Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.9% → 87.5%

Time: 6.0s

Alternatives: 8

Speedup: 0.7×

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: 80.9% 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: 87.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{M \cdot D}{2 \cdot d}\\ \mathbf{if}\;{t\_0}^{2} \cdot \frac{h}{\ell} \leq 5 \cdot 10^{-8}:\\ \;\;\;\;w0 \cdot \sqrt{1 - t\_0 \cdot \left(t\_0 \cdot \frac{h}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (/ (* M D) (* 2.0 d))))
   (if (<= (* (pow t_0 2.0) (/ h l)) 5e-8)
     (* w0 (sqrt (- 1.0 (* t_0 (* t_0 (/ h l))))))
     w0)))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = (M * D) / (2.0 * d);
	double tmp;
	if ((pow(t_0, 2.0) * (h / l)) <= 5e-8) {
		tmp = w0 * sqrt((1.0 - (t_0 * (t_0 * (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) :: t_0
    real(8) :: tmp
    t_0 = (m * d) / (2.0d0 * d_1)
    if (((t_0 ** 2.0d0) * (h / l)) <= 5d-8) then
        tmp = w0 * sqrt((1.0d0 - (t_0 * (t_0 * (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 t_0 = (M * D) / (2.0 * d);
	double tmp;
	if ((Math.pow(t_0, 2.0) * (h / l)) <= 5e-8) {
		tmp = w0 * Math.sqrt((1.0 - (t_0 * (t_0 * (h / l)))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	t_0 = (M * D) / (2.0 * d)
	tmp = 0
	if (math.pow(t_0, 2.0) * (h / l)) <= 5e-8:
		tmp = w0 * math.sqrt((1.0 - (t_0 * (t_0 * (h / l)))))
	else:
		tmp = w0
	return tmp

Julia

function code(w0, M, D, h, l, d)
	t_0 = Float64(Float64(M * D) / Float64(2.0 * d))
	tmp = 0.0
	if (Float64((t_0 ^ 2.0) * Float64(h / l)) <= 5e-8)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(t_0 * Float64(t_0 * Float64(h / l))))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	t_0 = (M * D) / (2.0 * d);
	tmp = 0.0;
	if (((t_0 ^ 2.0) * (h / l)) <= 5e-8)
		tmp = w0 * sqrt((1.0 - (t_0 * (t_0 * (h / l)))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := Block[{t$95$0 = N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Power[t$95$0, 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], 5e-8], N[(w0 * N[Sqrt[N[(1.0 - N[(t$95$0 * N[(t$95$0 * N[(h / l), $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)) < 4.9999999999999998e-8

   1. Initial program 87.9%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      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. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 87.5

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

   4. Applied rewrites 87.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. lift-/.f64 89.0

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

   6. Applied rewrites 89.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. *-commutative N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 88.0

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

   8. Applied rewrites 88.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. times-frac N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 89.3

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

   10. Applied rewrites 89.3%

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

   if 4.9999999999999998e-8 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 2.0%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 66.9%

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

Alternative 2: 80.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(-0.125, \frac{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right) \cdot h}{d \cdot \left(\ell \cdot d\right)}, 1\right)\\ \mathbf{elif}\;t\_0 \leq -2 \cdot 10^{-13}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{-0.25 \cdot \left(\left(M \cdot M\right) \cdot h\right)}{\left(d \cdot d\right) \cdot \ell}, D \cdot D, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l))))
   (if (<= t_0 (- INFINITY))
     (* w0 (fma -0.125 (/ (* (* (* D M) (* D M)) h) (* d (* l d))) 1.0))
     (if (<= t_0 -2e-13)
       (*
        w0
        (sqrt (fma (/ (* -0.25 (* (* M M) h)) (* (* d d) l)) (* D D) 1.0)))
       w0))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = pow(((M * D) / (2.0 * d)), 2.0) * (h / l);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = w0 * fma(-0.125, ((((D * M) * (D * M)) * h) / (d * (l * d))), 1.0);
	} else if (t_0 <= -2e-13) {
		tmp = w0 * sqrt(fma(((-0.25 * ((M * M) * h)) / ((d * d) * l)), (D * D), 1.0));
	} else {
		tmp = w0;
	}
	return tmp;
}

Julia

function code(w0, M, D, h, l, d)
	t_0 = Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(w0 * fma(-0.125, Float64(Float64(Float64(Float64(D * M) * Float64(D * M)) * h) / Float64(d * Float64(l * d))), 1.0));
	elseif (t_0 <= -2e-13)
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(-0.25 * Float64(Float64(M * M) * h)) / Float64(Float64(d * d) * l)), Float64(D * D), 1.0)));
	else
		tmp = w0;
	end
	return tmp
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := Block[{t$95$0 = N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(w0 * N[(-0.125 * N[(N[(N[(N[(D * M), $MachinePrecision] * N[(D * M), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] / N[(d * N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -2e-13], N[(w0 * N[Sqrt[N[(N[(N[(-0.25 * N[(N[(M * M), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[(D * D), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 55.0%

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

   3. Taylor expanded in M around 0

      \[\leadsto w0 \cdot \color{blue}{\left(1 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)} \]
   4. 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. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. pow-prod-down N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 50.2

          \[\leadsto w0 \cdot \mathsf{fma}\left(-0.125, \frac{{\left(D \cdot M\right)}^{2} \cdot h}{\left(d \cdot d\right) \cdot \ell}, 1\right) \]

   5. Applied rewrites 50.2%

      \[\leadsto w0 \cdot \color{blue}{\mathsf{fma}\left(-0.125, \frac{{\left(D \cdot M\right)}^{2} \cdot h}{\left(d \cdot d\right) \cdot \ell}, 1\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 50.2

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

   7. Applied rewrites 50.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 51.8

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

   9. Applied rewrites 51.8%

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

   if -inf.0 < (*.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 99.1%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      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. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 94.0

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

   4. Applied rewrites 94.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. lift-/.f64 94.0

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

   6. Applied rewrites 94.0%

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

   7. Taylor expanded in D around inf

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

   9. Applied rewrites 22.1%

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

   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 88.4%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 95.9%

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

Alternative 3: 80.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+300}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(-0.125, \frac{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right) \cdot h}{d \cdot \left(\ell \cdot d\right)}, 1\right)\\ \mathbf{elif}\;t\_0 \leq -5000000000000:\\ \;\;\;\;w0 \cdot \sqrt{\left(\left(\left(D \cdot D\right) \cdot \frac{h}{\left(d \cdot d\right) \cdot \ell}\right) \cdot -0.25\right) \cdot \left(M \cdot M\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l))))
   (if (<= t_0 -1e+300)
     (* w0 (fma -0.125 (/ (* (* (* D M) (* D M)) h) (* d (* l d))) 1.0))
     (if (<= t_0 -5000000000000.0)
       (* w0 (sqrt (* (* (* (* D D) (/ h (* (* d d) l))) -0.25) (* M M))))
       w0))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = pow(((M * D) / (2.0 * d)), 2.0) * (h / l);
	double tmp;
	if (t_0 <= -1e+300) {
		tmp = w0 * fma(-0.125, ((((D * M) * (D * M)) * h) / (d * (l * d))), 1.0);
	} else if (t_0 <= -5000000000000.0) {
		tmp = w0 * sqrt(((((D * D) * (h / ((d * d) * l))) * -0.25) * (M * M)));
	} else {
		tmp = w0;
	}
	return tmp;
}

Julia

function code(w0, M, D, h, l, d)
	t_0 = Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l))
	tmp = 0.0
	if (t_0 <= -1e+300)
		tmp = Float64(w0 * fma(-0.125, Float64(Float64(Float64(Float64(D * M) * Float64(D * M)) * h) / Float64(d * Float64(l * d))), 1.0));
	elseif (t_0 <= -5000000000000.0)
		tmp = Float64(w0 * sqrt(Float64(Float64(Float64(Float64(D * D) * Float64(h / Float64(Float64(d * d) * l))) * -0.25) * Float64(M * M))));
	else
		tmp = w0;
	end
	return tmp
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := Block[{t$95$0 = N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+300], N[(w0 * N[(-0.125 * N[(N[(N[(N[(D * M), $MachinePrecision] * N[(D * M), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] / N[(d * N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -5000000000000.0], N[(w0 * N[Sqrt[N[(N[(N[(N[(D * D), $MachinePrecision] * N[(h / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision] * N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -1.0000000000000001e300

   1. Initial program 55.1%

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

   3. Taylor expanded in M around 0

      \[\leadsto w0 \cdot \color{blue}{\left(1 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)} \]
   4. 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. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. pow-prod-down N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 50.1

          \[\leadsto w0 \cdot \mathsf{fma}\left(-0.125, \frac{{\left(D \cdot M\right)}^{2} \cdot h}{\left(d \cdot d\right) \cdot \ell}, 1\right) \]

   5. Applied rewrites 50.1%

      \[\leadsto w0 \cdot \color{blue}{\mathsf{fma}\left(-0.125, \frac{{\left(D \cdot M\right)}^{2} \cdot h}{\left(d \cdot d\right) \cdot \ell}, 1\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 50.1

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

   7. Applied rewrites 50.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 51.7

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

   9. Applied rewrites 51.7%

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

   if -1.0000000000000001e300 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -5e12

   1. Initial program 99.0%

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

   3. Taylor expanded in M around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. pow-flip N/A

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

      5. lower-pow.f64 N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

   5. Applied rewrites 21.4%

      \[\leadsto w0 \cdot \sqrt{\color{blue}{\left({M}^{-2} - \frac{0.25 \cdot \left(\left(D \cdot D\right) \cdot h\right)}{\left(d \cdot d\right) \cdot \ell}\right) \cdot \left(M \cdot M\right)}} \]

   6. Taylor expanded in M around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 21.1

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

   8. Applied rewrites 21.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 22.5

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

   10. Applied rewrites 22.5%

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

   if -5e12 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 88.5%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 95.4%

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

Alternative 4: 86.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{M \cdot D}{2 \cdot d}\\ \mathbf{if}\;{t\_0}^{2} \cdot \frac{h}{\ell} \leq 5 \cdot 10^{-8}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right) \cdot \left(t\_0 \cdot \frac{h}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (/ (* M D) (* 2.0 d))))
   (if (<= (* (pow t_0 2.0) (/ h l)) 5e-8)
     (* w0 (sqrt (- 1.0 (* (* (/ D d) (* 0.5 M)) (* t_0 (/ h l))))))
     w0)))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = (M * D) / (2.0 * d);
	double tmp;
	if ((pow(t_0, 2.0) * (h / l)) <= 5e-8) {
		tmp = w0 * sqrt((1.0 - (((D / d) * (0.5 * M)) * (t_0 * (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) :: t_0
    real(8) :: tmp
    t_0 = (m * d) / (2.0d0 * d_1)
    if (((t_0 ** 2.0d0) * (h / l)) <= 5d-8) then
        tmp = w0 * sqrt((1.0d0 - (((d / d_1) * (0.5d0 * m)) * (t_0 * (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 t_0 = (M * D) / (2.0 * d);
	double tmp;
	if ((Math.pow(t_0, 2.0) * (h / l)) <= 5e-8) {
		tmp = w0 * Math.sqrt((1.0 - (((D / d) * (0.5 * M)) * (t_0 * (h / l)))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	t_0 = (M * D) / (2.0 * d)
	tmp = 0
	if (math.pow(t_0, 2.0) * (h / l)) <= 5e-8:
		tmp = w0 * math.sqrt((1.0 - (((D / d) * (0.5 * M)) * (t_0 * (h / l)))))
	else:
		tmp = w0
	return tmp

Julia

function code(w0, M, D, h, l, d)
	t_0 = Float64(Float64(M * D) / Float64(2.0 * d))
	tmp = 0.0
	if (Float64((t_0 ^ 2.0) * Float64(h / l)) <= 5e-8)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(D / d) * Float64(0.5 * M)) * Float64(t_0 * Float64(h / l))))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	t_0 = (M * D) / (2.0 * d);
	tmp = 0.0;
	if (((t_0 ^ 2.0) * (h / l)) <= 5e-8)
		tmp = w0 * sqrt((1.0 - (((D / d) * (0.5 * M)) * (t_0 * (h / l)))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := Block[{t$95$0 = N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Power[t$95$0, 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], 5e-8], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(D / d), $MachinePrecision] * N[(0.5 * M), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(h / l), $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)) < 4.9999999999999998e-8

   1. Initial program 87.9%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      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. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 87.5

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

   4. Applied rewrites 87.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. lift-/.f64 89.0

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

   6. Applied rewrites 89.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. *-commutative N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 88.0

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

   8. Applied rewrites 88.0%

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

   9. Taylor expanded in M around 0

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

       1. lower-*.f64 88.0

          \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{D}{d} \cdot \left(0.5 \cdot \color{blue}{M}\right)\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)} \]

   11. Applied rewrites 88.0%

       \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{D}{d} \cdot \color{blue}{\left(0.5 \cdot M\right)}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)} \]

   if 4.9999999999999998e-8 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 2.0%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 66.9%

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

Alternative 5: 85.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{M \cdot D}{2 \cdot d}\\ \mathbf{if}\;{t\_0}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{-13}:\\ \;\;\;\;w0 \cdot \sqrt{1 - t\_0 \cdot \frac{0.5 \cdot \left(\left(h \cdot M\right) \cdot D\right)}{\ell \cdot d}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (/ (* M D) (* 2.0 d))))
   (if (<= (* (pow t_0 2.0) (/ h l)) -2e-13)
     (* w0 (sqrt (- 1.0 (* t_0 (/ (* 0.5 (* (* h M) D)) (* l d))))))
     w0)))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = (M * D) / (2.0 * d);
	double tmp;
	if ((pow(t_0, 2.0) * (h / l)) <= -2e-13) {
		tmp = w0 * sqrt((1.0 - (t_0 * ((0.5 * ((h * M) * D)) / (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) :: t_0
    real(8) :: tmp
    t_0 = (m * d) / (2.0d0 * d_1)
    if (((t_0 ** 2.0d0) * (h / l)) <= (-2d-13)) then
        tmp = w0 * sqrt((1.0d0 - (t_0 * ((0.5d0 * ((h * m) * d)) / (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 t_0 = (M * D) / (2.0 * d);
	double tmp;
	if ((Math.pow(t_0, 2.0) * (h / l)) <= -2e-13) {
		tmp = w0 * Math.sqrt((1.0 - (t_0 * ((0.5 * ((h * M) * D)) / (l * d)))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	t_0 = (M * D) / (2.0 * d)
	tmp = 0
	if (math.pow(t_0, 2.0) * (h / l)) <= -2e-13:
		tmp = w0 * math.sqrt((1.0 - (t_0 * ((0.5 * ((h * M) * D)) / (l * d)))))
	else:
		tmp = w0
	return tmp

Julia

function code(w0, M, D, h, l, d)
	t_0 = Float64(Float64(M * D) / Float64(2.0 * d))
	tmp = 0.0
	if (Float64((t_0 ^ 2.0) * Float64(h / l)) <= -2e-13)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(t_0 * Float64(Float64(0.5 * Float64(Float64(h * M) * D)) / Float64(l * d))))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	t_0 = (M * D) / (2.0 * d);
	tmp = 0.0;
	if (((t_0 ^ 2.0) * (h / l)) <= -2e-13)
		tmp = w0 * sqrt((1.0 - (t_0 * ((0.5 * ((h * M) * D)) / (l * d)))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := Block[{t$95$0 = N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Power[t$95$0, 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -2e-13], N[(w0 * N[Sqrt[N[(1.0 - N[(t$95$0 * N[(N[(0.5 * N[(N[(h * M), $MachinePrecision] * D), $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)) < -2.0000000000000001e-13

   1. Initial program 64.3%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      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. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 64.6

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

   4. Applied rewrites 64.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. lift-/.f64 68.8

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

   6. Applied rewrites 68.8%

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

   7. Taylor expanded in M around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 61.9

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{D}{d} \cdot \frac{M}{2}\right) \cdot \frac{0.5 \cdot \left(\left(h \cdot M\right) \cdot D\right)}{\ell \cdot \color{blue}{d}}} \]

   9. Applied rewrites 61.9%

      \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{D}{d} \cdot \frac{M}{2}\right) \cdot \color{blue}{\frac{0.5 \cdot \left(\left(h \cdot M\right) \cdot D\right)}{\ell \cdot d}}} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. *-commutative N/A

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

       4. lift-/.f64 N/A

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

       5. times-frac N/A

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

       6. lift-*.f64 N/A

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

       7. lift-/.f64 N/A

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

       8. lift-*.f64 61.9

          \[\leadsto w0 \cdot \sqrt{1 - \frac{M \cdot D}{\color{blue}{2 \cdot d}} \cdot \frac{0.5 \cdot \left(\left(h \cdot M\right) \cdot D\right)}{\ell \cdot d}} \]

   11. Applied rewrites 61.9%

       \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \frac{0.5 \cdot \left(\left(h \cdot M\right) \cdot D\right)}{\ell \cdot d}} \]

   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 88.4%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 95.9%

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

Alternative 6: 85.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{-13}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right) \cdot \frac{0.5 \cdot \left(\left(h \cdot M\right) \cdot D\right)}{\ell \cdot d}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -2e-13)
   (*
    w0
    (sqrt (- 1.0 (* (* (/ D d) (* 0.5 M)) (/ (* 0.5 (* (* h M) D)) (* 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)) <= -2e-13) {
		tmp = w0 * sqrt((1.0 - (((D / d) * (0.5 * M)) * ((0.5 * ((h * M) * D)) / (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)) <= (-2d-13)) then
        tmp = w0 * sqrt((1.0d0 - (((d / d_1) * (0.5d0 * m)) * ((0.5d0 * ((h * m) * d)) / (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)) <= -2e-13) {
		tmp = w0 * Math.sqrt((1.0 - (((D / d) * (0.5 * M)) * ((0.5 * ((h * M) * D)) / (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)) <= -2e-13:
		tmp = w0 * math.sqrt((1.0 - (((D / d) * (0.5 * M)) * ((0.5 * ((h * M) * D)) / (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)) <= -2e-13)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(D / d) * Float64(0.5 * M)) * Float64(Float64(0.5 * Float64(Float64(h * M) * D)) / 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)) <= -2e-13)
		tmp = w0 * sqrt((1.0 - (((D / d) * (0.5 * M)) * ((0.5 * ((h * M) * D)) / (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], -2e-13], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(D / d), $MachinePrecision] * N[(0.5 * M), $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 * N[(N[(h * M), $MachinePrecision] * D), $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)) < -2.0000000000000001e-13

   1. Initial program 64.3%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      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. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 64.6

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

   4. Applied rewrites 64.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. lift-/.f64 68.8

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

   6. Applied rewrites 68.8%

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

   7. Taylor expanded in M around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 61.9

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{D}{d} \cdot \frac{M}{2}\right) \cdot \frac{0.5 \cdot \left(\left(h \cdot M\right) \cdot D\right)}{\ell \cdot \color{blue}{d}}} \]

   9. Applied rewrites 61.9%

      \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{D}{d} \cdot \frac{M}{2}\right) \cdot \color{blue}{\frac{0.5 \cdot \left(\left(h \cdot M\right) \cdot D\right)}{\ell \cdot d}}} \]

   10. Taylor expanded in M around 0

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

       1. lower-*.f64 61.9

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

   12. Applied rewrites 61.9%

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

   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 88.4%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 95.9%

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

Alternative 7: 79.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(M \cdot D\right) \cdot \left(\left(M \cdot D\right) \cdot \left(h \cdot w0\right)\right)}{\left(d \cdot d\right) \cdot \ell}, -0.125, w0\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -2e-13)
   (fma (/ (* (* M D) (* (* M D) (* h w0))) (* (* 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)) <= -2e-13) {
		tmp = fma((((M * D) * ((M * D) * (h * w0))) / ((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)) <= -2e-13)
		tmp = fma(Float64(Float64(Float64(M * D) * Float64(Float64(M * D) * Float64(h * w0))) / Float64(Float64(d * d) * l)), -0.125, 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], -2e-13], N[(N[(N[(N[(M * D), $MachinePrecision] * N[(N[(M * D), $MachinePrecision] * N[(h * w0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * -0.125 + 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 64.3%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. pow-prod-down N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 41.2

          \[\leadsto \mathsf{fma}\left(\frac{{\left(D \cdot M\right)}^{2} \cdot \left(h \cdot w0\right)}{\left(d \cdot d\right) \cdot \ell}, -0.125, w0\right) \]

   5. Applied rewrites 41.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(D \cdot M\right)}^{2} \cdot \left(h \cdot w0\right)}{\left(d \cdot d\right) \cdot \ell}, -0.125, w0\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 41.2

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

   7. Applied rewrites 41.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 42.4

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

   9. Applied rewrites 42.4%

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

   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 88.4%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 95.9%

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

Alternative 8: 68.0% accurate, 157.0× 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 80.9%

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

3. Taylor expanded in M around 0

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

5. Applied rewrites 68.0%

   \[\leadsto \color{blue}{w0} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2025087 
(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))))))