Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.9% → 87.7%

Time: 6.6s

Alternatives: 11

Speedup: 1.8×

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: 87.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ w0\_s \cdot \begin{array}{l} \mathbf{if}\;w0\_m \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \leq 10^{+246}:\\ \;\;\;\;w0\_m \cdot \sqrt{1 - {\left(\frac{M \cdot D}{d + d}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0\_m \cdot \sqrt{1 - \frac{\frac{M}{2} \cdot \left(\left(\left(\frac{D}{d} \cdot \frac{M}{2}\right) \cdot \frac{D}{d}\right) \cdot h\right)}{\ell}}\\ \end{array} \end{array} \]

FPCore

w0\_m = (fabs.f64 w0)
w0\_s = (copysign.f64 #s(literal 1 binary64) w0)
(FPCore (w0_s w0_m M D h l d)
 :precision binary64
 (*
  w0_s
  (if (<=
       (* w0_m (sqrt (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)))))
       1e+246)
    (* w0_m (sqrt (- 1.0 (* (pow (/ (* M D) (+ d d)) 2.0) (/ h l)))))
    (*
     w0_m
     (sqrt
      (- 1.0 (/ (* (/ M 2.0) (* (* (* (/ D d) (/ M 2.0)) (/ D d)) h)) l)))))))

C

w0\_m = fabs(w0);
w0\_s = copysign(1.0, w0);
double code(double w0_s, double w0_m, double M, double D, double h, double l, double d) {
	double tmp;
	if ((w0_m * sqrt((1.0 - (pow(((M * D) / (2.0 * d)), 2.0) * (h / l))))) <= 1e+246) {
		tmp = w0_m * sqrt((1.0 - (pow(((M * D) / (d + d)), 2.0) * (h / l))));
	} else {
		tmp = w0_m * sqrt((1.0 - (((M / 2.0) * ((((D / d) * (M / 2.0)) * (D / d)) * h)) / l)));
	}
	return w0_s * tmp;
}

Fortran

w0\_m =     private
w0\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(w0_s, w0_m, m, d, h, l, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: w0_s
    real(8), intent (in) :: w0_m
    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 ((w0_m * sqrt((1.0d0 - ((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l))))) <= 1d+246) then
        tmp = w0_m * sqrt((1.0d0 - ((((m * d) / (d_1 + d_1)) ** 2.0d0) * (h / l))))
    else
        tmp = w0_m * sqrt((1.0d0 - (((m / 2.0d0) * ((((d / d_1) * (m / 2.0d0)) * (d / d_1)) * h)) / l)))
    end if
    code = w0_s * tmp
end function

Java

w0\_m = Math.abs(w0);
w0\_s = Math.copySign(1.0, w0);
public static double code(double w0_s, double w0_m, double M, double D, double h, double l, double d) {
	double tmp;
	if ((w0_m * Math.sqrt((1.0 - (Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l))))) <= 1e+246) {
		tmp = w0_m * Math.sqrt((1.0 - (Math.pow(((M * D) / (d + d)), 2.0) * (h / l))));
	} else {
		tmp = w0_m * Math.sqrt((1.0 - (((M / 2.0) * ((((D / d) * (M / 2.0)) * (D / d)) * h)) / l)));
	}
	return w0_s * tmp;
}

Python

w0\_m = math.fabs(w0)
w0\_s = math.copysign(1.0, w0)
def code(w0_s, w0_m, M, D, h, l, d):
	tmp = 0
	if (w0_m * math.sqrt((1.0 - (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l))))) <= 1e+246:
		tmp = w0_m * math.sqrt((1.0 - (math.pow(((M * D) / (d + d)), 2.0) * (h / l))))
	else:
		tmp = w0_m * math.sqrt((1.0 - (((M / 2.0) * ((((D / d) * (M / 2.0)) * (D / d)) * h)) / l)))
	return w0_s * tmp

Julia

w0\_m = abs(w0)
w0\_s = copysign(1.0, w0)
function code(w0_s, w0_m, M, D, h, l, d)
	tmp = 0.0
	if (Float64(w0_m * sqrt(Float64(1.0 - Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l))))) <= 1e+246)
		tmp = Float64(w0_m * sqrt(Float64(1.0 - Float64((Float64(Float64(M * D) / Float64(d + d)) ^ 2.0) * Float64(h / l)))));
	else
		tmp = Float64(w0_m * sqrt(Float64(1.0 - Float64(Float64(Float64(M / 2.0) * Float64(Float64(Float64(Float64(D / d) * Float64(M / 2.0)) * Float64(D / d)) * h)) / l))));
	end
	return Float64(w0_s * tmp)
end

MATLAB

w0\_m = abs(w0);
w0\_s = sign(w0) * abs(1.0);
function tmp_2 = code(w0_s, w0_m, M, D, h, l, d)
	tmp = 0.0;
	if ((w0_m * sqrt((1.0 - ((((M * D) / (2.0 * d)) ^ 2.0) * (h / l))))) <= 1e+246)
		tmp = w0_m * sqrt((1.0 - ((((M * D) / (d + d)) ^ 2.0) * (h / l))));
	else
		tmp = w0_m * sqrt((1.0 - (((M / 2.0) * ((((D / d) * (M / 2.0)) * (D / d)) * h)) / l)));
	end
	tmp_2 = w0_s * tmp;
end

Wolfram

w0\_m = N[Abs[w0], $MachinePrecision]
w0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[w0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[w0$95$s_, w0$95$m_, M_, D_, h_, l_, d_] := N[(w0$95$s * If[LessEqual[N[(w0$95$m * 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], 1e+246], N[(w0$95$m * N[Sqrt[N[(1.0 - N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(d + d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0$95$m * N[Sqrt[N[(1.0 - N[(N[(N[(M / 2.0), $MachinePrecision] * N[(N[(N[(N[(D / d), $MachinePrecision] * N[(M / 2.0), $MachinePrecision]), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 w0 (sqrt.f64 (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))))) < 1.00000000000000007e246

   1. Initial program 91.8%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 91.8

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

   4. Applied rewrites 91.8%

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

   if 1.00000000000000007e246 < (*.f64 w0 (sqrt.f64 (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)))))

   1. Initial program 54.9%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 78.8

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

   4. Applied rewrites 78.8%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 78.8

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-/.f64 78.8

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

   6. Applied rewrites 78.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 81.8

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 81.8

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

   8. Applied rewrites 81.8%

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

Alternative 2: 86.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ w0\_s \cdot \begin{array}{l} \mathbf{if}\;w0\_m \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \leq \infty:\\ \;\;\;\;w0\_m \cdot \sqrt{1 - \left(M \cdot \frac{D}{2 \cdot d}\right) \cdot \left(\left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right) \cdot \frac{h}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot w0\_m\right)}{\ell}, -0.125, w0\_m\right)\\ \end{array} \end{array} \]

FPCore

w0\_m = (fabs.f64 w0)
w0\_s = (copysign.f64 #s(literal 1 binary64) w0)
(FPCore (w0_s w0_m M D h l d)
 :precision binary64
 (*
  w0_s
  (if (<=
       (* w0_m (sqrt (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)))))
       INFINITY)
    (*
     w0_m
     (sqrt
      (- 1.0 (* (* M (/ D (* 2.0 d))) (* (* (/ D d) (* 0.5 M)) (/ h l))))))
    (fma (/ (* (* (/ D d) (/ D d)) (* (* (* M M) h) w0_m)) l) -0.125 w0_m))))

C

w0\_m = fabs(w0);
w0\_s = copysign(1.0, w0);
double code(double w0_s, double w0_m, double M, double D, double h, double l, double d) {
	double tmp;
	if ((w0_m * sqrt((1.0 - (pow(((M * D) / (2.0 * d)), 2.0) * (h / l))))) <= ((double) INFINITY)) {
		tmp = w0_m * sqrt((1.0 - ((M * (D / (2.0 * d))) * (((D / d) * (0.5 * M)) * (h / l)))));
	} else {
		tmp = fma(((((D / d) * (D / d)) * (((M * M) * h) * w0_m)) / l), -0.125, w0_m);
	}
	return w0_s * tmp;
}

Julia

w0\_m = abs(w0)
w0\_s = copysign(1.0, w0)
function code(w0_s, w0_m, M, D, h, l, d)
	tmp = 0.0
	if (Float64(w0_m * sqrt(Float64(1.0 - Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l))))) <= Inf)
		tmp = Float64(w0_m * sqrt(Float64(1.0 - Float64(Float64(M * Float64(D / Float64(2.0 * d))) * Float64(Float64(Float64(D / d) * Float64(0.5 * M)) * Float64(h / l))))));
	else
		tmp = fma(Float64(Float64(Float64(Float64(D / d) * Float64(D / d)) * Float64(Float64(Float64(M * M) * h) * w0_m)) / l), -0.125, w0_m);
	end
	return Float64(w0_s * tmp)
end

Wolfram

w0\_m = N[Abs[w0], $MachinePrecision]
w0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[w0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[w0$95$s_, w0$95$m_, M_, D_, h_, l_, d_] := N[(w0$95$s * If[LessEqual[N[(w0$95$m * 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], Infinity], N[(w0$95$m * N[Sqrt[N[(1.0 - N[(N[(M * N[(D / N[(2.0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(D / d), $MachinePrecision] * N[(0.5 * M), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M * M), $MachinePrecision] * h), $MachinePrecision] * w0$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * -0.125 + w0$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 w0 (sqrt.f64 (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))))) < +inf.0

   1. Initial program 88.2%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 86.4

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

   4. Applied rewrites 86.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. associate-*r/ N/A

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

      9. frac-times N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. associate-*l* N/A

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

      15. 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. Applied rewrites 89.6%

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

   7. Taylor expanded in M around 0

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

      1. lower-*.f64 89.6

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

   9. Applied rewrites 89.6%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. *-commutative N/A

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

       4. lift-/.f64 N/A

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

       5. times-frac N/A

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

       6. associate-/l* N/A

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

       7. lower-*.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. lower-*.f64 89.6

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

   11. Applied rewrites 89.6%

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

   if +inf.0 < (*.f64 w0 (sqrt.f64 (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)))))

   1. Initial program 0.0%

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

   3. Taylor expanded in M around 0

      \[\leadsto \color{blue}{w0 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}} \]
   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 44.7

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

   5. Applied rewrites 44.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(D \cdot M\right)}^{2} \cdot \left(h \cdot w0\right)}{\left(d \cdot d\right) \cdot \ell}, -0.125, w0\right)} \]
   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. lift-*.f64 N/A

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

      6. lift-*.f64 44.7

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

   7. Applied rewrites 44.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.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. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. pow2 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. unpow-prod-down N/A

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

      12. associate-*r* N/A

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

      13. associate-*r* N/A

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

      14. pow2 N/A

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

      15. frac-times N/A

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

   9. Applied rewrites 68.8%

      \[\leadsto \mathsf{fma}\left(\frac{\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot w0\right)}{\ell}, -0.125, w0\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 85.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ w0\_s \cdot \begin{array}{l} \mathbf{if}\;\sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \leq 1:\\ \;\;\;\;w0\_m\\ \mathbf{else}:\\ \;\;\;\;w0\_m \cdot \sqrt{1 - \left(M \cdot \frac{D}{2 \cdot d}\right) \cdot \frac{0.5 \cdot \left(\left(h \cdot M\right) \cdot D\right)}{\ell \cdot d}}\\ \end{array} \end{array} \]

FPCore

w0\_m = (fabs.f64 w0)
w0\_s = (copysign.f64 #s(literal 1 binary64) w0)
(FPCore (w0_s w0_m M D h l d)
 :precision binary64
 (*
  w0_s
  (if (<= (sqrt (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)))) 1.0)
    w0_m
    (*
     w0_m
     (sqrt
      (- 1.0 (* (* M (/ D (* 2.0 d))) (/ (* 0.5 (* (* h M) D)) (* l d)))))))))

C

w0\_m = fabs(w0);
w0\_s = copysign(1.0, w0);
double code(double w0_s, double w0_m, double M, double D, double h, double l, double d) {
	double tmp;
	if (sqrt((1.0 - (pow(((M * D) / (2.0 * d)), 2.0) * (h / l)))) <= 1.0) {
		tmp = w0_m;
	} else {
		tmp = w0_m * sqrt((1.0 - ((M * (D / (2.0 * d))) * ((0.5 * ((h * M) * D)) / (l * d)))));
	}
	return w0_s * tmp;
}

Fortran

w0\_m =     private
w0\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(w0_s, w0_m, m, d, h, l, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: w0_s
    real(8), intent (in) :: w0_m
    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 (sqrt((1.0d0 - ((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l)))) <= 1.0d0) then
        tmp = w0_m
    else
        tmp = w0_m * sqrt((1.0d0 - ((m * (d / (2.0d0 * d_1))) * ((0.5d0 * ((h * m) * d)) / (l * d_1)))))
    end if
    code = w0_s * tmp
end function

Java

w0\_m = Math.abs(w0);
w0\_s = Math.copySign(1.0, w0);
public static double code(double w0_s, double w0_m, double M, double D, double h, double l, double d) {
	double tmp;
	if (Math.sqrt((1.0 - (Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)))) <= 1.0) {
		tmp = w0_m;
	} else {
		tmp = w0_m * Math.sqrt((1.0 - ((M * (D / (2.0 * d))) * ((0.5 * ((h * M) * D)) / (l * d)))));
	}
	return w0_s * tmp;
}

Python

w0\_m = math.fabs(w0)
w0\_s = math.copysign(1.0, w0)
def code(w0_s, w0_m, M, D, h, l, d):
	tmp = 0
	if math.sqrt((1.0 - (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)))) <= 1.0:
		tmp = w0_m
	else:
		tmp = w0_m * math.sqrt((1.0 - ((M * (D / (2.0 * d))) * ((0.5 * ((h * M) * D)) / (l * d)))))
	return w0_s * tmp

Julia

w0\_m = abs(w0)
w0\_s = copysign(1.0, w0)
function code(w0_s, w0_m, M, D, h, l, d)
	tmp = 0.0
	if (sqrt(Float64(1.0 - Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)))) <= 1.0)
		tmp = w0_m;
	else
		tmp = Float64(w0_m * sqrt(Float64(1.0 - Float64(Float64(M * Float64(D / Float64(2.0 * d))) * Float64(Float64(0.5 * Float64(Float64(h * M) * D)) / Float64(l * d))))));
	end
	return Float64(w0_s * tmp)
end

MATLAB

w0\_m = abs(w0);
w0\_s = sign(w0) * abs(1.0);
function tmp_2 = code(w0_s, w0_m, M, D, h, l, d)
	tmp = 0.0;
	if (sqrt((1.0 - ((((M * D) / (2.0 * d)) ^ 2.0) * (h / l)))) <= 1.0)
		tmp = w0_m;
	else
		tmp = w0_m * sqrt((1.0 - ((M * (D / (2.0 * d))) * ((0.5 * ((h * M) * D)) / (l * d)))));
	end
	tmp_2 = w0_s * tmp;
end

Wolfram

w0\_m = N[Abs[w0], $MachinePrecision]
w0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[w0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[w0$95$s_, w0$95$m_, M_, D_, h_, l_, d_] := N[(w0$95$s * If[LessEqual[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], 1.0], w0$95$m, N[(w0$95$m * N[Sqrt[N[(1.0 - N[(N[(M * N[(D / N[(2.0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 * N[(N[(h * M), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.7%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 99.7%

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

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

   1. Initial program 58.7%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 67.5

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

   4. Applied rewrites 67.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. associate-*r/ N/A

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

      9. frac-times N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. associate-*l* N/A

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

      15. 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. Applied rewrites 63.3%

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

   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 64.5

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

   9. Applied rewrites 64.5%

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

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

       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

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

       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. associate-/l* N/A

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

       7. lower-*.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. lower-*.f64 64.5

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

   11. Applied rewrites 64.5%

       \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)} \cdot \frac{0.5 \cdot \left(\left(h \cdot M\right) \cdot D\right)}{\ell \cdot d}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 77.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ w0\_s \cdot \begin{array}{l} \mathbf{if}\;\sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \leq 1:\\ \;\;\;\;w0\_m\\ \mathbf{else}:\\ \;\;\;\;w0\_m \cdot \mathsf{fma}\left(-0.125, \frac{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right) \cdot h}{\left(d \cdot d\right) \cdot \ell}, 1\right)\\ \end{array} \end{array} \]

FPCore

w0\_m = (fabs.f64 w0)
w0\_s = (copysign.f64 #s(literal 1 binary64) w0)
(FPCore (w0_s w0_m M D h l d)
 :precision binary64
 (*
  w0_s
  (if (<= (sqrt (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)))) 1.0)
    w0_m
    (* w0_m (fma -0.125 (/ (* (* (* D M) (* D M)) h) (* (* d d) l)) 1.0)))))

C

w0\_m = fabs(w0);
w0\_s = copysign(1.0, w0);
double code(double w0_s, double w0_m, double M, double D, double h, double l, double d) {
	double tmp;
	if (sqrt((1.0 - (pow(((M * D) / (2.0 * d)), 2.0) * (h / l)))) <= 1.0) {
		tmp = w0_m;
	} else {
		tmp = w0_m * fma(-0.125, ((((D * M) * (D * M)) * h) / ((d * d) * l)), 1.0);
	}
	return w0_s * tmp;
}

Julia

w0\_m = abs(w0)
w0\_s = copysign(1.0, w0)
function code(w0_s, w0_m, M, D, h, l, d)
	tmp = 0.0
	if (sqrt(Float64(1.0 - Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)))) <= 1.0)
		tmp = w0_m;
	else
		tmp = Float64(w0_m * fma(-0.125, Float64(Float64(Float64(Float64(D * M) * Float64(D * M)) * h) / Float64(Float64(d * d) * l)), 1.0));
	end
	return Float64(w0_s * tmp)
end

Wolfram

w0\_m = N[Abs[w0], $MachinePrecision]
w0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[w0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[w0$95$s_, w0$95$m_, M_, D_, h_, l_, d_] := N[(w0$95$s * If[LessEqual[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], 1.0], w0$95$m, N[(w0$95$m * N[(-0.125 * N[(N[(N[(N[(D * M), $MachinePrecision] * N[(D * M), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.7%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 99.7%

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

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

   1. Initial program 58.7%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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 52.2

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

   5. Applied rewrites 52.2%

      \[\leadsto w0 \cdot \color{blue}{\mathsf{fma}\left(-0.125, \frac{{\left(D \cdot M\right)}^{2} \cdot h}{\left(d \cdot d\right) \cdot \ell}, 1\right)} \]
   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 52.2

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

   7. Applied rewrites 52.2%

      \[\leadsto w0 \cdot \mathsf{fma}\left(-0.125, \frac{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right) \cdot h}{\left(\color{blue}{d} \cdot d\right) \cdot \ell}, 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 81.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ w0\_s \cdot \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+51}:\\ \;\;\;\;w0\_m \cdot \sqrt{-0.25 \cdot \frac{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot h}{d \cdot \left(\ell \cdot d\right)}}\\ \mathbf{else}:\\ \;\;\;\;w0\_m\\ \end{array} \end{array} \]

FPCore

w0\_m = (fabs.f64 w0)
w0\_s = (copysign.f64 #s(literal 1 binary64) w0)
(FPCore (w0_s w0_m M D h l d)
 :precision binary64
 (*
  w0_s
  (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -5e+51)
    (* w0_m (sqrt (* -0.25 (/ (* (* (* M D) (* M D)) h) (* d (* l d))))))
    w0_m)))

C

w0\_m = fabs(w0);
w0\_s = copysign(1.0, w0);
double code(double w0_s, double w0_m, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -5e+51) {
		tmp = w0_m * sqrt((-0.25 * ((((M * D) * (M * D)) * h) / (d * (l * d)))));
	} else {
		tmp = w0_m;
	}
	return w0_s * tmp;
}

Fortran

w0\_m =     private
w0\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(w0_s, w0_m, m, d, h, l, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: w0_s
    real(8), intent (in) :: w0_m
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l)) <= (-5d+51)) then
        tmp = w0_m * sqrt(((-0.25d0) * ((((m * d) * (m * d)) * h) / (d_1 * (l * d_1)))))
    else
        tmp = w0_m
    end if
    code = w0_s * tmp
end function

Java

w0\_m = Math.abs(w0);
w0\_s = Math.copySign(1.0, w0);
public static double code(double w0_s, double w0_m, double M, double D, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -5e+51) {
		tmp = w0_m * Math.sqrt((-0.25 * ((((M * D) * (M * D)) * h) / (d * (l * d)))));
	} else {
		tmp = w0_m;
	}
	return w0_s * tmp;
}

Python

w0\_m = math.fabs(w0)
w0\_s = math.copysign(1.0, w0)
def code(w0_s, w0_m, M, D, h, l, d):
	tmp = 0
	if (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -5e+51:
		tmp = w0_m * math.sqrt((-0.25 * ((((M * D) * (M * D)) * h) / (d * (l * d)))))
	else:
		tmp = w0_m
	return w0_s * tmp

Julia

w0\_m = abs(w0)
w0\_s = copysign(1.0, w0)
function code(w0_s, w0_m, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -5e+51)
		tmp = Float64(w0_m * sqrt(Float64(-0.25 * Float64(Float64(Float64(Float64(M * D) * Float64(M * D)) * h) / Float64(d * Float64(l * d))))));
	else
		tmp = w0_m;
	end
	return Float64(w0_s * tmp)
end

MATLAB

w0\_m = abs(w0);
w0\_s = sign(w0) * abs(1.0);
function tmp_2 = code(w0_s, w0_m, M, D, h, l, d)
	tmp = 0.0;
	if (((((M * D) / (2.0 * d)) ^ 2.0) * (h / l)) <= -5e+51)
		tmp = w0_m * sqrt((-0.25 * ((((M * D) * (M * D)) * h) / (d * (l * d)))));
	else
		tmp = w0_m;
	end
	tmp_2 = w0_s * tmp;
end

Wolfram

w0\_m = N[Abs[w0], $MachinePrecision]
w0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[w0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[w0$95$s_, w0$95$m_, M_, D_, h_, l_, d_] := N[(w0$95$s * If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -5e+51], N[(w0$95$m * N[Sqrt[N[(-0.25 * N[(N[(N[(N[(M * D), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] / N[(d * N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0$95$m]), $MachinePrecision]

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)) < -5e51

   1. Initial program 68.4%

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

   3. Taylor expanded in M around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. pow-prod-down N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 56.8

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

   5. Applied rewrites 56.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 58.1

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

   7. Applied rewrites 58.1%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 58.1

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

   9. Applied rewrites 58.1%

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

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

   1. Initial program 90.2%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 96.3%

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

Alternative 6: 79.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ w0\_s \cdot \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\frac{M \cdot D}{d \cdot d} \cdot \frac{\left(\left(h \cdot M\right) \cdot D\right) \cdot w0\_m}{\ell}, -0.125, w0\_m\right)\\ \mathbf{else}:\\ \;\;\;\;w0\_m\\ \end{array} \end{array} \]

FPCore

w0\_m = (fabs.f64 w0)
w0\_s = (copysign.f64 #s(literal 1 binary64) w0)
(FPCore (w0_s w0_m M D h l d)
 :precision binary64
 (*
  w0_s
  (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -2e-6)
    (fma (* (/ (* M D) (* d d)) (/ (* (* (* h M) D) w0_m) l)) -0.125 w0_m)
    w0_m)))

C

w0\_m = fabs(w0);
w0\_s = copysign(1.0, w0);
double code(double w0_s, double w0_m, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -2e-6) {
		tmp = fma((((M * D) / (d * d)) * ((((h * M) * D) * w0_m) / l)), -0.125, w0_m);
	} else {
		tmp = w0_m;
	}
	return w0_s * tmp;
}

Julia

w0\_m = abs(w0)
w0\_s = copysign(1.0, w0)
function code(w0_s, w0_m, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -2e-6)
		tmp = fma(Float64(Float64(Float64(M * D) / Float64(d * d)) * Float64(Float64(Float64(Float64(h * M) * D) * w0_m) / l)), -0.125, w0_m);
	else
		tmp = w0_m;
	end
	return Float64(w0_s * tmp)
end

Wolfram

w0\_m = N[Abs[w0], $MachinePrecision]
w0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[w0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[w0$95$s_, w0$95$m_, M_, D_, h_, l_, d_] := N[(w0$95$s * 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-6], N[(N[(N[(N[(M * D), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(h * M), $MachinePrecision] * D), $MachinePrecision] * w0$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * -0.125 + w0$95$m), $MachinePrecision], w0$95$m]), $MachinePrecision]

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)) < -1.99999999999999991e-6

   1. Initial program 68.7%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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 47.2

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

   5. Applied rewrites 47.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(D \cdot M\right)}^{2} \cdot \left(h \cdot w0\right)}{\left(d \cdot d\right) \cdot \ell}, -0.125, w0\right)} \]
   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. lift-*.f64 N/A

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

      6. lift-*.f64 47.2

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

   7. Applied rewrites 47.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.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. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 47.4

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

   9. Applied rewrites 47.4%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. pow2 N/A

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

       7. times-frac N/A

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

       8. lower-*.f64 N/A

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

       9. lower-/.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 N/A

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

       12. pow2 N/A

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

       13. lift-*.f64 N/A

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

       14. lower-/.f64 49.9

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

   11. Applied rewrites 50.2%

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

   if -1.99999999999999991e-6 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 90.2%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 96.6%

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

Alternative 7: 79.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ w0\_s \cdot \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -1 \cdot 10^{+124}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(D \cdot M\right) \cdot \left(\left(D \cdot M\right) \cdot \left(h \cdot w0\_m\right)\right)}{\left(\ell \cdot d\right) \cdot d}, -0.125, w0\_m\right)\\ \mathbf{else}:\\ \;\;\;\;w0\_m\\ \end{array} \end{array} \]

FPCore

w0\_m = (fabs.f64 w0)
w0\_s = (copysign.f64 #s(literal 1 binary64) w0)
(FPCore (w0_s w0_m M D h l d)
 :precision binary64
 (*
  w0_s
  (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -1e+124)
    (fma (/ (* (* D M) (* (* D M) (* h w0_m))) (* (* l d) d)) -0.125 w0_m)
    w0_m)))

C

w0\_m = fabs(w0);
w0\_s = copysign(1.0, w0);
double code(double w0_s, double w0_m, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -1e+124) {
		tmp = fma((((D * M) * ((D * M) * (h * w0_m))) / ((l * d) * d)), -0.125, w0_m);
	} else {
		tmp = w0_m;
	}
	return w0_s * tmp;
}

Julia

w0\_m = abs(w0)
w0\_s = copysign(1.0, w0)
function code(w0_s, w0_m, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -1e+124)
		tmp = fma(Float64(Float64(Float64(D * M) * Float64(Float64(D * M) * Float64(h * w0_m))) / Float64(Float64(l * d) * d)), -0.125, w0_m);
	else
		tmp = w0_m;
	end
	return Float64(w0_s * tmp)
end

Wolfram

w0\_m = N[Abs[w0], $MachinePrecision]
w0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[w0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[w0$95$s_, w0$95$m_, M_, D_, h_, l_, d_] := N[(w0$95$s * If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -1e+124], N[(N[(N[(N[(D * M), $MachinePrecision] * N[(N[(D * M), $MachinePrecision] * N[(h * w0$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(l * d), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] * -0.125 + w0$95$m), $MachinePrecision], w0$95$m]), $MachinePrecision]

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)) < -9.99999999999999948e123

   1. Initial program 66.2%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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 50.4

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

   5. Applied rewrites 50.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(D \cdot M\right)}^{2} \cdot \left(h \cdot w0\right)}{\left(d \cdot d\right) \cdot \ell}, -0.125, w0\right)} \]
   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. lift-*.f64 N/A

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

      6. lift-*.f64 50.4

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

   7. Applied rewrites 50.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.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. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 50.6

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

   9. Applied rewrites 50.6%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 50.8

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

   11. Applied rewrites 50.8%

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

   if -9.99999999999999948e123 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 90.5%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 93.3%

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

Alternative 8: 78.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ w0\_s \cdot \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -1 \cdot 10^{+124}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(D \cdot M\right) \cdot \left(\left(D \cdot M\right) \cdot \left(h \cdot w0\_m\right)\right)}{\left(d \cdot d\right) \cdot \ell}, -0.125, w0\_m\right)\\ \mathbf{else}:\\ \;\;\;\;w0\_m\\ \end{array} \end{array} \]

FPCore

w0\_m = (fabs.f64 w0)
w0\_s = (copysign.f64 #s(literal 1 binary64) w0)
(FPCore (w0_s w0_m M D h l d)
 :precision binary64
 (*
  w0_s
  (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -1e+124)
    (fma (/ (* (* D M) (* (* D M) (* h w0_m))) (* (* d d) l)) -0.125 w0_m)
    w0_m)))

C

w0\_m = fabs(w0);
w0\_s = copysign(1.0, w0);
double code(double w0_s, double w0_m, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -1e+124) {
		tmp = fma((((D * M) * ((D * M) * (h * w0_m))) / ((d * d) * l)), -0.125, w0_m);
	} else {
		tmp = w0_m;
	}
	return w0_s * tmp;
}

Julia

w0\_m = abs(w0)
w0\_s = copysign(1.0, w0)
function code(w0_s, w0_m, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -1e+124)
		tmp = fma(Float64(Float64(Float64(D * M) * Float64(Float64(D * M) * Float64(h * w0_m))) / Float64(Float64(d * d) * l)), -0.125, w0_m);
	else
		tmp = w0_m;
	end
	return Float64(w0_s * tmp)
end

Wolfram

w0\_m = N[Abs[w0], $MachinePrecision]
w0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[w0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[w0$95$s_, w0$95$m_, M_, D_, h_, l_, d_] := N[(w0$95$s * If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -1e+124], N[(N[(N[(N[(D * M), $MachinePrecision] * N[(N[(D * M), $MachinePrecision] * N[(h * w0$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * -0.125 + w0$95$m), $MachinePrecision], w0$95$m]), $MachinePrecision]

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)) < -9.99999999999999948e123

   1. Initial program 66.2%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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 50.4

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

   5. Applied rewrites 50.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(D \cdot M\right)}^{2} \cdot \left(h \cdot w0\right)}{\left(d \cdot d\right) \cdot \ell}, -0.125, w0\right)} \]
   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. lift-*.f64 N/A

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

      6. lift-*.f64 50.4

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

   7. Applied rewrites 50.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.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. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 50.6

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

   9. Applied rewrites 50.6%

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

   if -9.99999999999999948e123 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 90.5%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 93.3%

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

Alternative 9: 78.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ w0\_s \cdot \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\frac{D \cdot \left(M \cdot \left(\left(\left(h \cdot M\right) \cdot D\right) \cdot w0\_m\right)\right)}{\left(d \cdot d\right) \cdot \ell}, -0.125, w0\_m\right)\\ \mathbf{else}:\\ \;\;\;\;w0\_m\\ \end{array} \end{array} \]

FPCore

w0\_m = (fabs.f64 w0)
w0\_s = (copysign.f64 #s(literal 1 binary64) w0)
(FPCore (w0_s w0_m M D h l d)
 :precision binary64
 (*
  w0_s
  (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -2e-6)
    (fma (/ (* D (* M (* (* (* h M) D) w0_m))) (* (* d d) l)) -0.125 w0_m)
    w0_m)))

C

w0\_m = fabs(w0);
w0\_s = copysign(1.0, w0);
double code(double w0_s, double w0_m, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -2e-6) {
		tmp = fma(((D * (M * (((h * M) * D) * w0_m))) / ((d * d) * l)), -0.125, w0_m);
	} else {
		tmp = w0_m;
	}
	return w0_s * tmp;
}

Julia

w0\_m = abs(w0)
w0\_s = copysign(1.0, w0)
function code(w0_s, w0_m, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -2e-6)
		tmp = fma(Float64(Float64(D * Float64(M * Float64(Float64(Float64(h * M) * D) * w0_m))) / Float64(Float64(d * d) * l)), -0.125, w0_m);
	else
		tmp = w0_m;
	end
	return Float64(w0_s * tmp)
end

Wolfram

w0\_m = N[Abs[w0], $MachinePrecision]
w0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[w0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[w0$95$s_, w0$95$m_, M_, D_, h_, l_, d_] := N[(w0$95$s * 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-6], N[(N[(N[(D * N[(M * N[(N[(N[(h * M), $MachinePrecision] * D), $MachinePrecision] * w0$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * -0.125 + w0$95$m), $MachinePrecision], w0$95$m]), $MachinePrecision]

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)) < -1.99999999999999991e-6

   1. Initial program 68.7%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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 47.2

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

   5. Applied rewrites 47.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(D \cdot M\right)}^{2} \cdot \left(h \cdot w0\right)}{\left(d \cdot d\right) \cdot \ell}, -0.125, w0\right)} \]
   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. lift-*.f64 N/A

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

      6. lift-*.f64 47.2

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

   7. Applied rewrites 47.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.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. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 47.4

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

   9. Applied rewrites 47.4%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 47.4

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. associate-*r* N/A

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

       10. associate-*r* N/A

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

       11. lower-*.f64 N/A

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

       12. *-commutative N/A

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

       13. lift-*.f64 N/A

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

       14. *-commutative N/A

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

       15. lift-*.f64 47.7

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

   11. Applied rewrites 47.7%

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

   if -1.99999999999999991e-6 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 90.2%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 96.6%

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

Alternative 10: 82.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ w0\_s \cdot \left(w0\_m \cdot \sqrt{1 - \left(\frac{D}{d} \cdot \frac{M}{2}\right) \cdot \left(\left(\frac{D}{d} \cdot \frac{h \cdot M}{\ell}\right) \cdot 0.5\right)}\right) \end{array} \]

FPCore

w0\_m = (fabs.f64 w0)
w0\_s = (copysign.f64 #s(literal 1 binary64) w0)
(FPCore (w0_s w0_m M D h l d)
 :precision binary64
 (*
  w0_s
  (*
   w0_m
   (sqrt
    (- 1.0 (* (* (/ D d) (/ M 2.0)) (* (* (/ D d) (/ (* h M) l)) 0.5)))))))

C

w0\_m = fabs(w0);
w0\_s = copysign(1.0, w0);
double code(double w0_s, double w0_m, double M, double D, double h, double l, double d) {
	return w0_s * (w0_m * sqrt((1.0 - (((D / d) * (M / 2.0)) * (((D / d) * ((h * M) / l)) * 0.5)))));
}

Fortran

w0\_m =     private
w0\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(w0_s, w0_m, m, d, h, l, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: w0_s
    real(8), intent (in) :: w0_m
    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_s * (w0_m * sqrt((1.0d0 - (((d / d_1) * (m / 2.0d0)) * (((d / d_1) * ((h * m) / l)) * 0.5d0)))))
end function

Java

w0\_m = Math.abs(w0);
w0\_s = Math.copySign(1.0, w0);
public static double code(double w0_s, double w0_m, double M, double D, double h, double l, double d) {
	return w0_s * (w0_m * Math.sqrt((1.0 - (((D / d) * (M / 2.0)) * (((D / d) * ((h * M) / l)) * 0.5)))));
}

Python

w0\_m = math.fabs(w0)
w0\_s = math.copysign(1.0, w0)
def code(w0_s, w0_m, M, D, h, l, d):
	return w0_s * (w0_m * math.sqrt((1.0 - (((D / d) * (M / 2.0)) * (((D / d) * ((h * M) / l)) * 0.5)))))

Julia

w0\_m = abs(w0)
w0\_s = copysign(1.0, w0)
function code(w0_s, w0_m, M, D, h, l, d)
	return Float64(w0_s * Float64(w0_m * sqrt(Float64(1.0 - Float64(Float64(Float64(D / d) * Float64(M / 2.0)) * Float64(Float64(Float64(D / d) * Float64(Float64(h * M) / l)) * 0.5))))))
end

MATLAB

w0\_m = abs(w0);
w0\_s = sign(w0) * abs(1.0);
function tmp = code(w0_s, w0_m, M, D, h, l, d)
	tmp = w0_s * (w0_m * sqrt((1.0 - (((D / d) * (M / 2.0)) * (((D / d) * ((h * M) / l)) * 0.5)))));
end

Wolfram

w0\_m = N[Abs[w0], $MachinePrecision]
w0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[w0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[w0$95$s_, w0$95$m_, M_, D_, h_, l_, d_] := N[(w0$95$s * N[(w0$95$m * N[Sqrt[N[(1.0 - N[(N[(N[(D / d), $MachinePrecision] * N[(M / 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(D / d), $MachinePrecision] * N[(N[(h * M), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.7%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. associate-*r/ N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-pow.f64 N/A

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

   11. times-frac N/A

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

   12. lower-*.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-/.f64 86.5

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

4. Applied rewrites 86.5%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-pow.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. frac-times N/A

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

   8. associate-*r/ N/A

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

   9. frac-times N/A

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

   10. lift-/.f64 N/A

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

   11. lift-/.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. pow2 N/A

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

   14. associate-*l* N/A

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

   15. 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. Applied rewrites 84.8%

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

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 79.5

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

9. Applied rewrites 79.5%

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

    1. lift-/.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. associate-/l* N/A

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

    4. lift-*.f64 N/A

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

    5. *-commutative N/A

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

    6. lift-*.f64 N/A

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

    7. *-commutative N/A

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

    8. lift-*.f64 N/A

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

    9. *-commutative N/A

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

    10. *-commutative N/A

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

    11. lower-*.f64 N/A

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

    12. times-frac N/A

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

    13. lift-/.f64 N/A

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

    14. lower-*.f64 N/A

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

    15. lower-/.f64 N/A

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

    16. *-commutative N/A

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

    17. lift-*.f64 82.9

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

11. Applied rewrites 82.9%

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

Alternative 11: 67.2% accurate, 157.0× speedup

Math

\[\begin{array}{l} w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ w0\_s \cdot w0\_m \end{array} \]

FPCore

w0\_m = (fabs.f64 w0)
w0\_s = (copysign.f64 #s(literal 1 binary64) w0)
(FPCore (w0_s w0_m M D h l d) :precision binary64 (* w0_s w0_m))

C

w0\_m = fabs(w0);
w0\_s = copysign(1.0, w0);
double code(double w0_s, double w0_m, double M, double D, double h, double l, double d) {
	return w0_s * w0_m;
}

Fortran

w0\_m =     private
w0\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(w0_s, w0_m, m, d, h, l, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: w0_s
    real(8), intent (in) :: w0_m
    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_s * w0_m
end function

Java

w0\_m = Math.abs(w0);
w0\_s = Math.copySign(1.0, w0);
public static double code(double w0_s, double w0_m, double M, double D, double h, double l, double d) {
	return w0_s * w0_m;
}

Python

w0\_m = math.fabs(w0)
w0\_s = math.copysign(1.0, w0)
def code(w0_s, w0_m, M, D, h, l, d):
	return w0_s * w0_m

Julia

w0\_m = abs(w0)
w0\_s = copysign(1.0, w0)
function code(w0_s, w0_m, M, D, h, l, d)
	return Float64(w0_s * w0_m)
end

MATLAB

w0\_m = abs(w0);
w0\_s = sign(w0) * abs(1.0);
function tmp = code(w0_s, w0_m, M, D, h, l, d)
	tmp = w0_s * w0_m;
end

Wolfram

w0\_m = N[Abs[w0], $MachinePrecision]
w0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[w0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[w0$95$s_, w0$95$m_, M_, D_, h_, l_, d_] := N[(w0$95$s * w0$95$m), $MachinePrecision]

Derivation

1. Initial program 82.7%

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

3. Taylor expanded in M around 0

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

5. Applied rewrites 64.9%

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

Reproduce

herbie shell --seed 2025054 
(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))))))