmp

Julia

d_m = abs(d)
function code(d_m, h, l, M, D)
	t_0 = Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d_m)) ^ 2.0)) * Float64(h / l)))
	t_1 = Float64(d_m / h) ^ Float64(1.0 / 2.0)
	t_2 = Float64(D / Float64(d_m + d_m))
	tmp = 0.0
	if (Float64(Float64(t_1 * (Float64(d_m / l) ^ Float64(1.0 / 2.0))) * t_0) <= 2e+221)
		tmp = Float64(Float64(t_1 * Float64(sqrt(d_m) / sqrt(l))) * t_0);
	else
		tmp = Float64(Float64(d_m * sqrt(Float64(1.0 / Float64(h * l)))) * Float64(1.0 - Float64(Float64(Float64(Float64(Float64(Float64(M * t_2) * t_2) * M) * 0.5) * h) / l)));
	end
	return tmp
end

MATLAB

d_m = abs(d);
function tmp_2 = code(d_m, h, l, M, D)
	t_0 = 1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d_m)) ^ 2.0)) * (h / l));
	t_1 = (d_m / h) ^ (1.0 / 2.0);
	t_2 = D / (d_m + d_m);
	tmp = 0.0;
	if (((t_1 * ((d_m / l) ^ (1.0 / 2.0))) * t_0) <= 2e+221)
		tmp = (t_1 * (sqrt(d_m) / sqrt(l))) * t_0;
	else
		tmp = (d_m * sqrt((1.0 / (h * l)))) * (1.0 - ((((((M * t_2) * t_2) * M) * 0.5) * h) / l));
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
code[d$95$m_, h_, l_, M_, D_] := Block[{t$95$0 = N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(d$95$m / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(D / N[(d$95$m + d$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(t$95$1 * N[Power[N[(d$95$m / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], 2e+221], N[(N[(t$95$1 * N[(N[Sqrt[d$95$m], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(d$95$m * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(N[(N[(M * t$95$2), $MachinePrecision] * t$95$2), $MachinePrecision] * M), $MachinePrecision] * 0.5), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $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)))) < 2.0000000000000001e221

   1. Initial program 36.2%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      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 36.0

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

   3. Applied rewrites 36.0%

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

   if 2.0000000000000001e221 < (*.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 36.2%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. frac-times N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 43.0

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

   3. Applied rewrites 43.0%

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

   4. Taylor expanded in d around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 69.6

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

   6. Applied rewrites 69.6%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. count-2-rev N/A

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

      7. lift-+.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 69.1

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

      15. lift-/.f64 N/A

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

      16. metadata-eval 69.1

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 69.1

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 69.1

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

   8. Applied rewrites 69.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

   10. Applied rewrites 72.9%

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

Alternative 2: 74.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} t_0 := \frac{D}{d\_m + d\_m}\\ t_1 := t\_0 \cdot M\\ \mathbf{if}\;\left({\left(\frac{d\_m}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d\_m}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d\_m}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 10^{+27}:\\ \;\;\;\;\left(d\_m \cdot \frac{\sqrt{\frac{h}{\ell}}}{h}\right) \cdot \left(1 - \left(\left(0.5 \cdot t\_1\right) \cdot t\_1\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(d\_m \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - \frac{\left(\left(\left(\left(M \cdot t\_0\right) \cdot t\_0\right) \cdot M\right) \cdot 0.5\right) \cdot h}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
(FPCore (d_m h l M D)
 :precision binary64
 (let* ((t_0 (/ D (+ d_m d_m))) (t_1 (* t_0 M)))
   (if (<=
        (*
         (* (pow (/ d_m h) (/ 1.0 2.0)) (pow (/ d_m l) (/ 1.0 2.0)))
         (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d_m)) 2.0)) (/ h l))))
        1e+27)
     (* (* d_m (/ (sqrt (/ h l)) h)) (- 1.0 (* (* (* 0.5 t_1) t_1) (/ h l))))
     (*
      (* d_m (sqrt (/ 1.0 (* h l))))
      (- 1.0 (/ (* (* (* (* (* M t_0) t_0) M) 0.5) h) l))))))

C

d_m = fabs(d);
double code(double d_m, double h, double l, double M, double D) {
	double t_0 = D / (d_m + d_m);
	double t_1 = t_0 * M;
	double tmp;
	if (((pow((d_m / h), (1.0 / 2.0)) * pow((d_m / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d_m)), 2.0)) * (h / l)))) <= 1e+27) {
		tmp = (d_m * (sqrt((h / l)) / h)) * (1.0 - (((0.5 * t_1) * t_1) * (h / l)));
	} else {
		tmp = (d_m * sqrt((1.0 / (h * l)))) * (1.0 - ((((((M * t_0) * t_0) * M) * 0.5) * h) / l));
	}
	return tmp;
}

Fortran

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

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

real(8) function code(d_m, h, l, m, d)
use fmin_fmax_functions
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = d / (d_m + d_m)
    t_1 = t_0 * m
    if (((((d_m / h) ** (1.0d0 / 2.0d0)) * ((d_m / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d) / (2.0d0 * d_m)) ** 2.0d0)) * (h / l)))) <= 1d+27) then
        tmp = (d_m * (sqrt((h / l)) / h)) * (1.0d0 - (((0.5d0 * t_1) * t_1) * (h / l)))
    else
        tmp = (d_m * sqrt((1.0d0 / (h * l)))) * (1.0d0 - ((((((m * t_0) * t_0) * m) * 0.5d0) * h) / l))
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
public static double code(double d_m, double h, double l, double M, double D) {
	double t_0 = D / (d_m + d_m);
	double t_1 = t_0 * M;
	double tmp;
	if (((Math.pow((d_m / h), (1.0 / 2.0)) * Math.pow((d_m / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d_m)), 2.0)) * (h / l)))) <= 1e+27) {
		tmp = (d_m * (Math.sqrt((h / l)) / h)) * (1.0 - (((0.5 * t_1) * t_1) * (h / l)));
	} else {
		tmp = (d_m * Math.sqrt((1.0 / (h * l)))) * (1.0 - ((((((M * t_0) * t_0) * M) * 0.5) * h) / l));
	}
	return tmp;
}

Python

d_m = math.fabs(d)
def code(d_m, h, l, M, D):
	t_0 = D / (d_m + d_m)
	t_1 = t_0 * M
	tmp = 0
	if ((math.pow((d_m / h), (1.0 / 2.0)) * math.pow((d_m / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d_m)), 2.0)) * (h / l)))) <= 1e+27:
		tmp = (d_m * (math.sqrt((h / l)) / h)) * (1.0 - (((0.5 * t_1) * t_1) * (h / l)))
	else:
		tmp = (d_m * math.sqrt((1.0 / (h * l)))) * (1.0 - ((((((M * t_0) * t_0) * M) * 0.5) * h) / l))
	return tmp

Julia

d_m = abs(d)
function code(d_m, h, l, M, D)
	t_0 = Float64(D / Float64(d_m + d_m))
	t_1 = Float64(t_0 * M)
	tmp = 0.0
	if (Float64(Float64((Float64(d_m / h) ^ Float64(1.0 / 2.0)) * (Float64(d_m / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d_m)) ^ 2.0)) * Float64(h / l)))) <= 1e+27)
		tmp = Float64(Float64(d_m * Float64(sqrt(Float64(h / l)) / h)) * Float64(1.0 - Float64(Float64(Float64(0.5 * t_1) * t_1) * Float64(h / l))));
	else
		tmp = Float64(Float64(d_m * sqrt(Float64(1.0 / Float64(h * l)))) * Float64(1.0 - Float64(Float64(Float64(Float64(Float64(Float64(M * t_0) * t_0) * M) * 0.5) * h) / l)));
	end
	return tmp
end

MATLAB

d_m = abs(d);
function tmp_2 = code(d_m, h, l, M, D)
	t_0 = D / (d_m + d_m);
	t_1 = t_0 * M;
	tmp = 0.0;
	if (((((d_m / h) ^ (1.0 / 2.0)) * ((d_m / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d_m)) ^ 2.0)) * (h / l)))) <= 1e+27)
		tmp = (d_m * (sqrt((h / l)) / h)) * (1.0 - (((0.5 * t_1) * t_1) * (h / l)));
	else
		tmp = (d_m * sqrt((1.0 / (h * l)))) * (1.0 - ((((((M * t_0) * t_0) * M) * 0.5) * h) / l));
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
code[d$95$m_, h_, l_, M_, D_] := Block[{t$95$0 = N[(D / N[(d$95$m + d$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * M), $MachinePrecision]}, If[LessEqual[N[(N[(N[Power[N[(d$95$m / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d$95$m / 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$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+27], N[(N[(d$95$m * N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(0.5 * t$95$1), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d$95$m * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(N[(N[(M * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision] * M), $MachinePrecision] * 0.5), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $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)))) < 1e27

   1. Initial program 36.2%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. frac-times N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 43.0

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

   3. Applied rewrites 43.0%

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

   4. Taylor expanded in d around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 69.6

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

   6. Applied rewrites 69.6%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. count-2-rev N/A

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

      7. lift-+.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 69.1

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

      15. lift-/.f64 N/A

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

      16. metadata-eval 69.1

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 69.1

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 69.1

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

   8. Applied rewrites 69.1%

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

   9. Taylor expanded in h around 0

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

       1. lower-/.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-/.f64 38.1

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

   11. Applied rewrites 38.1%

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

   if 1e27 < (*.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 36.2%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. frac-times N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 43.0

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

   3. Applied rewrites 43.0%

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

   4. Taylor expanded in d around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 69.6

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

   6. Applied rewrites 69.6%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. count-2-rev N/A

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

      7. lift-+.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 69.1

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

      15. lift-/.f64 N/A

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

      16. metadata-eval 69.1

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 69.1

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 69.1

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

   8. Applied rewrites 69.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

   10. Applied rewrites 72.9%

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

Alternative 3: 72.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} t_0 := \frac{D}{d\_m + d\_m}\\ \left(d\_m \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - \frac{\left(\left(\left(\left(M \cdot t\_0\right) \cdot t\_0\right) \cdot M\right) \cdot 0.5\right) \cdot h}{\ell}\right) \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
(FPCore (d_m h l M D)
 :precision binary64
 (let* ((t_0 (/ D (+ d_m d_m))))
   (*
    (* d_m (sqrt (/ 1.0 (* h l))))
    (- 1.0 (/ (* (* (* (* (* M t_0) t_0) M) 0.5) h) l)))))

C

d_m = fabs(d);
double code(double d_m, double h, double l, double M, double D) {
	double t_0 = D / (d_m + d_m);
	return (d_m * sqrt((1.0 / (h * l)))) * (1.0 - ((((((M * t_0) * t_0) * M) * 0.5) * h) / l));
}

Fortran

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

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

real(8) function code(d_m, h, l, m, d)
use fmin_fmax_functions
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8) :: t_0
    t_0 = d / (d_m + d_m)
    code = (d_m * sqrt((1.0d0 / (h * l)))) * (1.0d0 - ((((((m * t_0) * t_0) * m) * 0.5d0) * h) / l))
end function

Java

d_m = Math.abs(d);
public static double code(double d_m, double h, double l, double M, double D) {
	double t_0 = D / (d_m + d_m);
	return (d_m * Math.sqrt((1.0 / (h * l)))) * (1.0 - ((((((M * t_0) * t_0) * M) * 0.5) * h) / l));
}

Python

d_m = math.fabs(d)
def code(d_m, h, l, M, D):
	t_0 = D / (d_m + d_m)
	return (d_m * math.sqrt((1.0 / (h * l)))) * (1.0 - ((((((M * t_0) * t_0) * M) * 0.5) * h) / l))

Julia

d_m = abs(d)
function code(d_m, h, l, M, D)
	t_0 = Float64(D / Float64(d_m + d_m))
	return Float64(Float64(d_m * sqrt(Float64(1.0 / Float64(h * l)))) * Float64(1.0 - Float64(Float64(Float64(Float64(Float64(Float64(M * t_0) * t_0) * M) * 0.5) * h) / l)))
end

MATLAB

d_m = abs(d);
function tmp = code(d_m, h, l, M, D)
	t_0 = D / (d_m + d_m);
	tmp = (d_m * sqrt((1.0 / (h * l)))) * (1.0 - ((((((M * t_0) * t_0) * M) * 0.5) * h) / l));
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
code[d$95$m_, h_, l_, M_, D_] := Block[{t$95$0 = N[(D / N[(d$95$m + d$95$m), $MachinePrecision]), $MachinePrecision]}, N[(N[(d$95$m * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(N[(N[(M * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision] * M), $MachinePrecision] * 0.5), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 36.2%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-pow.f64 N/A

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

   4. pow-prod-down N/A

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

   5. lift-/.f64 N/A

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

   6. metadata-eval N/A

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

   7. unpow1/2 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. frac-times N/A

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

   12. lower-/.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 43.0

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

3. Applied rewrites 43.0%

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

4. Taylor expanded in d around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-*.f64 69.6

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

6. Applied rewrites 69.6%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. count-2-rev N/A

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

   7. lift-+.f64 N/A

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

   8. associate-/l* N/A

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

   9. lift-/.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. pow2 N/A

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

   12. associate-*r* N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 69.1

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

   15. lift-/.f64 N/A

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

   16. metadata-eval 69.1

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

   17. lift-*.f64 N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 69.1

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

   20. lift-*.f64 N/A

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

   21. *-commutative N/A

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

   22. lower-*.f64 69.1

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

8. Applied rewrites 69.1%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

10. Applied rewrites 72.9%

    \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \frac{D}{d + d}\right) \cdot M\right) \cdot 0.5\right) \cdot h}{\ell}}\right) \]
11. Add Preprocessing

Alternative 4: 56.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} t_0 := d\_m \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{if}\;M \cdot D \leq 2 \cdot 10^{-158}:\\ \;\;\;\;t\_0 \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(1 - \left(\left(0.25 \cdot \frac{D \cdot M}{d\_m}\right) \cdot \left(\frac{D}{d\_m + d\_m} \cdot M\right)\right) \cdot \frac{h}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
(FPCore (d_m h l M D)
 :precision binary64
 (let* ((t_0 (* d_m (sqrt (/ 1.0 (* h l))))))
   (if (<= (* M D) 2e-158)
     (* t_0 1.0)
     (*
      t_0
      (-
       1.0
       (* (* (* 0.25 (/ (* D M) d_m)) (* (/ D (+ d_m d_m)) M)) (/ h l)))))))

C

d_m = fabs(d);
double code(double d_m, double h, double l, double M, double D) {
	double t_0 = d_m * sqrt((1.0 / (h * l)));
	double tmp;
	if ((M * D) <= 2e-158) {
		tmp = t_0 * 1.0;
	} else {
		tmp = t_0 * (1.0 - (((0.25 * ((D * M) / d_m)) * ((D / (d_m + d_m)) * M)) * (h / l)));
	}
	return tmp;
}

Fortran

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

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

real(8) function code(d_m, h, l, m, d)
use fmin_fmax_functions
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = d_m * sqrt((1.0d0 / (h * l)))
    if ((m * d) <= 2d-158) then
        tmp = t_0 * 1.0d0
    else
        tmp = t_0 * (1.0d0 - (((0.25d0 * ((d * m) / d_m)) * ((d / (d_m + d_m)) * m)) * (h / l)))
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
public static double code(double d_m, double h, double l, double M, double D) {
	double t_0 = d_m * Math.sqrt((1.0 / (h * l)));
	double tmp;
	if ((M * D) <= 2e-158) {
		tmp = t_0 * 1.0;
	} else {
		tmp = t_0 * (1.0 - (((0.25 * ((D * M) / d_m)) * ((D / (d_m + d_m)) * M)) * (h / l)));
	}
	return tmp;
}

Python

d_m = math.fabs(d)
def code(d_m, h, l, M, D):
	t_0 = d_m * math.sqrt((1.0 / (h * l)))
	tmp = 0
	if (M * D) <= 2e-158:
		tmp = t_0 * 1.0
	else:
		tmp = t_0 * (1.0 - (((0.25 * ((D * M) / d_m)) * ((D / (d_m + d_m)) * M)) * (h / l)))
	return tmp

Julia

d_m = abs(d)
function code(d_m, h, l, M, D)
	t_0 = Float64(d_m * sqrt(Float64(1.0 / Float64(h * l))))
	tmp = 0.0
	if (Float64(M * D) <= 2e-158)
		tmp = Float64(t_0 * 1.0);
	else
		tmp = Float64(t_0 * Float64(1.0 - Float64(Float64(Float64(0.25 * Float64(Float64(D * M) / d_m)) * Float64(Float64(D / Float64(d_m + d_m)) * M)) * Float64(h / l))));
	end
	return tmp
end

MATLAB

d_m = abs(d);
function tmp_2 = code(d_m, h, l, M, D)
	t_0 = d_m * sqrt((1.0 / (h * l)));
	tmp = 0.0;
	if ((M * D) <= 2e-158)
		tmp = t_0 * 1.0;
	else
		tmp = t_0 * (1.0 - (((0.25 * ((D * M) / d_m)) * ((D / (d_m + d_m)) * M)) * (h / l)));
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
code[d$95$m_, h_, l_, M_, D_] := Block[{t$95$0 = N[(d$95$m * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(M * D), $MachinePrecision], 2e-158], N[(t$95$0 * 1.0), $MachinePrecision], N[(t$95$0 * N[(1.0 - N[(N[(N[(0.25 * N[(N[(D * M), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(D / N[(d$95$m + d$95$m), $MachinePrecision]), $MachinePrecision] * M), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M D) < 2.00000000000000013e-158

   1. Initial program 36.2%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. frac-times N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 43.0

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

   3. Applied rewrites 43.0%

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

   4. Taylor expanded in d around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 69.6

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

   6. Applied rewrites 69.6%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. count-2-rev N/A

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

      7. lift-+.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 69.1

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

      15. lift-/.f64 N/A

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

      16. metadata-eval 69.1

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 69.1

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 69.1

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

   8. Applied rewrites 69.1%

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

   9. Taylor expanded in d around inf

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

   11. Applied rewrites 42.7%

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

   if 2.00000000000000013e-158 < (*.f64 M D)

   1. Initial program 36.2%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. frac-times N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 43.0

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

   3. Applied rewrites 43.0%

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

   4. Taylor expanded in d around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 69.6

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

   6. Applied rewrites 69.6%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. count-2-rev N/A

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

      7. lift-+.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 69.1

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

      15. lift-/.f64 N/A

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

      16. metadata-eval 69.1

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 69.1

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 69.1

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

   8. Applied rewrites 69.1%

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

   9. Taylor expanded in d around 0

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

       1. lower-*.f64 N/A

          \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - \left(\left(\frac{1}{4} \cdot \color{blue}{\frac{D \cdot M}{d}}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot \frac{h}{\ell}\right) \]

       2. lower-/.f64 N/A

          \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - \left(\left(\frac{1}{4} \cdot \frac{D \cdot M}{\color{blue}{d}}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot \frac{h}{\ell}\right) \]

       3. lower-*.f64 68.8

          \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - \left(\left(0.25 \cdot \frac{D \cdot M}{d}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot \frac{h}{\ell}\right) \]

   11. Applied rewrites 68.8%

       \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - \left(\color{blue}{\left(0.25 \cdot \frac{D \cdot M}{d}\right)} \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot \frac{h}{\ell}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 46.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} \mathbf{if}\;M \cdot D \leq 1.95 \cdot 10^{+177}:\\ \;\;\;\;\left(d\_m \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \sqrt{\frac{{d\_m}^{2}}{h \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
(FPCore (d_m h l M D)
 :precision binary64
 (if (<= (* M D) 1.95e+177)
   (* (* d_m (sqrt (/ 1.0 (* h l)))) 1.0)
   (* -1.0 (sqrt (/ (pow d_m 2.0) (* h l))))))

C

d_m = fabs(d);
double code(double d_m, double h, double l, double M, double D) {
	double tmp;
	if ((M * D) <= 1.95e+177) {
		tmp = (d_m * sqrt((1.0 / (h * l)))) * 1.0;
	} else {
		tmp = -1.0 * sqrt((pow(d_m, 2.0) / (h * l)));
	}
	return tmp;
}

Fortran

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

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

real(8) function code(d_m, h, l, m, d)
use fmin_fmax_functions
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((m * d) <= 1.95d+177) then
        tmp = (d_m * sqrt((1.0d0 / (h * l)))) * 1.0d0
    else
        tmp = (-1.0d0) * sqrt(((d_m ** 2.0d0) / (h * l)))
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
public static double code(double d_m, double h, double l, double M, double D) {
	double tmp;
	if ((M * D) <= 1.95e+177) {
		tmp = (d_m * Math.sqrt((1.0 / (h * l)))) * 1.0;
	} else {
		tmp = -1.0 * Math.sqrt((Math.pow(d_m, 2.0) / (h * l)));
	}
	return tmp;
}

Python

d_m = math.fabs(d)
def code(d_m, h, l, M, D):
	tmp = 0
	if (M * D) <= 1.95e+177:
		tmp = (d_m * math.sqrt((1.0 / (h * l)))) * 1.0
	else:
		tmp = -1.0 * math.sqrt((math.pow(d_m, 2.0) / (h * l)))
	return tmp

Julia

d_m = abs(d)
function code(d_m, h, l, M, D)
	tmp = 0.0
	if (Float64(M * D) <= 1.95e+177)
		tmp = Float64(Float64(d_m * sqrt(Float64(1.0 / Float64(h * l)))) * 1.0);
	else
		tmp = Float64(-1.0 * sqrt(Float64((d_m ^ 2.0) / Float64(h * l))));
	end
	return tmp
end

MATLAB

d_m = abs(d);
function tmp_2 = code(d_m, h, l, M, D)
	tmp = 0.0;
	if ((M * D) <= 1.95e+177)
		tmp = (d_m * sqrt((1.0 / (h * l)))) * 1.0;
	else
		tmp = -1.0 * sqrt(((d_m ^ 2.0) / (h * l)));
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
code[d$95$m_, h_, l_, M_, D_] := If[LessEqual[N[(M * D), $MachinePrecision], 1.95e+177], N[(N[(d$95$m * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(-1.0 * N[Sqrt[N[(N[Power[d$95$m, 2.0], $MachinePrecision] / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M D) < 1.95e177

   1. Initial program 36.2%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. frac-times N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 43.0

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

   3. Applied rewrites 43.0%

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

   4. Taylor expanded in d around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 69.6

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

   6. Applied rewrites 69.6%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. count-2-rev N/A

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

      7. lift-+.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 69.1

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

      15. lift-/.f64 N/A

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

      16. metadata-eval 69.1

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 69.1

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 69.1

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

   8. Applied rewrites 69.1%

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

   9. Taylor expanded in d around inf

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

   11. Applied rewrites 42.7%

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

   if 1.95e177 < (*.f64 M D)

   1. Initial program 36.2%

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

   2. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 18.9

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

   4. Applied rewrites 18.9%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lower-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. sqrt-unprod N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 21.7

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

   6. Applied rewrites 21.7%

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

   7. Taylor expanded in h around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-*.f64 12.2

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

   9. Applied rewrites 12.2%

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

Alternative 6: 46.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d\_m}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d\_m}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d\_m}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -5 \cdot 10^{+77}:\\ \;\;\;\;-1 \cdot \frac{d\_m \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{else}:\\ \;\;\;\;\left(d\_m \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
(FPCore (d_m h l M D)
 :precision binary64
 (if (<=
      (*
       (* (pow (/ d_m h) (/ 1.0 2.0)) (pow (/ d_m l) (/ 1.0 2.0)))
       (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d_m)) 2.0)) (/ h l))))
      -5e+77)
   (* -1.0 (/ (* d_m (sqrt (/ h l))) h))
   (* (* d_m (sqrt (/ 1.0 (* h l)))) 1.0)))

C

d_m = fabs(d);
double code(double d_m, double h, double l, double M, double D) {
	double tmp;
	if (((pow((d_m / h), (1.0 / 2.0)) * pow((d_m / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d_m)), 2.0)) * (h / l)))) <= -5e+77) {
		tmp = -1.0 * ((d_m * sqrt((h / l))) / h);
	} else {
		tmp = (d_m * sqrt((1.0 / (h * l)))) * 1.0;
	}
	return tmp;
}

Fortran

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

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

real(8) function code(d_m, h, l, m, d)
use fmin_fmax_functions
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8) :: tmp
    if (((((d_m / h) ** (1.0d0 / 2.0d0)) * ((d_m / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d) / (2.0d0 * d_m)) ** 2.0d0)) * (h / l)))) <= (-5d+77)) then
        tmp = (-1.0d0) * ((d_m * sqrt((h / l))) / h)
    else
        tmp = (d_m * sqrt((1.0d0 / (h * l)))) * 1.0d0
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
public static double code(double d_m, double h, double l, double M, double D) {
	double tmp;
	if (((Math.pow((d_m / h), (1.0 / 2.0)) * Math.pow((d_m / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d_m)), 2.0)) * (h / l)))) <= -5e+77) {
		tmp = -1.0 * ((d_m * Math.sqrt((h / l))) / h);
	} else {
		tmp = (d_m * Math.sqrt((1.0 / (h * l)))) * 1.0;
	}
	return tmp;
}

Python

d_m = math.fabs(d)
def code(d_m, h, l, M, D):
	tmp = 0
	if ((math.pow((d_m / h), (1.0 / 2.0)) * math.pow((d_m / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d_m)), 2.0)) * (h / l)))) <= -5e+77:
		tmp = -1.0 * ((d_m * math.sqrt((h / l))) / h)
	else:
		tmp = (d_m * math.sqrt((1.0 / (h * l)))) * 1.0
	return tmp

Julia

d_m = abs(d)
function code(d_m, h, l, M, D)
	tmp = 0.0
	if (Float64(Float64((Float64(d_m / h) ^ Float64(1.0 / 2.0)) * (Float64(d_m / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d_m)) ^ 2.0)) * Float64(h / l)))) <= -5e+77)
		tmp = Float64(-1.0 * Float64(Float64(d_m * sqrt(Float64(h / l))) / h));
	else
		tmp = Float64(Float64(d_m * sqrt(Float64(1.0 / Float64(h * l)))) * 1.0);
	end
	return tmp
end

MATLAB

d_m = abs(d);
function tmp_2 = code(d_m, h, l, M, D)
	tmp = 0.0;
	if (((((d_m / h) ^ (1.0 / 2.0)) * ((d_m / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d_m)) ^ 2.0)) * (h / l)))) <= -5e+77)
		tmp = -1.0 * ((d_m * sqrt((h / l))) / h);
	else
		tmp = (d_m * sqrt((1.0 / (h * l)))) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
code[d$95$m_, h_, l_, M_, D_] := If[LessEqual[N[(N[(N[Power[N[(d$95$m / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d$95$m / 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$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e+77], N[(-1.0 * N[(N[(d$95$m * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision], N[(N[(d$95$m * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $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)))) < -5.00000000000000004e77

   1. Initial program 36.2%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. frac-times N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 43.0

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

   3. Applied rewrites 43.0%

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

   4. Taylor expanded in d around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 10.1

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

   6. Applied rewrites 10.1%

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

   7. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 25.6

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

   9. Applied rewrites 25.6%

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

   if -5.00000000000000004e77 < (*.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 36.2%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. frac-times N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 43.0

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

   3. Applied rewrites 43.0%

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

   4. Taylor expanded in d around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 69.6

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

   6. Applied rewrites 69.6%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. count-2-rev N/A

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

      7. lift-+.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 69.1

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

      15. lift-/.f64 N/A

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

      16. metadata-eval 69.1

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 69.1

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 69.1

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

   8. Applied rewrites 69.1%

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

   9. Taylor expanded in d around inf

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

   11. Applied rewrites 42.7%

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

Alternative 7: 44.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d\_m}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d\_m}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d\_m}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -5 \cdot 10^{+77}:\\ \;\;\;\;-\sqrt{\frac{1}{\ell \cdot h}} \cdot d\_m\\ \mathbf{else}:\\ \;\;\;\;\left(d\_m \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
(FPCore (d_m h l M D)
 :precision binary64
 (if (<=
      (*
       (* (pow (/ d_m h) (/ 1.0 2.0)) (pow (/ d_m l) (/ 1.0 2.0)))
       (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d_m)) 2.0)) (/ h l))))
      -5e+77)
   (- (* (sqrt (/ 1.0 (* l h))) d_m))
   (* (* d_m (sqrt (/ 1.0 (* h l)))) 1.0)))

C

d_m = fabs(d);
double code(double d_m, double h, double l, double M, double D) {
	double tmp;
	if (((pow((d_m / h), (1.0 / 2.0)) * pow((d_m / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d_m)), 2.0)) * (h / l)))) <= -5e+77) {
		tmp = -(sqrt((1.0 / (l * h))) * d_m);
	} else {
		tmp = (d_m * sqrt((1.0 / (h * l)))) * 1.0;
	}
	return tmp;
}

Fortran

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

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

real(8) function code(d_m, h, l, m, d)
use fmin_fmax_functions
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8) :: tmp
    if (((((d_m / h) ** (1.0d0 / 2.0d0)) * ((d_m / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d) / (2.0d0 * d_m)) ** 2.0d0)) * (h / l)))) <= (-5d+77)) then
        tmp = -(sqrt((1.0d0 / (l * h))) * d_m)
    else
        tmp = (d_m * sqrt((1.0d0 / (h * l)))) * 1.0d0
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
public static double code(double d_m, double h, double l, double M, double D) {
	double tmp;
	if (((Math.pow((d_m / h), (1.0 / 2.0)) * Math.pow((d_m / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d_m)), 2.0)) * (h / l)))) <= -5e+77) {
		tmp = -(Math.sqrt((1.0 / (l * h))) * d_m);
	} else {
		tmp = (d_m * Math.sqrt((1.0 / (h * l)))) * 1.0;
	}
	return tmp;
}

Python

d_m = math.fabs(d)
def code(d_m, h, l, M, D):
	tmp = 0
	if ((math.pow((d_m / h), (1.0 / 2.0)) * math.pow((d_m / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d_m)), 2.0)) * (h / l)))) <= -5e+77:
		tmp = -(math.sqrt((1.0 / (l * h))) * d_m)
	else:
		tmp = (d_m * math.sqrt((1.0 / (h * l)))) * 1.0
	return tmp

Julia

d_m = abs(d)
function code(d_m, h, l, M, D)
	tmp = 0.0
	if (Float64(Float64((Float64(d_m / h) ^ Float64(1.0 / 2.0)) * (Float64(d_m / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d_m)) ^ 2.0)) * Float64(h / l)))) <= -5e+77)
		tmp = Float64(-Float64(sqrt(Float64(1.0 / Float64(l * h))) * d_m));
	else
		tmp = Float64(Float64(d_m * sqrt(Float64(1.0 / Float64(h * l)))) * 1.0);
	end
	return tmp
end

MATLAB

d_m = abs(d);
function tmp_2 = code(d_m, h, l, M, D)
	tmp = 0.0;
	if (((((d_m / h) ^ (1.0 / 2.0)) * ((d_m / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d_m)) ^ 2.0)) * (h / l)))) <= -5e+77)
		tmp = -(sqrt((1.0 / (l * h))) * d_m);
	else
		tmp = (d_m * sqrt((1.0 / (h * l)))) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
code[d$95$m_, h_, l_, M_, D_] := If[LessEqual[N[(N[(N[Power[N[(d$95$m / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d$95$m / 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$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e+77], (-N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * d$95$m), $MachinePrecision]), N[(N[(d$95$m * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $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)))) < -5.00000000000000004e77

   1. Initial program 36.2%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. frac-times N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 43.0

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

   3. Applied rewrites 43.0%

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

   4. Taylor expanded in d around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 10.1

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

   6. Applied rewrites 10.1%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 10.1

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 10.1

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 10.1

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

   8. Applied rewrites 10.1%

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

   if -5.00000000000000004e77 < (*.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 36.2%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. frac-times N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 43.0

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

   3. Applied rewrites 43.0%

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

   4. Taylor expanded in d around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 69.6

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

   6. Applied rewrites 69.6%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. count-2-rev N/A

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

      7. lift-+.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 69.1

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

      15. lift-/.f64 N/A

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

      16. metadata-eval 69.1

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 69.1

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 69.1

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

   8. Applied rewrites 69.1%

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

   9. Taylor expanded in d around inf

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

   11. Applied rewrites 42.7%

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

Alternative 8: 27.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d\_m}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d\_m}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d\_m}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -5 \cdot 10^{+77}:\\ \;\;\;\;-\sqrt{\frac{1}{\ell \cdot h}} \cdot d\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{d\_m \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
(FPCore (d_m h l M D)
 :precision binary64
 (if (<=
      (*
       (* (pow (/ d_m h) (/ 1.0 2.0)) (pow (/ d_m l) (/ 1.0 2.0)))
       (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d_m)) 2.0)) (/ h l))))
      -5e+77)
   (- (* (sqrt (/ 1.0 (* l h))) d_m))
   (/ (* d_m (sqrt (/ h l))) h)))

C

d_m = fabs(d);
double code(double d_m, double h, double l, double M, double D) {
	double tmp;
	if (((pow((d_m / h), (1.0 / 2.0)) * pow((d_m / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d_m)), 2.0)) * (h / l)))) <= -5e+77) {
		tmp = -(sqrt((1.0 / (l * h))) * d_m);
	} else {
		tmp = (d_m * sqrt((h / l))) / h;
	}
	return tmp;
}

Fortran

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

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

real(8) function code(d_m, h, l, m, d)
use fmin_fmax_functions
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8) :: tmp
    if (((((d_m / h) ** (1.0d0 / 2.0d0)) * ((d_m / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d) / (2.0d0 * d_m)) ** 2.0d0)) * (h / l)))) <= (-5d+77)) then
        tmp = -(sqrt((1.0d0 / (l * h))) * d_m)
    else
        tmp = (d_m * sqrt((h / l))) / h
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
public static double code(double d_m, double h, double l, double M, double D) {
	double tmp;
	if (((Math.pow((d_m / h), (1.0 / 2.0)) * Math.pow((d_m / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d_m)), 2.0)) * (h / l)))) <= -5e+77) {
		tmp = -(Math.sqrt((1.0 / (l * h))) * d_m);
	} else {
		tmp = (d_m * Math.sqrt((h / l))) / h;
	}
	return tmp;
}

Python

d_m = math.fabs(d)
def code(d_m, h, l, M, D):
	tmp = 0
	if ((math.pow((d_m / h), (1.0 / 2.0)) * math.pow((d_m / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d_m)), 2.0)) * (h / l)))) <= -5e+77:
		tmp = -(math.sqrt((1.0 / (l * h))) * d_m)
	else:
		tmp = (d_m * math.sqrt((h / l))) / h
	return tmp

Julia

d_m = abs(d)
function code(d_m, h, l, M, D)
	tmp = 0.0
	if (Float64(Float64((Float64(d_m / h) ^ Float64(1.0 / 2.0)) * (Float64(d_m / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d_m)) ^ 2.0)) * Float64(h / l)))) <= -5e+77)
		tmp = Float64(-Float64(sqrt(Float64(1.0 / Float64(l * h))) * d_m));
	else
		tmp = Float64(Float64(d_m * sqrt(Float64(h / l))) / h);
	end
	return tmp
end

MATLAB

d_m = abs(d);
function tmp_2 = code(d_m, h, l, M, D)
	tmp = 0.0;
	if (((((d_m / h) ^ (1.0 / 2.0)) * ((d_m / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d_m)) ^ 2.0)) * (h / l)))) <= -5e+77)
		tmp = -(sqrt((1.0 / (l * h))) * d_m);
	else
		tmp = (d_m * sqrt((h / l))) / h;
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
code[d$95$m_, h_, l_, M_, D_] := If[LessEqual[N[(N[(N[Power[N[(d$95$m / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d$95$m / 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$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e+77], (-N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * d$95$m), $MachinePrecision]), N[(N[(d$95$m * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $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)))) < -5.00000000000000004e77

   1. Initial program 36.2%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. frac-times N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 43.0

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

   3. Applied rewrites 43.0%

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

   4. Taylor expanded in d around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 10.1

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

   6. Applied rewrites 10.1%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 10.1

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 10.1

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 10.1

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

   8. Applied rewrites 10.1%

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

   if -5.00000000000000004e77 < (*.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 36.2%

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

   2. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 18.9

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

   4. Applied rewrites 18.9%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lower-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. sqrt-unprod N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 21.7

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

   6. Applied rewrites 21.7%

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

   7. Taylor expanded in d around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 24.7

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

   9. Applied rewrites 24.7%

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

Alternative 9: 24.7% accurate, 7.4× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \frac{d\_m \cdot \sqrt{\frac{h}{\ell}}}{h} \end{array} \]

FPCore

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

C

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

Fortran

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

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

real(8) function code(d_m, h, l, m, d)
use fmin_fmax_functions
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    code = (d_m * sqrt((h / l))) / h
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

d_m = N[Abs[d], $MachinePrecision]
code[d$95$m_, h_, l_, M_, D_] := N[(N[(d$95$m * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]

Derivation

1. Initial program 36.2%

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

2. Taylor expanded in h around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-/.f64 18.9

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

4. Applied rewrites 18.9%

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

   1. lift-/.f64 N/A

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

   2. mult-flip N/A

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

   3. lower-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. sqrt-unprod N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-/.f64 21.7

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

6. Applied rewrites 21.7%

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

7. Taylor expanded in d around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-/.f64 24.7

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

9. Applied rewrites 24.7%

   \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{\color{blue}{h}} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025142 
(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)))))