Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.4% → 89.1%

Time: 6.9s

Alternatives: 16

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * sqrt((1.0 - (pow(((M * D) / (2.0 * d)), 2.0) * (h / l))));
}

Fortran

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(w0, m, d, h, l, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    code = w0 * sqrt((1.0d0 - ((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l))))
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * Math.sqrt((1.0 - (Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l))));
}

Python

def code(w0, M, D, h, l, d):
	return w0 * math.sqrt((1.0 - (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l))))

Julia

function code(w0, M, D, h, l, d)
	return Float64(w0 * sqrt(Float64(1.0 - Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)))))
end

MATLAB

function tmp = code(w0, M, D, h, l, d)
	tmp = w0 * sqrt((1.0 - ((((M * D) / (2.0 * d)) ^ 2.0) * (h / l))));
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := N[(w0 * N[Sqrt[N[(1.0 - N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 81.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * sqrt((1.0 - (pow(((M * D) / (2.0 * d)), 2.0) * (h / l))));
}

Fortran

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(w0, m, d, h, l, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    code = w0 * sqrt((1.0d0 - ((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l))))
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * Math.sqrt((1.0 - (Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l))));
}

Python

def code(w0, M, D, h, l, d):
	return w0 * math.sqrt((1.0 - (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l))))

Julia

function code(w0, M, D, h, l, d)
	return Float64(w0 * sqrt(Float64(1.0 - Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)))))
end

MATLAB

function tmp = code(w0, M, D, h, l, d)
	tmp = w0 * sqrt((1.0 - ((((M * D) / (2.0 * d)) ^ 2.0) * (h / l))));
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := N[(w0 * N[Sqrt[N[(1.0 - N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 89.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\\ t_1 := \frac{D}{d + d}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{+110}:\\ \;\;\;\;w0 \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(t\_1 \cdot M\right) \cdot \frac{\left(t\_1 \cdot h\right) \cdot M}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (sqrt (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)))))
        (t_1 (/ D (+ d d))))
   (if (<= t_0 5e+110)
     (* w0 t_0)
     (* w0 (sqrt (- 1.0 (* (* t_1 M) (/ (* (* t_1 h) M) l))))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = sqrt((1.0 - (pow(((M * D) / (2.0 * d)), 2.0) * (h / l))));
	double t_1 = D / (d + d);
	double tmp;
	if (t_0 <= 5e+110) {
		tmp = w0 * t_0;
	} else {
		tmp = w0 * sqrt((1.0 - ((t_1 * M) * (((t_1 * h) * M) / l))));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(w0, m, d, h, l, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt((1.0d0 - ((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l))))
    t_1 = d / (d_1 + d_1)
    if (t_0 <= 5d+110) then
        tmp = w0 * t_0
    else
        tmp = w0 * sqrt((1.0d0 - ((t_1 * m) * (((t_1 * h) * m) / l))))
    end if
    code = tmp
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = Math.sqrt((1.0 - (Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l))));
	double t_1 = D / (d + d);
	double tmp;
	if (t_0 <= 5e+110) {
		tmp = w0 * t_0;
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((t_1 * M) * (((t_1 * h) * M) / l))));
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	t_0 = math.sqrt((1.0 - (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l))))
	t_1 = D / (d + d)
	tmp = 0
	if t_0 <= 5e+110:
		tmp = w0 * t_0
	else:
		tmp = w0 * math.sqrt((1.0 - ((t_1 * M) * (((t_1 * h) * M) / l))))
	return tmp

Julia

function code(w0, M, D, h, l, d)
	t_0 = sqrt(Float64(1.0 - Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l))))
	t_1 = Float64(D / Float64(d + d))
	tmp = 0.0
	if (t_0 <= 5e+110)
		tmp = Float64(w0 * t_0);
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(t_1 * M) * Float64(Float64(Float64(t_1 * h) * M) / l)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	t_0 = sqrt((1.0 - ((((M * D) / (2.0 * d)) ^ 2.0) * (h / l))));
	t_1 = D / (d + d);
	tmp = 0.0;
	if (t_0 <= 5e+110)
		tmp = w0 * t_0;
	else
		tmp = w0 * sqrt((1.0 - ((t_1 * M) * (((t_1 * h) * M) / l))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := Block[{t$95$0 = N[Sqrt[N[(1.0 - N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(D / N[(d + d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e+110], N[(w0 * t$95$0), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(t$95$1 * M), $MachinePrecision] * N[(N[(N[(t$95$1 * h), $MachinePrecision] * M), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)))) < 4.99999999999999978e110

   1. Initial program 99.8%

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

   if 4.99999999999999978e110 < (sqrt.f64 (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))))

   1. Initial program 44.8%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

   3. Applied rewrites 62.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lower-*.f64 67.5

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

      13. lift-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. lift-+.f64 67.5

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

   5. Applied rewrites 67.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 68.9

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 66.2

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

   7. Applied rewrites 66.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 67.9

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

   9. Applied rewrites 67.9%

      \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{\color{blue}{\left(\frac{D}{d + d} \cdot h\right) \cdot M}}{\ell}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 88.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{D}{d + d}\\ t_1 := M \cdot t\_0\\ t_2 := \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{if}\;t\_2 \leq 1:\\ \;\;\;\;w0\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+110}:\\ \;\;\;\;\sqrt{1 - \left(t\_1 \cdot t\_1\right) \cdot \frac{h}{\ell}} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(t\_0 \cdot M\right) \cdot \frac{\left(t\_0 \cdot h\right) \cdot M}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (/ D (+ d d)))
        (t_1 (* M t_0))
        (t_2 (sqrt (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l))))))
   (if (<= t_2 1.0)
     w0
     (if (<= t_2 5e+110)
       (* (sqrt (- 1.0 (* (* t_1 t_1) (/ h l)))) w0)
       (* w0 (sqrt (- 1.0 (* (* t_0 M) (/ (* (* t_0 h) M) l)))))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = D / (d + d);
	double t_1 = M * t_0;
	double t_2 = sqrt((1.0 - (pow(((M * D) / (2.0 * d)), 2.0) * (h / l))));
	double tmp;
	if (t_2 <= 1.0) {
		tmp = w0;
	} else if (t_2 <= 5e+110) {
		tmp = sqrt((1.0 - ((t_1 * t_1) * (h / l)))) * w0;
	} else {
		tmp = w0 * sqrt((1.0 - ((t_0 * M) * (((t_0 * h) * M) / l))));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(w0, m, d, h, l, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = d / (d_1 + d_1)
    t_1 = m * t_0
    t_2 = sqrt((1.0d0 - ((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l))))
    if (t_2 <= 1.0d0) then
        tmp = w0
    else if (t_2 <= 5d+110) then
        tmp = sqrt((1.0d0 - ((t_1 * t_1) * (h / l)))) * w0
    else
        tmp = w0 * sqrt((1.0d0 - ((t_0 * m) * (((t_0 * h) * m) / l))))
    end if
    code = tmp
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = D / (d + d);
	double t_1 = M * t_0;
	double t_2 = Math.sqrt((1.0 - (Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l))));
	double tmp;
	if (t_2 <= 1.0) {
		tmp = w0;
	} else if (t_2 <= 5e+110) {
		tmp = Math.sqrt((1.0 - ((t_1 * t_1) * (h / l)))) * w0;
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((t_0 * M) * (((t_0 * h) * M) / l))));
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	t_0 = D / (d + d)
	t_1 = M * t_0
	t_2 = math.sqrt((1.0 - (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l))))
	tmp = 0
	if t_2 <= 1.0:
		tmp = w0
	elif t_2 <= 5e+110:
		tmp = math.sqrt((1.0 - ((t_1 * t_1) * (h / l)))) * w0
	else:
		tmp = w0 * math.sqrt((1.0 - ((t_0 * M) * (((t_0 * h) * M) / l))))
	return tmp

Julia

function code(w0, M, D, h, l, d)
	t_0 = Float64(D / Float64(d + d))
	t_1 = Float64(M * t_0)
	t_2 = sqrt(Float64(1.0 - Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l))))
	tmp = 0.0
	if (t_2 <= 1.0)
		tmp = w0;
	elseif (t_2 <= 5e+110)
		tmp = Float64(sqrt(Float64(1.0 - Float64(Float64(t_1 * t_1) * Float64(h / l)))) * w0);
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(t_0 * M) * Float64(Float64(Float64(t_0 * h) * M) / l)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	t_0 = D / (d + d);
	t_1 = M * t_0;
	t_2 = sqrt((1.0 - ((((M * D) / (2.0 * d)) ^ 2.0) * (h / l))));
	tmp = 0.0;
	if (t_2 <= 1.0)
		tmp = w0;
	elseif (t_2 <= 5e+110)
		tmp = sqrt((1.0 - ((t_1 * t_1) * (h / l)))) * w0;
	else
		tmp = w0 * sqrt((1.0 - ((t_0 * M) * (((t_0 * h) * M) / l))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := Block[{t$95$0 = N[(D / N[(d + d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(M * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 - N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 1.0], w0, If[LessEqual[t$95$2, 5e+110], N[(N[Sqrt[N[(1.0 - N[(N[(t$95$1 * t$95$1), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(t$95$0 * M), $MachinePrecision] * N[(N[(N[(t$95$0 * h), $MachinePrecision] * M), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.9%

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

   2. Taylor expanded in M around 0

      \[\leadsto \color{blue}{w0} \]
   3. Step-by-step derivation

   4. Applied rewrites 99.6%

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

   if 1 < (sqrt.f64 (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)))) < 4.99999999999999978e110

   1. Initial program 99.0%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

   3. Applied rewrites 93.2%

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

   if 4.99999999999999978e110 < (sqrt.f64 (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))))

   1. Initial program 44.8%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

   3. Applied rewrites 62.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lower-*.f64 67.5

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

      13. lift-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. lift-+.f64 67.5

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

   5. Applied rewrites 67.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 68.9

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 66.2

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

   7. Applied rewrites 66.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 67.9

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

   9. Applied rewrites 67.9%

      \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{\color{blue}{\left(\frac{D}{d + d} \cdot h\right) \cdot M}}{\ell}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 86.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{D}{d + d}\\ w0 \cdot \sqrt{1 - \left(t\_0 \cdot M\right) \cdot \frac{t\_0 \cdot \left(h \cdot M\right)}{\ell}} \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (/ D (+ d d))))
   (* w0 (sqrt (- 1.0 (* (* t_0 M) (/ (* t_0 (* h M)) l)))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = D / (d + d);
	return w0 * sqrt((1.0 - ((t_0 * M) * ((t_0 * (h * M)) / 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(w0, m, d, h, l, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: t_0
    t_0 = d / (d_1 + d_1)
    code = w0 * sqrt((1.0d0 - ((t_0 * m) * ((t_0 * (h * m)) / l))))
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = D / (d + d);
	return w0 * Math.sqrt((1.0 - ((t_0 * M) * ((t_0 * (h * M)) / l))));
}

Python

def code(w0, M, D, h, l, d):
	t_0 = D / (d + d)
	return w0 * math.sqrt((1.0 - ((t_0 * M) * ((t_0 * (h * M)) / l))))

Julia

function code(w0, M, D, h, l, d)
	t_0 = Float64(D / Float64(d + d))
	return Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(t_0 * M) * Float64(Float64(t_0 * Float64(h * M)) / l)))))
end

MATLAB

function tmp = code(w0, M, D, h, l, d)
	t_0 = D / (d + d);
	tmp = w0 * sqrt((1.0 - ((t_0 * M) * ((t_0 * (h * M)) / l))));
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := Block[{t$95$0 = N[(D / N[(d + d), $MachinePrecision]), $MachinePrecision]}, N[(w0 * N[Sqrt[N[(1.0 - N[(N[(t$95$0 * M), $MachinePrecision] * N[(N[(t$95$0 * N[(h * M), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 81.4%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. associate-*r/ N/A

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

   8. lower-/.f64 N/A

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

3. Applied rewrites 86.1%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. lower-*.f64 87.8

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

   13. lift-*.f64 N/A

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

   14. lift-+.f64 N/A

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

   15. lift-/.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 N/A

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

   18. lift-/.f64 N/A

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

   19. lift-+.f64 87.8

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

5. Applied rewrites 87.8%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 88.6

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. associate-*l* N/A

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

   11. lower-*.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. lift-+.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 85.0

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

7. Applied rewrites 85.0%

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

Alternative 4: 85.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := M \cdot \frac{D}{d + d}\\ \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{-13}:\\ \;\;\;\;\sqrt{1 - \left(t\_0 \cdot t\_0\right) \cdot \frac{h}{\ell}} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (* M (/ D (+ d d)))))
   (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -2e-13)
     (* (sqrt (- 1.0 (* (* t_0 t_0) (/ h l)))) w0)
     w0)))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = M * (D / (d + d));
	double tmp;
	if ((pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -2e-13) {
		tmp = sqrt((1.0 - ((t_0 * t_0) * (h / l)))) * w0;
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(w0, m, d, h, l, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = m * (d / (d_1 + d_1))
    if (((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l)) <= (-2d-13)) then
        tmp = sqrt((1.0d0 - ((t_0 * t_0) * (h / l)))) * w0
    else
        tmp = w0
    end if
    code = tmp
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = M * (D / (d + d));
	double tmp;
	if ((Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -2e-13) {
		tmp = Math.sqrt((1.0 - ((t_0 * t_0) * (h / l)))) * w0;
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	t_0 = M * (D / (d + d))
	tmp = 0
	if (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -2e-13:
		tmp = math.sqrt((1.0 - ((t_0 * t_0) * (h / l)))) * w0
	else:
		tmp = w0
	return tmp

Julia

function code(w0, M, D, h, l, d)
	t_0 = Float64(M * Float64(D / Float64(d + d)))
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -2e-13)
		tmp = Float64(sqrt(Float64(1.0 - Float64(Float64(t_0 * t_0) * Float64(h / l)))) * w0);
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	t_0 = M * (D / (d + d));
	tmp = 0.0;
	if (((((M * D) / (2.0 * d)) ^ 2.0) * (h / l)) <= -2e-13)
		tmp = sqrt((1.0 - ((t_0 * t_0) * (h / l)))) * w0;
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := Block[{t$95$0 = N[(M * N[(D / N[(d + d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -2e-13], N[(N[Sqrt[N[(1.0 - N[(N[(t$95$0 * t$95$0), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision], w0]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -2.0000000000000001e-13

   1. Initial program 65.3%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

   3. Applied rewrites 65.4%

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

   if -2.0000000000000001e-13 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 88.8%

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

   2. Taylor expanded in M around 0

      \[\leadsto \color{blue}{w0} \]
   3. Step-by-step derivation

   4. Applied rewrites 96.2%

      \[\leadsto \color{blue}{w0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 85.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{-12}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{0.5 \cdot \left(\left(M \cdot D\right) \cdot h\right)}{\ell \cdot d}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -5e-12)
   (*
    w0
    (sqrt (- 1.0 (* (* (/ D (+ d d)) M) (/ (* 0.5 (* (* M D) h)) (* l d))))))
   w0))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -5e-12) {
		tmp = w0 * sqrt((1.0 - (((D / (d + d)) * M) * ((0.5 * ((M * D) * h)) / (l * d)))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(w0, m, d, h, l, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l)) <= (-5d-12)) then
        tmp = w0 * sqrt((1.0d0 - (((d / (d_1 + d_1)) * m) * ((0.5d0 * ((m * d) * h)) / (l * d_1)))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -5e-12) {
		tmp = w0 * Math.sqrt((1.0 - (((D / (d + d)) * M) * ((0.5 * ((M * D) * h)) / (l * d)))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -5e-12:
		tmp = w0 * math.sqrt((1.0 - (((D / (d + d)) * M) * ((0.5 * ((M * D) * h)) / (l * d)))))
	else:
		tmp = w0
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -5e-12)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(D / Float64(d + d)) * M) * Float64(Float64(0.5 * Float64(Float64(M * D) * h)) / Float64(l * d))))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (((((M * D) / (2.0 * d)) ^ 2.0) * (h / l)) <= -5e-12)
		tmp = w0 * sqrt((1.0 - (((D / (d + d)) * M) * ((0.5 * ((M * D) * h)) / (l * d)))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -5e-12], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(D / N[(d + d), $MachinePrecision]), $MachinePrecision] * M), $MachinePrecision] * N[(N[(0.5 * N[(N[(M * D), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -4.9999999999999997e-12

   1. Initial program 65.3%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

   3. Applied rewrites 65.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lower-*.f64 66.3

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

      13. lift-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. lift-+.f64 66.3

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

   5. Applied rewrites 66.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 68.7

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 64.7

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

   7. Applied rewrites 64.7%

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

   8. Taylor expanded in M around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 62.4

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{0.5 \cdot \left(\left(M \cdot D\right) \cdot h\right)}{\ell \cdot \color{blue}{d}}} \]

   10. Applied rewrites 62.4%

       \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \color{blue}{\frac{0.5 \cdot \left(\left(M \cdot D\right) \cdot h\right)}{\ell \cdot d}}} \]

   if -4.9999999999999997e-12 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 88.8%

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

   2. Taylor expanded in M around 0

      \[\leadsto \color{blue}{w0} \]
   3. Step-by-step derivation

   4. Applied rewrites 96.2%

      \[\leadsto \color{blue}{w0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 84.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{-13}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\frac{\frac{\left(M \cdot \left(M \cdot D\right)\right) \cdot D}{d}}{d} \cdot 0.25\right) \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -2e-13)
   (* w0 (sqrt (- 1.0 (* (* (/ (/ (* (* M (* M D)) D) d) d) 0.25) (/ h l)))))
   w0))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -2e-13) {
		tmp = w0 * sqrt((1.0 - ((((((M * (M * D)) * D) / d) / d) * 0.25) * (h / l))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(w0, m, d, h, l, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l)) <= (-2d-13)) then
        tmp = w0 * sqrt((1.0d0 - ((((((m * (m * d)) * d) / d_1) / d_1) * 0.25d0) * (h / l))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -2e-13) {
		tmp = w0 * Math.sqrt((1.0 - ((((((M * (M * D)) * D) / d) / d) * 0.25) * (h / l))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -2e-13:
		tmp = w0 * math.sqrt((1.0 - ((((((M * (M * D)) * D) / d) / d) * 0.25) * (h / l))))
	else:
		tmp = w0
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -2e-13)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(Float64(Float64(Float64(M * Float64(M * D)) * D) / d) / d) * 0.25) * Float64(h / l)))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (((((M * D) / (2.0 * d)) ^ 2.0) * (h / l)) <= -2e-13)
		tmp = w0 * sqrt((1.0 - ((((((M * (M * D)) * D) / d) / d) * 0.25) * (h / l))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -2e-13], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(N[(N[(N[(M * N[(M * D), $MachinePrecision]), $MachinePrecision] * D), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision] * 0.25), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -2.0000000000000001e-13

   1. Initial program 65.3%

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

   2. Taylor expanded in M around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\left(M \cdot M\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \frac{1}{4}\right) \cdot \frac{h}{\ell}} \]

      11. lower-*.f64 40.7

          \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\left(M \cdot M\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot 0.25\right) \cdot \frac{h}{\ell}} \]

   4. Applied rewrites 40.7%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(\frac{\left(M \cdot M\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot 0.25\right)} \cdot \frac{h}{\ell}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\left(M \cdot M\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \frac{1}{4}\right) \cdot \frac{h}{\ell}} \]

      2. lift-/.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\left(M \cdot M\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \frac{1}{4}\right) \cdot \frac{h}{\ell}} \]

      3. associate-/r* N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\frac{\left(M \cdot M\right) \cdot \left(D \cdot D\right)}{d}}{d} \cdot \frac{1}{4}\right) \cdot \frac{h}{\ell}} \]

      4. lower-/.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\frac{\left(M \cdot M\right) \cdot \left(D \cdot D\right)}{d}}{d} \cdot \frac{1}{4}\right) \cdot \frac{h}{\ell}} \]

      5. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\frac{\left(M \cdot M\right) \cdot \left(D \cdot D\right)}{d}}{d} \cdot \frac{1}{4}\right) \cdot \frac{h}{\ell}} \]

      6. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\frac{\left(M \cdot M\right) \cdot \left(D \cdot D\right)}{d}}{d} \cdot \frac{1}{4}\right) \cdot \frac{h}{\ell}} \]

      7. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\frac{\left(M \cdot M\right) \cdot \left(D \cdot D\right)}{d}}{d} \cdot \frac{1}{4}\right) \cdot \frac{h}{\ell}} \]

      8. pow2 N/A

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

      9. pow2 N/A

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

      10. *-commutative N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. pow2 N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. pow2 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\frac{\left(\left(M \cdot M\right) \cdot D\right) \cdot D}{d}}{d} \cdot \frac{1}{4}\right) \cdot \frac{h}{\ell}} \]

      18. lift-*.f64 50.9

          \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\frac{\left(\left(M \cdot M\right) \cdot D\right) \cdot D}{d}}{d} \cdot 0.25\right) \cdot \frac{h}{\ell}} \]

   6. Applied rewrites 50.9%

      \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\frac{\left(\left(M \cdot M\right) \cdot D\right) \cdot D}{d}}{d} \cdot 0.25\right) \cdot \frac{h}{\ell}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\frac{\left(\left(M \cdot M\right) \cdot D\right) \cdot D}{d}}{d} \cdot \frac{1}{4}\right) \cdot \frac{h}{\ell}} \]

      2. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\frac{\left(\left(M \cdot M\right) \cdot D\right) \cdot D}{d}}{d} \cdot \frac{1}{4}\right) \cdot \frac{h}{\ell}} \]

      3. associate-*l* N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\frac{\left(M \cdot \left(M \cdot D\right)\right) \cdot D}{d}}{d} \cdot \frac{1}{4}\right) \cdot \frac{h}{\ell}} \]

      4. *-commutative N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\frac{\left(M \cdot \left(D \cdot M\right)\right) \cdot D}{d}}{d} \cdot \frac{1}{4}\right) \cdot \frac{h}{\ell}} \]

      5. lower-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\frac{\left(M \cdot \left(D \cdot M\right)\right) \cdot D}{d}}{d} \cdot \frac{1}{4}\right) \cdot \frac{h}{\ell}} \]

      6. *-commutative N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\frac{\left(M \cdot \left(M \cdot D\right)\right) \cdot D}{d}}{d} \cdot \frac{1}{4}\right) \cdot \frac{h}{\ell}} \]

      7. lower-*.f64 56.7

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\frac{\left(M \cdot \left(M \cdot D\right)\right) \cdot D}{d}}{d} \cdot 0.25\right) \cdot \frac{h}{\ell}} \]

   8. Applied rewrites 56.7%

      \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\frac{\left(M \cdot \left(M \cdot D\right)\right) \cdot D}{d}}{d} \cdot 0.25\right) \cdot \frac{h}{\ell}} \]

   if -2.0000000000000001e-13 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 88.8%

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

   2. Taylor expanded in M around 0

      \[\leadsto \color{blue}{w0} \]
   3. Step-by-step derivation

   4. Applied rewrites 96.2%

      \[\leadsto \color{blue}{w0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 83.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{-13}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\frac{\left(\left(M \cdot D\right) \cdot M\right) \cdot \frac{D}{d}}{d} \cdot 0.25\right) \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -2e-13)
   (* w0 (sqrt (- 1.0 (* (* (/ (* (* (* M D) M) (/ D d)) d) 0.25) (/ h l)))))
   w0))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -2e-13) {
		tmp = w0 * sqrt((1.0 - ((((((M * D) * M) * (D / d)) / d) * 0.25) * (h / l))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(w0, m, d, h, l, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l)) <= (-2d-13)) then
        tmp = w0 * sqrt((1.0d0 - ((((((m * d) * m) * (d / d_1)) / d_1) * 0.25d0) * (h / l))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -2e-13) {
		tmp = w0 * Math.sqrt((1.0 - ((((((M * D) * M) * (D / d)) / d) * 0.25) * (h / l))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -2e-13:
		tmp = w0 * math.sqrt((1.0 - ((((((M * D) * M) * (D / d)) / d) * 0.25) * (h / l))))
	else:
		tmp = w0
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -2e-13)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(Float64(Float64(Float64(M * D) * M) * Float64(D / d)) / d) * 0.25) * Float64(h / l)))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (((((M * D) / (2.0 * d)) ^ 2.0) * (h / l)) <= -2e-13)
		tmp = w0 * sqrt((1.0 - ((((((M * D) * M) * (D / d)) / d) * 0.25) * (h / l))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -2e-13], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(N[(N[(N[(M * D), $MachinePrecision] * M), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * 0.25), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -2.0000000000000001e-13

   1. Initial program 65.3%

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

   2. Taylor expanded in M around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\left(M \cdot M\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \frac{1}{4}\right) \cdot \frac{h}{\ell}} \]

      11. lower-*.f64 40.7

          \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\left(M \cdot M\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot 0.25\right) \cdot \frac{h}{\ell}} \]

   4. Applied rewrites 40.7%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(\frac{\left(M \cdot M\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot 0.25\right)} \cdot \frac{h}{\ell}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\left(M \cdot M\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \frac{1}{4}\right) \cdot \frac{h}{\ell}} \]

      2. lift-/.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\left(M \cdot M\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \frac{1}{4}\right) \cdot \frac{h}{\ell}} \]

      3. associate-/r* N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\frac{\left(M \cdot M\right) \cdot \left(D \cdot D\right)}{d}}{d} \cdot \frac{1}{4}\right) \cdot \frac{h}{\ell}} \]

      4. lower-/.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\frac{\left(M \cdot M\right) \cdot \left(D \cdot D\right)}{d}}{d} \cdot \frac{1}{4}\right) \cdot \frac{h}{\ell}} \]

      5. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\frac{\left(M \cdot M\right) \cdot \left(D \cdot D\right)}{d}}{d} \cdot \frac{1}{4}\right) \cdot \frac{h}{\ell}} \]

      6. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\frac{\left(M \cdot M\right) \cdot \left(D \cdot D\right)}{d}}{d} \cdot \frac{1}{4}\right) \cdot \frac{h}{\ell}} \]

      7. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\frac{\left(M \cdot M\right) \cdot \left(D \cdot D\right)}{d}}{d} \cdot \frac{1}{4}\right) \cdot \frac{h}{\ell}} \]

      8. pow2 N/A

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

      9. pow2 N/A

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

      10. *-commutative N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. pow2 N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. pow2 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\frac{\left(\left(M \cdot M\right) \cdot D\right) \cdot D}{d}}{d} \cdot \frac{1}{4}\right) \cdot \frac{h}{\ell}} \]

      18. lift-*.f64 50.9

          \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\frac{\left(\left(M \cdot M\right) \cdot D\right) \cdot D}{d}}{d} \cdot 0.25\right) \cdot \frac{h}{\ell}} \]

   6. Applied rewrites 50.9%

      \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\frac{\left(\left(M \cdot M\right) \cdot D\right) \cdot D}{d}}{d} \cdot 0.25\right) \cdot \frac{h}{\ell}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\frac{\left(\left(M \cdot M\right) \cdot D\right) \cdot D}{d}}{d} \cdot \frac{1}{4}\right) \cdot \frac{h}{\ell}} \]

      2. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\frac{\left(\left(M \cdot M\right) \cdot D\right) \cdot D}{d}}{d} \cdot \frac{1}{4}\right) \cdot \frac{h}{\ell}} \]

      3. associate-*l* N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\frac{\left(M \cdot \left(M \cdot D\right)\right) \cdot D}{d}}{d} \cdot \frac{1}{4}\right) \cdot \frac{h}{\ell}} \]

      4. *-commutative N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\frac{\left(M \cdot \left(D \cdot M\right)\right) \cdot D}{d}}{d} \cdot \frac{1}{4}\right) \cdot \frac{h}{\ell}} \]

      5. lower-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\frac{\left(M \cdot \left(D \cdot M\right)\right) \cdot D}{d}}{d} \cdot \frac{1}{4}\right) \cdot \frac{h}{\ell}} \]

      6. *-commutative N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\frac{\left(M \cdot \left(M \cdot D\right)\right) \cdot D}{d}}{d} \cdot \frac{1}{4}\right) \cdot \frac{h}{\ell}} \]

      7. lower-*.f64 56.7

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\frac{\left(M \cdot \left(M \cdot D\right)\right) \cdot D}{d}}{d} \cdot 0.25\right) \cdot \frac{h}{\ell}} \]

   8. Applied rewrites 56.7%

      \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\frac{\left(M \cdot \left(M \cdot D\right)\right) \cdot D}{d}}{d} \cdot 0.25\right) \cdot \frac{h}{\ell}} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\frac{\left(M \cdot \left(M \cdot D\right)\right) \cdot D}{d}}{d} \cdot \frac{1}{4}\right) \cdot \frac{h}{\ell}} \]

      2. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\frac{\left(M \cdot \left(M \cdot D\right)\right) \cdot D}{d}}{d} \cdot \frac{1}{4}\right) \cdot \frac{h}{\ell}} \]

      3. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\frac{\left(M \cdot \left(M \cdot D\right)\right) \cdot D}{d}}{d} \cdot \frac{1}{4}\right) \cdot \frac{h}{\ell}} \]

      4. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\frac{\left(M \cdot \left(M \cdot D\right)\right) \cdot D}{d}}{d} \cdot \frac{1}{4}\right) \cdot \frac{h}{\ell}} \]

      5. associate-/l* N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\left(M \cdot \left(M \cdot D\right)\right) \cdot \frac{D}{d}}{d} \cdot \frac{1}{4}\right) \cdot \frac{h}{\ell}} \]

      6. lower-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\left(M \cdot \left(M \cdot D\right)\right) \cdot \frac{D}{d}}{d} \cdot \frac{1}{4}\right) \cdot \frac{h}{\ell}} \]

      7. *-commutative N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\left(\left(M \cdot D\right) \cdot M\right) \cdot \frac{D}{d}}{d} \cdot \frac{1}{4}\right) \cdot \frac{h}{\ell}} \]

      8. *-commutative N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\left(\left(D \cdot M\right) \cdot M\right) \cdot \frac{D}{d}}{d} \cdot \frac{1}{4}\right) \cdot \frac{h}{\ell}} \]

      9. lower-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\left(\left(D \cdot M\right) \cdot M\right) \cdot \frac{D}{d}}{d} \cdot \frac{1}{4}\right) \cdot \frac{h}{\ell}} \]

      10. *-commutative N/A

          \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\left(\left(M \cdot D\right) \cdot M\right) \cdot \frac{D}{d}}{d} \cdot \frac{1}{4}\right) \cdot \frac{h}{\ell}} \]

      11. lift-*.f64 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\left(\left(M \cdot D\right) \cdot M\right) \cdot \frac{D}{d}}{d} \cdot \frac{1}{4}\right) \cdot \frac{h}{\ell}} \]

      12. lower-/.f64 59.0

          \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\left(\left(M \cdot D\right) \cdot M\right) \cdot \frac{D}{d}}{d} \cdot 0.25\right) \cdot \frac{h}{\ell}} \]

   10. Applied rewrites 59.0%

       \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\left(\left(M \cdot D\right) \cdot M\right) \cdot \frac{D}{d}}{d} \cdot 0.25\right) \cdot \frac{h}{\ell}} \]

   if -2.0000000000000001e-13 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 88.8%

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

   2. Taylor expanded in M around 0

      \[\leadsto \color{blue}{w0} \]
   3. Step-by-step derivation

   4. Applied rewrites 96.2%

      \[\leadsto \color{blue}{w0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 82.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{-13}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{d \cdot d} \cdot 0.25\right) \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -2e-13)
   (* w0 (sqrt (- 1.0 (* (* (/ (* (* D M) (* D M)) (* d d)) 0.25) (/ h l)))))
   w0))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -2e-13) {
		tmp = w0 * sqrt((1.0 - (((((D * M) * (D * M)) / (d * d)) * 0.25) * (h / l))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(w0, m, d, h, l, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l)) <= (-2d-13)) then
        tmp = w0 * sqrt((1.0d0 - (((((d * m) * (d * m)) / (d_1 * d_1)) * 0.25d0) * (h / l))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -2e-13) {
		tmp = w0 * Math.sqrt((1.0 - (((((D * M) * (D * M)) / (d * d)) * 0.25) * (h / l))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -2e-13:
		tmp = w0 * math.sqrt((1.0 - (((((D * M) * (D * M)) / (d * d)) * 0.25) * (h / l))))
	else:
		tmp = w0
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -2e-13)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(Float64(Float64(D * M) * Float64(D * M)) / Float64(d * d)) * 0.25) * Float64(h / l)))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (((((M * D) / (2.0 * d)) ^ 2.0) * (h / l)) <= -2e-13)
		tmp = w0 * sqrt((1.0 - (((((D * M) * (D * M)) / (d * d)) * 0.25) * (h / l))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -2e-13], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(N[(N[(D * M), $MachinePrecision] * N[(D * M), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -2.0000000000000001e-13

   1. Initial program 65.3%

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

   2. Taylor expanded in M around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\left(M \cdot M\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \frac{1}{4}\right) \cdot \frac{h}{\ell}} \]

      11. lower-*.f64 40.7

          \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\left(M \cdot M\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot 0.25\right) \cdot \frac{h}{\ell}} \]

   4. Applied rewrites 40.7%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(\frac{\left(M \cdot M\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot 0.25\right)} \cdot \frac{h}{\ell}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\left(M \cdot M\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \frac{1}{4}\right) \cdot \frac{h}{\ell}} \]

      2. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\left(M \cdot M\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \frac{1}{4}\right) \cdot \frac{h}{\ell}} \]

      3. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\left(M \cdot M\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \frac{1}{4}\right) \cdot \frac{h}{\ell}} \]

      4. unswap-sqr N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{d \cdot d} \cdot \frac{1}{4}\right) \cdot \frac{h}{\ell}} \]

      5. *-commutative N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\left(D \cdot M\right) \cdot \left(M \cdot D\right)}{d \cdot d} \cdot \frac{1}{4}\right) \cdot \frac{h}{\ell}} \]

      6. *-commutative N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{d \cdot d} \cdot \frac{1}{4}\right) \cdot \frac{h}{\ell}} \]

      7. lower-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{d \cdot d} \cdot \frac{1}{4}\right) \cdot \frac{h}{\ell}} \]

      8. lower-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{d \cdot d} \cdot \frac{1}{4}\right) \cdot \frac{h}{\ell}} \]

      9. lower-*.f64 52.4

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{d \cdot d} \cdot 0.25\right) \cdot \frac{h}{\ell}} \]

   6. Applied rewrites 52.4%

      \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{d \cdot d} \cdot 0.25\right) \cdot \frac{h}{\ell}} \]

   if -2.0000000000000001e-13 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 88.8%

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

   2. Taylor expanded in M around 0

      \[\leadsto \color{blue}{w0} \]
   3. Step-by-step derivation

   4. Applied rewrites 96.2%

      \[\leadsto \color{blue}{w0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 82.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{-13}:\\ \;\;\;\;w0 \cdot \sqrt{\frac{\mathsf{fma}\left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \frac{h}{d \cdot d}, -0.25, \ell\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -2e-13)
   (* w0 (sqrt (/ (fma (* (* (* M D) (* M D)) (/ h (* d d))) -0.25 l) l)))
   w0))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -2e-13) {
		tmp = w0 * sqrt((fma((((M * D) * (M * D)) * (h / (d * d))), -0.25, l) / l));
	} else {
		tmp = w0;
	}
	return tmp;
}

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -2e-13)
		tmp = Float64(w0 * sqrt(Float64(fma(Float64(Float64(Float64(M * D) * Float64(M * D)) * Float64(h / Float64(d * d))), -0.25, l) / l)));
	else
		tmp = w0;
	end
	return tmp
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -2e-13], N[(w0 * N[Sqrt[N[(N[(N[(N[(N[(M * D), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] * N[(h / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.25 + l), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -2.0000000000000001e-13

   1. Initial program 65.3%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

   3. Applied rewrites 65.2%

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

   4. Taylor expanded in l around 0

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

      1. metadata-eval N/A

         \[\leadsto w0 \cdot \sqrt{\frac{\ell - \left(\mathsf{neg}\left(\frac{-1}{4}\right)\right) \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}}{\ell}} \]

      2. fp-cancel-sign-sub-inv N/A

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

      3. lower-/.f64 N/A

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

   6. Applied rewrites 44.0%

      \[\leadsto w0 \cdot \sqrt{\color{blue}{\frac{\mathsf{fma}\left(\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot D}{d \cdot d}, -0.25, \ell\right)}{\ell}}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{\frac{\mathsf{fma}\left(\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot D}{d \cdot d}, \frac{-1}{4}, \ell\right)}{\ell}} \]

      2. lift-/.f64 N/A

         \[\leadsto w0 \cdot \sqrt{\frac{\mathsf{fma}\left(\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot D}{d \cdot d}, \frac{-1}{4}, \ell\right)}{\ell}} \]

      3. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{\frac{\mathsf{fma}\left(\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot D}{d \cdot d}, \frac{-1}{4}, \ell\right)}{\ell}} \]

      4. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{\frac{\mathsf{fma}\left(\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot D}{d \cdot d}, \frac{-1}{4}, \ell\right)}{\ell}} \]

      5. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{\frac{\mathsf{fma}\left(\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot D}{d \cdot d}, \frac{-1}{4}, \ell\right)}{\ell}} \]

      6. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{\frac{\mathsf{fma}\left(\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot D}{d \cdot d}, \frac{-1}{4}, \ell\right)}{\ell}} \]

      7. associate-*r* N/A

         \[\leadsto w0 \cdot \sqrt{\frac{\mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d}, \frac{-1}{4}, \ell\right)}{\ell}} \]

      8. pow2 N/A

         \[\leadsto w0 \cdot \sqrt{\frac{\mathsf{fma}\left(\frac{\left({M}^{2} \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d}, \frac{-1}{4}, \ell\right)}{\ell}} \]

      9. pow2 N/A

         \[\leadsto w0 \cdot \sqrt{\frac{\mathsf{fma}\left(\frac{\left({M}^{2} \cdot h\right) \cdot {D}^{2}}{d \cdot d}, \frac{-1}{4}, \ell\right)}{\ell}} \]

      10. *-commutative N/A

          \[\leadsto w0 \cdot \sqrt{\frac{\mathsf{fma}\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot d}, \frac{-1}{4}, \ell\right)}{\ell}} \]

      11. pow2 N/A

          \[\leadsto w0 \cdot \sqrt{\frac{\mathsf{fma}\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}, \frac{-1}{4}, \ell\right)}{\ell}} \]

      12. associate-/l* N/A

          \[\leadsto w0 \cdot \sqrt{\frac{\mathsf{fma}\left({D}^{2} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}, \frac{-1}{4}, \ell\right)}{\ell}} \]

      13. lower-*.f64 N/A

          \[\leadsto w0 \cdot \sqrt{\frac{\mathsf{fma}\left({D}^{2} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}, \frac{-1}{4}, \ell\right)}{\ell}} \]

      14. pow2 N/A

          \[\leadsto w0 \cdot \sqrt{\frac{\mathsf{fma}\left(\left(D \cdot D\right) \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}, \frac{-1}{4}, \ell\right)}{\ell}} \]

      15. lift-*.f64 N/A

          \[\leadsto w0 \cdot \sqrt{\frac{\mathsf{fma}\left(\left(D \cdot D\right) \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}, \frac{-1}{4}, \ell\right)}{\ell}} \]

      16. associate-/l* N/A

          \[\leadsto w0 \cdot \sqrt{\frac{\mathsf{fma}\left(\left(D \cdot D\right) \cdot \left({M}^{2} \cdot \frac{h}{{d}^{2}}\right), \frac{-1}{4}, \ell\right)}{\ell}} \]

      17. lower-*.f64 N/A

          \[\leadsto w0 \cdot \sqrt{\frac{\mathsf{fma}\left(\left(D \cdot D\right) \cdot \left({M}^{2} \cdot \frac{h}{{d}^{2}}\right), \frac{-1}{4}, \ell\right)}{\ell}} \]

      18. pow2 N/A

          \[\leadsto w0 \cdot \sqrt{\frac{\mathsf{fma}\left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{h}{{d}^{2}}\right), \frac{-1}{4}, \ell\right)}{\ell}} \]

      19. lift-*.f64 N/A

          \[\leadsto w0 \cdot \sqrt{\frac{\mathsf{fma}\left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{h}{{d}^{2}}\right), \frac{-1}{4}, \ell\right)}{\ell}} \]

      20. lower-/.f64 N/A

          \[\leadsto w0 \cdot \sqrt{\frac{\mathsf{fma}\left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{h}{{d}^{2}}\right), \frac{-1}{4}, \ell\right)}{\ell}} \]

      21. pow2 N/A

          \[\leadsto w0 \cdot \sqrt{\frac{\mathsf{fma}\left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{h}{d \cdot d}\right), \frac{-1}{4}, \ell\right)}{\ell}} \]

      22. lift-*.f64 39.4

          \[\leadsto w0 \cdot \sqrt{\frac{\mathsf{fma}\left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{h}{d \cdot d}\right), -0.25, \ell\right)}{\ell}} \]

   8. Applied rewrites 39.4%

      \[\leadsto w0 \cdot \sqrt{\frac{\mathsf{fma}\left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{h}{d \cdot d}\right), -0.25, \ell\right)}{\ell}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{\frac{\mathsf{fma}\left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{h}{d \cdot d}\right), \frac{-1}{4}, \ell\right)}{\ell}} \]

      2. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{\frac{\mathsf{fma}\left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{h}{d \cdot d}\right), \frac{-1}{4}, \ell\right)}{\ell}} \]

      3. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{\frac{\mathsf{fma}\left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{h}{d \cdot d}\right), \frac{-1}{4}, \ell\right)}{\ell}} \]

      4. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{\frac{\mathsf{fma}\left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{h}{d \cdot d}\right), \frac{-1}{4}, \ell\right)}{\ell}} \]

      5. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{\frac{\mathsf{fma}\left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{h}{d \cdot d}\right), \frac{-1}{4}, \ell\right)}{\ell}} \]

      6. lift-/.f64 N/A

         \[\leadsto w0 \cdot \sqrt{\frac{\mathsf{fma}\left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{h}{d \cdot d}\right), \frac{-1}{4}, \ell\right)}{\ell}} \]

      7. pow2 N/A

         \[\leadsto w0 \cdot \sqrt{\frac{\mathsf{fma}\left({D}^{2} \cdot \left(\left(M \cdot M\right) \cdot \frac{h}{d \cdot d}\right), \frac{-1}{4}, \ell\right)}{\ell}} \]

      8. pow2 N/A

         \[\leadsto w0 \cdot \sqrt{\frac{\mathsf{fma}\left({D}^{2} \cdot \left({M}^{2} \cdot \frac{h}{d \cdot d}\right), \frac{-1}{4}, \ell\right)}{\ell}} \]

      9. associate-*r* N/A

         \[\leadsto w0 \cdot \sqrt{\frac{\mathsf{fma}\left(\left({D}^{2} \cdot {M}^{2}\right) \cdot \frac{h}{d \cdot d}, \frac{-1}{4}, \ell\right)}{\ell}} \]

      10. lower-*.f64 N/A

          \[\leadsto w0 \cdot \sqrt{\frac{\mathsf{fma}\left(\left({D}^{2} \cdot {M}^{2}\right) \cdot \frac{h}{d \cdot d}, \frac{-1}{4}, \ell\right)}{\ell}} \]

      11. pow-prod-down N/A

          \[\leadsto w0 \cdot \sqrt{\frac{\mathsf{fma}\left({\left(D \cdot M\right)}^{2} \cdot \frac{h}{d \cdot d}, \frac{-1}{4}, \ell\right)}{\ell}} \]

      12. unpow2 N/A

          \[\leadsto w0 \cdot \sqrt{\frac{\mathsf{fma}\left(\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right) \cdot \frac{h}{d \cdot d}, \frac{-1}{4}, \ell\right)}{\ell}} \]

      13. lower-*.f64 N/A

          \[\leadsto w0 \cdot \sqrt{\frac{\mathsf{fma}\left(\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right) \cdot \frac{h}{d \cdot d}, \frac{-1}{4}, \ell\right)}{\ell}} \]

      14. *-commutative N/A

          \[\leadsto w0 \cdot \sqrt{\frac{\mathsf{fma}\left(\left(\left(M \cdot D\right) \cdot \left(D \cdot M\right)\right) \cdot \frac{h}{d \cdot d}, \frac{-1}{4}, \ell\right)}{\ell}} \]

      15. lift-*.f64 N/A

          \[\leadsto w0 \cdot \sqrt{\frac{\mathsf{fma}\left(\left(\left(M \cdot D\right) \cdot \left(D \cdot M\right)\right) \cdot \frac{h}{d \cdot d}, \frac{-1}{4}, \ell\right)}{\ell}} \]

      16. *-commutative N/A

          \[\leadsto w0 \cdot \sqrt{\frac{\mathsf{fma}\left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \frac{h}{d \cdot d}, \frac{-1}{4}, \ell\right)}{\ell}} \]

      17. lift-*.f64 N/A

          \[\leadsto w0 \cdot \sqrt{\frac{\mathsf{fma}\left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \frac{h}{d \cdot d}, \frac{-1}{4}, \ell\right)}{\ell}} \]

      18. lift-/.f64 N/A

          \[\leadsto w0 \cdot \sqrt{\frac{\mathsf{fma}\left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \frac{h}{d \cdot d}, \frac{-1}{4}, \ell\right)}{\ell}} \]

      19. lift-*.f64 50.7

          \[\leadsto w0 \cdot \sqrt{\frac{\mathsf{fma}\left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \frac{h}{d \cdot d}, -0.25, \ell\right)}{\ell}} \]

   10. Applied rewrites 50.7%

       \[\leadsto w0 \cdot \sqrt{\frac{\mathsf{fma}\left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \frac{h}{d \cdot d}, -0.25, \ell\right)}{\ell}} \]

   if -2.0000000000000001e-13 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 88.8%

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

   2. Taylor expanded in M around 0

      \[\leadsto \color{blue}{w0} \]
   3. Step-by-step derivation

   4. Applied rewrites 96.2%

      \[\leadsto \color{blue}{w0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 81.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{-12}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\left(d \cdot d\right) \cdot \ell} \cdot -0.25, h, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -5e-12)
   (* w0 (sqrt (fma (* (/ (* (* M D) (* M D)) (* (* d d) l)) -0.25) h 1.0)))
   w0))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -5e-12) {
		tmp = w0 * sqrt(fma(((((M * D) * (M * D)) / ((d * d) * l)) * -0.25), h, 1.0));
	} else {
		tmp = w0;
	}
	return tmp;
}

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -5e-12)
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(Float64(Float64(M * D) * Float64(M * D)) / Float64(Float64(d * d) * l)) * -0.25), h, 1.0)));
	else
		tmp = w0;
	end
	return tmp
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -5e-12], N[(w0 * N[Sqrt[N[(N[(N[(N[(N[(M * D), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision] * h + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -4.9999999999999997e-12

   1. Initial program 65.3%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

   3. Applied rewrites 65.2%

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

   4. Taylor expanded in l around 0

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

      1. metadata-eval N/A

         \[\leadsto w0 \cdot \sqrt{\frac{\ell - \left(\mathsf{neg}\left(\frac{-1}{4}\right)\right) \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}}{\ell}} \]

      2. fp-cancel-sign-sub-inv N/A

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

      3. lower-/.f64 N/A

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

   6. Applied rewrites 44.0%

      \[\leadsto w0 \cdot \sqrt{\color{blue}{\frac{\mathsf{fma}\left(\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot D}{d \cdot d}, -0.25, \ell\right)}{\ell}}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{\frac{\mathsf{fma}\left(\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot D}{d \cdot d}, \frac{-1}{4}, \ell\right)}{\ell}} \]

      2. lift-/.f64 N/A

         \[\leadsto w0 \cdot \sqrt{\frac{\mathsf{fma}\left(\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot D}{d \cdot d}, \frac{-1}{4}, \ell\right)}{\ell}} \]

      3. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{\frac{\mathsf{fma}\left(\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot D}{d \cdot d}, \frac{-1}{4}, \ell\right)}{\ell}} \]

      4. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{\frac{\mathsf{fma}\left(\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot D}{d \cdot d}, \frac{-1}{4}, \ell\right)}{\ell}} \]

      5. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{\frac{\mathsf{fma}\left(\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot D}{d \cdot d}, \frac{-1}{4}, \ell\right)}{\ell}} \]

      6. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{\frac{\mathsf{fma}\left(\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot D}{d \cdot d}, \frac{-1}{4}, \ell\right)}{\ell}} \]

      7. associate-*r* N/A

         \[\leadsto w0 \cdot \sqrt{\frac{\mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d}, \frac{-1}{4}, \ell\right)}{\ell}} \]

      8. pow2 N/A

         \[\leadsto w0 \cdot \sqrt{\frac{\mathsf{fma}\left(\frac{\left({M}^{2} \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d}, \frac{-1}{4}, \ell\right)}{\ell}} \]

      9. pow2 N/A

         \[\leadsto w0 \cdot \sqrt{\frac{\mathsf{fma}\left(\frac{\left({M}^{2} \cdot h\right) \cdot {D}^{2}}{d \cdot d}, \frac{-1}{4}, \ell\right)}{\ell}} \]

      10. *-commutative N/A

          \[\leadsto w0 \cdot \sqrt{\frac{\mathsf{fma}\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot d}, \frac{-1}{4}, \ell\right)}{\ell}} \]

      11. pow2 N/A

          \[\leadsto w0 \cdot \sqrt{\frac{\mathsf{fma}\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}, \frac{-1}{4}, \ell\right)}{\ell}} \]

      12. associate-/l* N/A

          \[\leadsto w0 \cdot \sqrt{\frac{\mathsf{fma}\left({D}^{2} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}, \frac{-1}{4}, \ell\right)}{\ell}} \]

      13. lower-*.f64 N/A

          \[\leadsto w0 \cdot \sqrt{\frac{\mathsf{fma}\left({D}^{2} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}, \frac{-1}{4}, \ell\right)}{\ell}} \]

      14. pow2 N/A

          \[\leadsto w0 \cdot \sqrt{\frac{\mathsf{fma}\left(\left(D \cdot D\right) \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}, \frac{-1}{4}, \ell\right)}{\ell}} \]

      15. lift-*.f64 N/A

          \[\leadsto w0 \cdot \sqrt{\frac{\mathsf{fma}\left(\left(D \cdot D\right) \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}, \frac{-1}{4}, \ell\right)}{\ell}} \]

      16. associate-/l* N/A

          \[\leadsto w0 \cdot \sqrt{\frac{\mathsf{fma}\left(\left(D \cdot D\right) \cdot \left({M}^{2} \cdot \frac{h}{{d}^{2}}\right), \frac{-1}{4}, \ell\right)}{\ell}} \]

      17. lower-*.f64 N/A

          \[\leadsto w0 \cdot \sqrt{\frac{\mathsf{fma}\left(\left(D \cdot D\right) \cdot \left({M}^{2} \cdot \frac{h}{{d}^{2}}\right), \frac{-1}{4}, \ell\right)}{\ell}} \]

      18. pow2 N/A

          \[\leadsto w0 \cdot \sqrt{\frac{\mathsf{fma}\left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{h}{{d}^{2}}\right), \frac{-1}{4}, \ell\right)}{\ell}} \]

      19. lift-*.f64 N/A

          \[\leadsto w0 \cdot \sqrt{\frac{\mathsf{fma}\left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{h}{{d}^{2}}\right), \frac{-1}{4}, \ell\right)}{\ell}} \]

      20. lower-/.f64 N/A

          \[\leadsto w0 \cdot \sqrt{\frac{\mathsf{fma}\left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{h}{{d}^{2}}\right), \frac{-1}{4}, \ell\right)}{\ell}} \]

      21. pow2 N/A

          \[\leadsto w0 \cdot \sqrt{\frac{\mathsf{fma}\left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{h}{d \cdot d}\right), \frac{-1}{4}, \ell\right)}{\ell}} \]

      22. lift-*.f64 39.4

          \[\leadsto w0 \cdot \sqrt{\frac{\mathsf{fma}\left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{h}{d \cdot d}\right), -0.25, \ell\right)}{\ell}} \]

   8. Applied rewrites 39.4%

      \[\leadsto w0 \cdot \sqrt{\frac{\mathsf{fma}\left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{h}{d \cdot d}\right), -0.25, \ell\right)}{\ell}} \]

   9. Taylor expanded in h around inf

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

       1. distribute-rgt-in N/A

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

       2. inv-pow N/A

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

       3. pow-plus N/A

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

       4. metadata-eval N/A

          \[\leadsto w0 \cdot \sqrt{\left(\frac{-1}{4} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}\right) \cdot h + {h}^{0}} \]

       5. metadata-eval N/A

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

       6. lower-fma.f64 N/A

          \[\leadsto w0 \cdot \sqrt{\mathsf{fma}\left(\frac{-1}{4} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}, h, 1\right)} \]

   11. Applied rewrites 50.9%

       \[\leadsto w0 \cdot \sqrt{\mathsf{fma}\left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\left(d \cdot d\right) \cdot \ell} \cdot -0.25, \color{blue}{h}, 1\right)} \]

   if -4.9999999999999997e-12 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 88.8%

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

   2. Taylor expanded in M around 0

      \[\leadsto \color{blue}{w0} \]
   3. Step-by-step derivation

   4. Applied rewrites 96.2%

      \[\leadsto \color{blue}{w0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 79.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+229}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(\frac{\left(M \cdot \left(M \cdot h\right)\right) \cdot \left(D \cdot D\right)}{d \cdot \left(d \cdot \ell\right)}, -0.125, 1\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -5e+229)
   (* w0 (fma (/ (* (* M (* M h)) (* D D)) (* d (* d l))) -0.125 1.0))
   w0))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -5e+229) {
		tmp = w0 * fma((((M * (M * h)) * (D * D)) / (d * (d * l))), -0.125, 1.0);
	} else {
		tmp = w0;
	}
	return tmp;
}

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -5e+229)
		tmp = Float64(w0 * fma(Float64(Float64(Float64(M * Float64(M * h)) * Float64(D * D)) / Float64(d * Float64(d * l))), -0.125, 1.0));
	else
		tmp = w0;
	end
	return tmp
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -5e+229], N[(w0 * N[(N[(N[(N[(M * N[(M * h), $MachinePrecision]), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision] / N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -5.0000000000000005e229

   1. Initial program 57.6%

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

   2. Taylor expanded in M around 0

      \[\leadsto w0 \cdot \color{blue}{\left(1 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto w0 \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} + \color{blue}{1}\right) \]

      2. *-commutative N/A

         \[\leadsto w0 \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} \cdot \frac{-1}{8} + 1\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto w0 \cdot \mathsf{fma}\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}, \color{blue}{\frac{-1}{8}}, 1\right) \]

      4. lower-/.f64 N/A

         \[\leadsto w0 \cdot \mathsf{fma}\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}, \frac{-1}{8}, 1\right) \]

      5. *-commutative N/A

         \[\leadsto w0 \cdot \mathsf{fma}\left(\frac{\left({M}^{2} \cdot h\right) \cdot {D}^{2}}{{d}^{2} \cdot \ell}, \frac{-1}{8}, 1\right) \]

      6. lower-*.f64 N/A

         \[\leadsto w0 \cdot \mathsf{fma}\left(\frac{\left({M}^{2} \cdot h\right) \cdot {D}^{2}}{{d}^{2} \cdot \ell}, \frac{-1}{8}, 1\right) \]

      7. lower-*.f64 N/A

         \[\leadsto w0 \cdot \mathsf{fma}\left(\frac{\left({M}^{2} \cdot h\right) \cdot {D}^{2}}{{d}^{2} \cdot \ell}, \frac{-1}{8}, 1\right) \]

      8. unpow2 N/A

         \[\leadsto w0 \cdot \mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot {D}^{2}}{{d}^{2} \cdot \ell}, \frac{-1}{8}, 1\right) \]

      9. lower-*.f64 N/A

         \[\leadsto w0 \cdot \mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot {D}^{2}}{{d}^{2} \cdot \ell}, \frac{-1}{8}, 1\right) \]

      10. unpow2 N/A

          \[\leadsto w0 \cdot \mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{{d}^{2} \cdot \ell}, \frac{-1}{8}, 1\right) \]

      11. lower-*.f64 N/A

          \[\leadsto w0 \cdot \mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{{d}^{2} \cdot \ell}, \frac{-1}{8}, 1\right) \]

      12. lower-*.f64 N/A

          \[\leadsto w0 \cdot \mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{{d}^{2} \cdot \ell}, \frac{-1}{8}, 1\right) \]

      13. unpow2 N/A

          \[\leadsto w0 \cdot \mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{\left(d \cdot d\right) \cdot \ell}, \frac{-1}{8}, 1\right) \]

      14. lower-*.f64 42.2

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

   4. Applied rewrites 42.2%

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

      1. lift-*.f64 N/A

         \[\leadsto w0 \cdot \mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{\left(d \cdot d\right) \cdot \ell}, \frac{-1}{8}, 1\right) \]

      2. lift-*.f64 N/A

         \[\leadsto w0 \cdot \mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{\left(d \cdot d\right) \cdot \ell}, \frac{-1}{8}, 1\right) \]

      3. associate-*l* N/A

         \[\leadsto w0 \cdot \mathsf{fma}\left(\frac{\left(M \cdot \left(M \cdot h\right)\right) \cdot \left(D \cdot D\right)}{\left(d \cdot d\right) \cdot \ell}, \frac{-1}{8}, 1\right) \]

      4. lower-*.f64 N/A

         \[\leadsto w0 \cdot \mathsf{fma}\left(\frac{\left(M \cdot \left(M \cdot h\right)\right) \cdot \left(D \cdot D\right)}{\left(d \cdot d\right) \cdot \ell}, \frac{-1}{8}, 1\right) \]

      5. lower-*.f64 43.5

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

   6. Applied rewrites 43.5%

      \[\leadsto w0 \cdot \mathsf{fma}\left(\frac{\left(M \cdot \left(M \cdot h\right)\right) \cdot \left(D \cdot D\right)}{\left(d \cdot d\right) \cdot \ell}, -0.125, 1\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto w0 \cdot \mathsf{fma}\left(\frac{\left(M \cdot \left(M \cdot h\right)\right) \cdot \left(D \cdot D\right)}{\left(d \cdot d\right) \cdot \ell}, \frac{-1}{8}, 1\right) \]

      2. lift-*.f64 N/A

         \[\leadsto w0 \cdot \mathsf{fma}\left(\frac{\left(M \cdot \left(M \cdot h\right)\right) \cdot \left(D \cdot D\right)}{\left(d \cdot d\right) \cdot \ell}, \frac{-1}{8}, 1\right) \]

      3. associate-*l* N/A

         \[\leadsto w0 \cdot \mathsf{fma}\left(\frac{\left(M \cdot \left(M \cdot h\right)\right) \cdot \left(D \cdot D\right)}{d \cdot \left(d \cdot \ell\right)}, \frac{-1}{8}, 1\right) \]

      4. lower-*.f64 N/A

         \[\leadsto w0 \cdot \mathsf{fma}\left(\frac{\left(M \cdot \left(M \cdot h\right)\right) \cdot \left(D \cdot D\right)}{d \cdot \left(d \cdot \ell\right)}, \frac{-1}{8}, 1\right) \]

      5. lower-*.f64 45.5

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

   8. Applied rewrites 45.5%

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

   if -5.0000000000000005e229 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 89.6%

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

   2. Taylor expanded in M around 0

      \[\leadsto \color{blue}{w0} \]
   3. Step-by-step derivation

   4. Applied rewrites 89.5%

      \[\leadsto \color{blue}{w0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 79.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{-12}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125, h, 1\right) \cdot w0\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -5e-12)
   (* (fma (* (/ (* (* M D) (* M D)) (* (* d d) l)) -0.125) h 1.0) w0)
   w0))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -5e-12) {
		tmp = fma(((((M * D) * (M * D)) / ((d * d) * l)) * -0.125), h, 1.0) * w0;
	} else {
		tmp = w0;
	}
	return tmp;
}

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -5e-12)
		tmp = Float64(fma(Float64(Float64(Float64(Float64(M * D) * Float64(M * D)) / Float64(Float64(d * d) * l)) * -0.125), h, 1.0) * w0);
	else
		tmp = w0;
	end
	return tmp
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -5e-12], N[(N[(N[(N[(N[(N[(M * D), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] * h + 1.0), $MachinePrecision] * w0), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -4.9999999999999997e-12

   1. Initial program 65.3%

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

   2. Taylor expanded in M around 0

      \[\leadsto w0 \cdot \color{blue}{\left(1 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto w0 \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} + \color{blue}{1}\right) \]

      2. *-commutative N/A

         \[\leadsto w0 \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} \cdot \frac{-1}{8} + 1\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto w0 \cdot \mathsf{fma}\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}, \color{blue}{\frac{-1}{8}}, 1\right) \]

      4. lower-/.f64 N/A

         \[\leadsto w0 \cdot \mathsf{fma}\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}, \frac{-1}{8}, 1\right) \]

      5. *-commutative N/A

         \[\leadsto w0 \cdot \mathsf{fma}\left(\frac{\left({M}^{2} \cdot h\right) \cdot {D}^{2}}{{d}^{2} \cdot \ell}, \frac{-1}{8}, 1\right) \]

      6. lower-*.f64 N/A

         \[\leadsto w0 \cdot \mathsf{fma}\left(\frac{\left({M}^{2} \cdot h\right) \cdot {D}^{2}}{{d}^{2} \cdot \ell}, \frac{-1}{8}, 1\right) \]

      7. lower-*.f64 N/A

         \[\leadsto w0 \cdot \mathsf{fma}\left(\frac{\left({M}^{2} \cdot h\right) \cdot {D}^{2}}{{d}^{2} \cdot \ell}, \frac{-1}{8}, 1\right) \]

      8. unpow2 N/A

         \[\leadsto w0 \cdot \mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot {D}^{2}}{{d}^{2} \cdot \ell}, \frac{-1}{8}, 1\right) \]

      9. lower-*.f64 N/A

         \[\leadsto w0 \cdot \mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot {D}^{2}}{{d}^{2} \cdot \ell}, \frac{-1}{8}, 1\right) \]

      10. unpow2 N/A

          \[\leadsto w0 \cdot \mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{{d}^{2} \cdot \ell}, \frac{-1}{8}, 1\right) \]

      11. lower-*.f64 N/A

          \[\leadsto w0 \cdot \mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{{d}^{2} \cdot \ell}, \frac{-1}{8}, 1\right) \]

      12. lower-*.f64 N/A

          \[\leadsto w0 \cdot \mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{{d}^{2} \cdot \ell}, \frac{-1}{8}, 1\right) \]

      13. unpow2 N/A

          \[\leadsto w0 \cdot \mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{\left(d \cdot d\right) \cdot \ell}, \frac{-1}{8}, 1\right) \]

      14. lower-*.f64 35.7

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

   4. Applied rewrites 35.7%

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

   5. Taylor expanded in M around inf

      \[\leadsto w0 \cdot \left(\frac{-1}{8} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto w0 \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} \cdot \frac{-1}{8}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto w0 \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} \cdot \frac{-1}{8}\right) \]

   7. Applied rewrites 38.7%

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

   9. Applied rewrites 35.2%

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

   10. Taylor expanded in h around inf

       \[\leadsto \left(h \cdot \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell} + \frac{1}{h}\right)}\right) \cdot w0 \]
   11. Step-by-step derivation

       1. associate-*l/ N/A

          \[\leadsto \left(h \cdot \left(\color{blue}{\frac{-1}{8}} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell} + \frac{1}{h}\right)\right) \cdot w0 \]

       2. associate-*r* N/A

          \[\leadsto \left(h \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell} + \frac{1}{h}\right)\right) \cdot w0 \]

       3. associate-*l/ N/A

          \[\leadsto \left(h \cdot \left(\color{blue}{\frac{-1}{8}} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell} + \frac{1}{h}\right)\right) \cdot w0 \]

       4. distribute-rgt-in N/A

          \[\leadsto \left(\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}\right) \cdot h + \frac{1}{h} \cdot \color{blue}{h}\right) \cdot w0 \]

       5. inv-pow N/A

          \[\leadsto \left(\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}\right) \cdot h + {h}^{-1} \cdot h\right) \cdot w0 \]

       6. pow-plus N/A

          \[\leadsto \left(\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}\right) \cdot h + {h}^{\left(-1 + \color{blue}{1}\right)}\right) \cdot w0 \]

       7. metadata-eval N/A

          \[\leadsto \left(\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}\right) \cdot h + {h}^{0}\right) \cdot w0 \]

       8. metadata-eval N/A

          \[\leadsto \left(\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}\right) \cdot h + 1\right) \cdot w0 \]

       9. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}, h, 1\right) \cdot w0 \]

   12. Applied rewrites 42.5%

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

   if -4.9999999999999997e-12 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 88.8%

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

   2. Taylor expanded in M around 0

      \[\leadsto \color{blue}{w0} \]
   3. Step-by-step derivation

   4. Applied rewrites 96.2%

      \[\leadsto \color{blue}{w0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 78.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+229}:\\ \;\;\;\;\left(\frac{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot h}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125\right) \cdot w0\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -5e+229)
   (* (* (/ (* (* (* M D) (* M D)) h) (* (* d d) l)) -0.125) w0)
   w0))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -5e+229) {
		tmp = (((((M * D) * (M * D)) * h) / ((d * d) * l)) * -0.125) * w0;
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(w0, m, d, h, l, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l)) <= (-5d+229)) then
        tmp = (((((m * d) * (m * d)) * h) / ((d_1 * d_1) * l)) * (-0.125d0)) * w0
    else
        tmp = w0
    end if
    code = tmp
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -5e+229) {
		tmp = (((((M * D) * (M * D)) * h) / ((d * d) * l)) * -0.125) * w0;
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -5e+229:
		tmp = (((((M * D) * (M * D)) * h) / ((d * d) * l)) * -0.125) * w0
	else:
		tmp = w0
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -5e+229)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(M * D) * Float64(M * D)) * h) / Float64(Float64(d * d) * l)) * -0.125) * w0);
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (((((M * D) / (2.0 * d)) ^ 2.0) * (h / l)) <= -5e+229)
		tmp = (((((M * D) * (M * D)) * h) / ((d * d) * l)) * -0.125) * w0;
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -5e+229], N[(N[(N[(N[(N[(N[(M * D), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] * w0), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -5.0000000000000005e229

   1. Initial program 57.6%

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

   2. Taylor expanded in M around 0

      \[\leadsto w0 \cdot \color{blue}{\left(1 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto w0 \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} + \color{blue}{1}\right) \]

      2. *-commutative N/A

         \[\leadsto w0 \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} \cdot \frac{-1}{8} + 1\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto w0 \cdot \mathsf{fma}\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}, \color{blue}{\frac{-1}{8}}, 1\right) \]

      4. lower-/.f64 N/A

         \[\leadsto w0 \cdot \mathsf{fma}\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}, \frac{-1}{8}, 1\right) \]

      5. *-commutative N/A

         \[\leadsto w0 \cdot \mathsf{fma}\left(\frac{\left({M}^{2} \cdot h\right) \cdot {D}^{2}}{{d}^{2} \cdot \ell}, \frac{-1}{8}, 1\right) \]

      6. lower-*.f64 N/A

         \[\leadsto w0 \cdot \mathsf{fma}\left(\frac{\left({M}^{2} \cdot h\right) \cdot {D}^{2}}{{d}^{2} \cdot \ell}, \frac{-1}{8}, 1\right) \]

      7. lower-*.f64 N/A

         \[\leadsto w0 \cdot \mathsf{fma}\left(\frac{\left({M}^{2} \cdot h\right) \cdot {D}^{2}}{{d}^{2} \cdot \ell}, \frac{-1}{8}, 1\right) \]

      8. unpow2 N/A

         \[\leadsto w0 \cdot \mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot {D}^{2}}{{d}^{2} \cdot \ell}, \frac{-1}{8}, 1\right) \]

      9. lower-*.f64 N/A

         \[\leadsto w0 \cdot \mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot {D}^{2}}{{d}^{2} \cdot \ell}, \frac{-1}{8}, 1\right) \]

      10. unpow2 N/A

          \[\leadsto w0 \cdot \mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{{d}^{2} \cdot \ell}, \frac{-1}{8}, 1\right) \]

      11. lower-*.f64 N/A

          \[\leadsto w0 \cdot \mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{{d}^{2} \cdot \ell}, \frac{-1}{8}, 1\right) \]

      12. lower-*.f64 N/A

          \[\leadsto w0 \cdot \mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{{d}^{2} \cdot \ell}, \frac{-1}{8}, 1\right) \]

      13. unpow2 N/A

          \[\leadsto w0 \cdot \mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{\left(d \cdot d\right) \cdot \ell}, \frac{-1}{8}, 1\right) \]

      14. lower-*.f64 42.2

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

   4. Applied rewrites 42.2%

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

   5. Taylor expanded in M around inf

      \[\leadsto w0 \cdot \left(\frac{-1}{8} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto w0 \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} \cdot \frac{-1}{8}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto w0 \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} \cdot \frac{-1}{8}\right) \]

   7. Applied rewrites 45.6%

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

   9. Applied rewrites 41.9%

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

       1. lift-*.f64 N/A

          \[\leadsto \left(\frac{\left(M \cdot M\right) \cdot \left(\left(D \cdot D\right) \cdot h\right)}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}\right) \cdot w0 \]

       2. lift-*.f64 N/A

          \[\leadsto \left(\frac{\left(M \cdot M\right) \cdot \left(\left(D \cdot D\right) \cdot h\right)}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}\right) \cdot w0 \]

       3. pow2 N/A

          \[\leadsto \left(\frac{{M}^{2} \cdot \left(\left(D \cdot D\right) \cdot h\right)}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}\right) \cdot w0 \]

       4. lift-*.f64 N/A

          \[\leadsto \left(\frac{{M}^{2} \cdot \left(\left(D \cdot D\right) \cdot h\right)}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}\right) \cdot w0 \]

       5. lift-*.f64 N/A

          \[\leadsto \left(\frac{{M}^{2} \cdot \left(\left(D \cdot D\right) \cdot h\right)}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}\right) \cdot w0 \]

       6. pow2 N/A

          \[\leadsto \left(\frac{{M}^{2} \cdot \left({D}^{2} \cdot h\right)}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}\right) \cdot w0 \]

       7. associate-*r* N/A

          \[\leadsto \left(\frac{\left({M}^{2} \cdot {D}^{2}\right) \cdot h}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}\right) \cdot w0 \]

       8. *-commutative N/A

          \[\leadsto \left(\frac{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}\right) \cdot w0 \]

       9. lower-*.f64 N/A

          \[\leadsto \left(\frac{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}\right) \cdot w0 \]

       10. pow-prod-down N/A

           \[\leadsto \left(\frac{{\left(D \cdot M\right)}^{2} \cdot h}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}\right) \cdot w0 \]

       11. unpow2 N/A

           \[\leadsto \left(\frac{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right) \cdot h}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}\right) \cdot w0 \]

       12. lower-*.f64 N/A

           \[\leadsto \left(\frac{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right) \cdot h}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}\right) \cdot w0 \]

       13. *-commutative N/A

           \[\leadsto \left(\frac{\left(\left(M \cdot D\right) \cdot \left(D \cdot M\right)\right) \cdot h}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}\right) \cdot w0 \]

       14. lift-*.f64 N/A

           \[\leadsto \left(\frac{\left(\left(M \cdot D\right) \cdot \left(D \cdot M\right)\right) \cdot h}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}\right) \cdot w0 \]

       15. *-commutative N/A

           \[\leadsto \left(\frac{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot h}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}\right) \cdot w0 \]

       16. lift-*.f64 48.7

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

   11. Applied rewrites 48.7%

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

   if -5.0000000000000005e229 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 89.6%

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

   2. Taylor expanded in M around 0

      \[\leadsto \color{blue}{w0} \]
   3. Step-by-step derivation

   4. Applied rewrites 89.5%

      \[\leadsto \color{blue}{w0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 77.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+270}:\\ \;\;\;\;\left(\frac{\left(M \cdot M\right) \cdot \left(\left(D \cdot D\right) \cdot h\right)}{d \cdot \left(\ell \cdot d\right)} \cdot -0.125\right) \cdot w0\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -5e+270)
   (* (* (/ (* (* M M) (* (* D D) h)) (* d (* l d))) -0.125) w0)
   w0))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -5e+270) {
		tmp = ((((M * M) * ((D * D) * h)) / (d * (l * d))) * -0.125) * w0;
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(w0, m, d, h, l, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l)) <= (-5d+270)) then
        tmp = ((((m * m) * ((d * d) * h)) / (d_1 * (l * d_1))) * (-0.125d0)) * w0
    else
        tmp = w0
    end if
    code = tmp
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -5e+270) {
		tmp = ((((M * M) * ((D * D) * h)) / (d * (l * d))) * -0.125) * w0;
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -5e+270:
		tmp = ((((M * M) * ((D * D) * h)) / (d * (l * d))) * -0.125) * w0
	else:
		tmp = w0
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -5e+270)
		tmp = Float64(Float64(Float64(Float64(Float64(M * M) * Float64(Float64(D * D) * h)) / Float64(d * Float64(l * d))) * -0.125) * w0);
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (((((M * D) / (2.0 * d)) ^ 2.0) * (h / l)) <= -5e+270)
		tmp = ((((M * M) * ((D * D) * h)) / (d * (l * d))) * -0.125) * w0;
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -5e+270], N[(N[(N[(N[(N[(M * M), $MachinePrecision] * N[(N[(D * D), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] / N[(d * N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] * w0), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -4.99999999999999976e270

   1. Initial program 56.2%

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

   2. Taylor expanded in M around 0

      \[\leadsto w0 \cdot \color{blue}{\left(1 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto w0 \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} + \color{blue}{1}\right) \]

      2. *-commutative N/A

         \[\leadsto w0 \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} \cdot \frac{-1}{8} + 1\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto w0 \cdot \mathsf{fma}\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}, \color{blue}{\frac{-1}{8}}, 1\right) \]

      4. lower-/.f64 N/A

         \[\leadsto w0 \cdot \mathsf{fma}\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}, \frac{-1}{8}, 1\right) \]

      5. *-commutative N/A

         \[\leadsto w0 \cdot \mathsf{fma}\left(\frac{\left({M}^{2} \cdot h\right) \cdot {D}^{2}}{{d}^{2} \cdot \ell}, \frac{-1}{8}, 1\right) \]

      6. lower-*.f64 N/A

         \[\leadsto w0 \cdot \mathsf{fma}\left(\frac{\left({M}^{2} \cdot h\right) \cdot {D}^{2}}{{d}^{2} \cdot \ell}, \frac{-1}{8}, 1\right) \]

      7. lower-*.f64 N/A

         \[\leadsto w0 \cdot \mathsf{fma}\left(\frac{\left({M}^{2} \cdot h\right) \cdot {D}^{2}}{{d}^{2} \cdot \ell}, \frac{-1}{8}, 1\right) \]

      8. unpow2 N/A

         \[\leadsto w0 \cdot \mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot {D}^{2}}{{d}^{2} \cdot \ell}, \frac{-1}{8}, 1\right) \]

      9. lower-*.f64 N/A

         \[\leadsto w0 \cdot \mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot {D}^{2}}{{d}^{2} \cdot \ell}, \frac{-1}{8}, 1\right) \]

      10. unpow2 N/A

          \[\leadsto w0 \cdot \mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{{d}^{2} \cdot \ell}, \frac{-1}{8}, 1\right) \]

      11. lower-*.f64 N/A

          \[\leadsto w0 \cdot \mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{{d}^{2} \cdot \ell}, \frac{-1}{8}, 1\right) \]

      12. lower-*.f64 N/A

          \[\leadsto w0 \cdot \mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{{d}^{2} \cdot \ell}, \frac{-1}{8}, 1\right) \]

      13. unpow2 N/A

          \[\leadsto w0 \cdot \mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{\left(d \cdot d\right) \cdot \ell}, \frac{-1}{8}, 1\right) \]

      14. lower-*.f64 43.3

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

   4. Applied rewrites 43.3%

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

   5. Taylor expanded in M around inf

      \[\leadsto w0 \cdot \left(\frac{-1}{8} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto w0 \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} \cdot \frac{-1}{8}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto w0 \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} \cdot \frac{-1}{8}\right) \]

   7. Applied rewrites 46.6%

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

   9. Applied rewrites 42.9%

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

       1. lift-*.f64 N/A

          \[\leadsto \left(\frac{\left(M \cdot M\right) \cdot \left(\left(D \cdot D\right) \cdot h\right)}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}\right) \cdot w0 \]

       2. lift-*.f64 N/A

          \[\leadsto \left(\frac{\left(M \cdot M\right) \cdot \left(\left(D \cdot D\right) \cdot h\right)}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}\right) \cdot w0 \]

       3. associate-*l* N/A

          \[\leadsto \left(\frac{\left(M \cdot M\right) \cdot \left(\left(D \cdot D\right) \cdot h\right)}{d \cdot \left(d \cdot \ell\right)} \cdot \frac{-1}{8}\right) \cdot w0 \]

       4. lower-*.f64 N/A

          \[\leadsto \left(\frac{\left(M \cdot M\right) \cdot \left(\left(D \cdot D\right) \cdot h\right)}{d \cdot \left(d \cdot \ell\right)} \cdot \frac{-1}{8}\right) \cdot w0 \]

       5. *-commutative N/A

          \[\leadsto \left(\frac{\left(M \cdot M\right) \cdot \left(\left(D \cdot D\right) \cdot h\right)}{d \cdot \left(\ell \cdot d\right)} \cdot \frac{-1}{8}\right) \cdot w0 \]

       6. lower-*.f64 44.9

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

   11. Applied rewrites 44.9%

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

   if -4.99999999999999976e270 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 89.7%

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

   2. Taylor expanded in M around 0

      \[\leadsto \color{blue}{w0} \]
   3. Step-by-step derivation

   4. Applied rewrites 88.6%

      \[\leadsto \color{blue}{w0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 77.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+270}:\\ \;\;\;\;\left(\left(\left(M \cdot M\right) \cdot \left(\left(D \cdot D\right) \cdot \frac{h}{\left(d \cdot d\right) \cdot \ell}\right)\right) \cdot -0.125\right) \cdot w0\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -5e+270)
   (* (* (* (* M M) (* (* D D) (/ h (* (* d d) l)))) -0.125) w0)
   w0))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -5e+270) {
		tmp = (((M * M) * ((D * D) * (h / ((d * d) * l)))) * -0.125) * w0;
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(w0, m, d, h, l, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l)) <= (-5d+270)) then
        tmp = (((m * m) * ((d * d) * (h / ((d_1 * d_1) * l)))) * (-0.125d0)) * w0
    else
        tmp = w0
    end if
    code = tmp
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -5e+270) {
		tmp = (((M * M) * ((D * D) * (h / ((d * d) * l)))) * -0.125) * w0;
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -5e+270:
		tmp = (((M * M) * ((D * D) * (h / ((d * d) * l)))) * -0.125) * w0
	else:
		tmp = w0
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -5e+270)
		tmp = Float64(Float64(Float64(Float64(M * M) * Float64(Float64(D * D) * Float64(h / Float64(Float64(d * d) * l)))) * -0.125) * w0);
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (((((M * D) / (2.0 * d)) ^ 2.0) * (h / l)) <= -5e+270)
		tmp = (((M * M) * ((D * D) * (h / ((d * d) * l)))) * -0.125) * w0;
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -5e+270], N[(N[(N[(N[(M * M), $MachinePrecision] * N[(N[(D * D), $MachinePrecision] * N[(h / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] * w0), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -4.99999999999999976e270

   1. Initial program 56.2%

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

   2. Taylor expanded in M around 0

      \[\leadsto w0 \cdot \color{blue}{\left(1 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto w0 \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} + \color{blue}{1}\right) \]

      2. *-commutative N/A

         \[\leadsto w0 \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} \cdot \frac{-1}{8} + 1\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto w0 \cdot \mathsf{fma}\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}, \color{blue}{\frac{-1}{8}}, 1\right) \]

      4. lower-/.f64 N/A

         \[\leadsto w0 \cdot \mathsf{fma}\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}, \frac{-1}{8}, 1\right) \]

      5. *-commutative N/A

         \[\leadsto w0 \cdot \mathsf{fma}\left(\frac{\left({M}^{2} \cdot h\right) \cdot {D}^{2}}{{d}^{2} \cdot \ell}, \frac{-1}{8}, 1\right) \]

      6. lower-*.f64 N/A

         \[\leadsto w0 \cdot \mathsf{fma}\left(\frac{\left({M}^{2} \cdot h\right) \cdot {D}^{2}}{{d}^{2} \cdot \ell}, \frac{-1}{8}, 1\right) \]

      7. lower-*.f64 N/A

         \[\leadsto w0 \cdot \mathsf{fma}\left(\frac{\left({M}^{2} \cdot h\right) \cdot {D}^{2}}{{d}^{2} \cdot \ell}, \frac{-1}{8}, 1\right) \]

      8. unpow2 N/A

         \[\leadsto w0 \cdot \mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot {D}^{2}}{{d}^{2} \cdot \ell}, \frac{-1}{8}, 1\right) \]

      9. lower-*.f64 N/A

         \[\leadsto w0 \cdot \mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot {D}^{2}}{{d}^{2} \cdot \ell}, \frac{-1}{8}, 1\right) \]

      10. unpow2 N/A

          \[\leadsto w0 \cdot \mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{{d}^{2} \cdot \ell}, \frac{-1}{8}, 1\right) \]

      11. lower-*.f64 N/A

          \[\leadsto w0 \cdot \mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{{d}^{2} \cdot \ell}, \frac{-1}{8}, 1\right) \]

      12. lower-*.f64 N/A

          \[\leadsto w0 \cdot \mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{{d}^{2} \cdot \ell}, \frac{-1}{8}, 1\right) \]

      13. unpow2 N/A

          \[\leadsto w0 \cdot \mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{\left(d \cdot d\right) \cdot \ell}, \frac{-1}{8}, 1\right) \]

      14. lower-*.f64 43.3

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

   4. Applied rewrites 43.3%

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

   5. Taylor expanded in M around inf

      \[\leadsto w0 \cdot \left(\frac{-1}{8} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto w0 \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} \cdot \frac{-1}{8}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto w0 \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} \cdot \frac{-1}{8}\right) \]

   7. Applied rewrites 46.6%

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

   9. Applied rewrites 42.9%

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

       1. lift-/.f64 N/A

          \[\leadsto \left(\frac{\left(M \cdot M\right) \cdot \left(\left(D \cdot D\right) \cdot h\right)}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}\right) \cdot w0 \]

       2. lift-*.f64 N/A

          \[\leadsto \left(\frac{\left(M \cdot M\right) \cdot \left(\left(D \cdot D\right) \cdot h\right)}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}\right) \cdot w0 \]

       3. lift-*.f64 N/A

          \[\leadsto \left(\frac{\left(M \cdot M\right) \cdot \left(\left(D \cdot D\right) \cdot h\right)}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}\right) \cdot w0 \]

       4. lift-*.f64 N/A

          \[\leadsto \left(\frac{\left(M \cdot M\right) \cdot \left(\left(D \cdot D\right) \cdot h\right)}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}\right) \cdot w0 \]

       5. lift-*.f64 N/A

          \[\leadsto \left(\frac{\left(M \cdot M\right) \cdot \left(\left(D \cdot D\right) \cdot h\right)}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}\right) \cdot w0 \]

       6. pow2 N/A

          \[\leadsto \left(\frac{{M}^{2} \cdot \left(\left(D \cdot D\right) \cdot h\right)}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}\right) \cdot w0 \]

       7. lift-*.f64 N/A

          \[\leadsto \left(\frac{{M}^{2} \cdot \left(\left(D \cdot D\right) \cdot h\right)}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}\right) \cdot w0 \]

       8. lift-*.f64 N/A

          \[\leadsto \left(\frac{{M}^{2} \cdot \left(\left(D \cdot D\right) \cdot h\right)}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}\right) \cdot w0 \]

       9. associate-/l* N/A

          \[\leadsto \left(\left({M}^{2} \cdot \frac{\left(D \cdot D\right) \cdot h}{\left(d \cdot d\right) \cdot \ell}\right) \cdot \frac{-1}{8}\right) \cdot w0 \]

       10. pow2 N/A

           \[\leadsto \left(\left({M}^{2} \cdot \frac{{D}^{2} \cdot h}{\left(d \cdot d\right) \cdot \ell}\right) \cdot \frac{-1}{8}\right) \cdot w0 \]

       11. pow2 N/A

           \[\leadsto \left(\left({M}^{2} \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}\right) \cdot \frac{-1}{8}\right) \cdot w0 \]

       12. lower-*.f64 N/A

           \[\leadsto \left(\left({M}^{2} \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}\right) \cdot \frac{-1}{8}\right) \cdot w0 \]

       13. pow2 N/A

           \[\leadsto \left(\left(\left(M \cdot M\right) \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}\right) \cdot \frac{-1}{8}\right) \cdot w0 \]

       14. lift-*.f64 N/A

           \[\leadsto \left(\left(\left(M \cdot M\right) \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}\right) \cdot \frac{-1}{8}\right) \cdot w0 \]

       15. associate-/l* N/A

           \[\leadsto \left(\left(\left(M \cdot M\right) \cdot \left({D}^{2} \cdot \frac{h}{{d}^{2} \cdot \ell}\right)\right) \cdot \frac{-1}{8}\right) \cdot w0 \]

       16. lower-*.f64 N/A

           \[\leadsto \left(\left(\left(M \cdot M\right) \cdot \left({D}^{2} \cdot \frac{h}{{d}^{2} \cdot \ell}\right)\right) \cdot \frac{-1}{8}\right) \cdot w0 \]

       17. pow2 N/A

           \[\leadsto \left(\left(\left(M \cdot M\right) \cdot \left(\left(D \cdot D\right) \cdot \frac{h}{{d}^{2} \cdot \ell}\right)\right) \cdot \frac{-1}{8}\right) \cdot w0 \]

       18. lift-*.f64 N/A

           \[\leadsto \left(\left(\left(M \cdot M\right) \cdot \left(\left(D \cdot D\right) \cdot \frac{h}{{d}^{2} \cdot \ell}\right)\right) \cdot \frac{-1}{8}\right) \cdot w0 \]

       19. lower-/.f64 N/A

           \[\leadsto \left(\left(\left(M \cdot M\right) \cdot \left(\left(D \cdot D\right) \cdot \frac{h}{{d}^{2} \cdot \ell}\right)\right) \cdot \frac{-1}{8}\right) \cdot w0 \]

       20. pow2 N/A

           \[\leadsto \left(\left(\left(M \cdot M\right) \cdot \left(\left(D \cdot D\right) \cdot \frac{h}{\left(d \cdot d\right) \cdot \ell}\right)\right) \cdot \frac{-1}{8}\right) \cdot w0 \]

       21. lift-*.f64 N/A

           \[\leadsto \left(\left(\left(M \cdot M\right) \cdot \left(\left(D \cdot D\right) \cdot \frac{h}{\left(d \cdot d\right) \cdot \ell}\right)\right) \cdot \frac{-1}{8}\right) \cdot w0 \]

       22. lift-*.f64 43.4

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

   11. Applied rewrites 43.4%

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

   if -4.99999999999999976e270 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 89.7%

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

   2. Taylor expanded in M around 0

      \[\leadsto \color{blue}{w0} \]
   3. Step-by-step derivation

   4. Applied rewrites 88.6%

      \[\leadsto \color{blue}{w0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 68.0% accurate, 39.8× speedup

Math

\[\begin{array}{l} \\ w0 \end{array} \]

FPCore

(FPCore (w0 M D h l d) :precision binary64 w0)

C

double code(double w0, double M, double D, double h, double l, double d) {
	return w0;
}

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(w0, m, d, h, l, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    code = w0
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	return w0;
}

Python

def code(w0, M, D, h, l, d):
	return w0

Julia

function code(w0, M, D, h, l, d)
	return w0
end

MATLAB

function tmp = code(w0, M, D, h, l, d)
	tmp = w0;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := w0

Derivation

1. Initial program 81.4%

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

2. Taylor expanded in M around 0

   \[\leadsto \color{blue}{w0} \]
3. Step-by-step derivation

4. Applied rewrites 68.0%

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

Reproduce

herbie shell --seed 2025108 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  :precision binary64
  (* w0 (sqrt (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l))))))