Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.1% → 88.2%

Time: 12.2s

Alternatives: 16

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: 81.1% 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: 88.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ \begin{array}{l} t_0 := \frac{D \cdot \frac{M\_m}{d}}{2}\\ \mathbf{if}\;{\left(\frac{M\_m \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 0.05:\\ \;\;\;\;w0 \cdot \sqrt{1 - t\_0 \cdot \left(t\_0 \cdot \frac{h}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{\frac{-M\_m}{d} \cdot \left(\left(D \cdot M\_m\right) \cdot h\right)}{2 \cdot \ell}, \frac{\frac{D}{d}}{2}, 1\right)}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D h l d)
 :precision binary64
 (let* ((t_0 (/ (* D (/ M_m d)) 2.0)))
   (if (<= (* (pow (/ (* M_m D) (* 2.0 d)) 2.0) (/ h l)) 0.05)
     (* w0 (sqrt (- 1.0 (* t_0 (* t_0 (/ h l))))))
     (*
      w0
      (sqrt
       (fma
        (/ (* (/ (- M_m) d) (* (* D M_m) h)) (* 2.0 l))
        (/ (/ D d) 2.0)
        1.0))))))

C

M_m = fabs(M);
assert(w0 < M_m && M_m < D && D < h && h < l && l < d);
double code(double w0, double M_m, double D, double h, double l, double d) {
	double t_0 = (D * (M_m / d)) / 2.0;
	double tmp;
	if ((pow(((M_m * D) / (2.0 * d)), 2.0) * (h / l)) <= 0.05) {
		tmp = w0 * sqrt((1.0 - (t_0 * (t_0 * (h / l)))));
	} else {
		tmp = w0 * sqrt(fma((((-M_m / d) * ((D * M_m) * h)) / (2.0 * l)), ((D / d) / 2.0), 1.0));
	}
	return tmp;
}

Julia

M_m = abs(M)
w0, M_m, D, h, l, d = sort([w0, M_m, D, h, l, d])
function code(w0, M_m, D, h, l, d)
	t_0 = Float64(Float64(D * Float64(M_m / d)) / 2.0)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= 0.05)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(t_0 * Float64(t_0 * Float64(h / l))))));
	else
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(Float64(Float64(-M_m) / d) * Float64(Float64(D * M_m) * h)) / Float64(2.0 * l)), Float64(Float64(D / d) / 2.0), 1.0)));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d_] := Block[{t$95$0 = N[(N[(D * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[N[(N[Power[N[(N[(M$95$m * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], 0.05], N[(w0 * N[Sqrt[N[(1.0 - N[(t$95$0 * N[(t$95$0 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(N[(N[(N[((-M$95$m) / d), $MachinePrecision] * N[(N[(D * M$95$m), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] / N[(2.0 * l), $MachinePrecision]), $MachinePrecision] * N[(N[(D / d), $MachinePrecision] / 2.0), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $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)) < 0.050000000000000003

   1. Initial program 81.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 80.2

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

   4. Applied rewrites 80.2%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 81.4

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 81.4

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

   6. Applied rewrites 81.4%

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

   if 0.050000000000000003 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 4.6%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

   4. Applied rewrites 14.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 73.2

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

   6. Applied rewrites 73.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 81.9

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

   8. Applied rewrites 81.9%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 0.05:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{D \cdot \frac{M}{d}}{2} \cdot \left(\frac{D \cdot \frac{M}{d}}{2} \cdot \frac{h}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{\frac{-M}{d} \cdot \left(\left(D \cdot M\right) \cdot h\right)}{2 \cdot \ell}, \frac{\frac{D}{d}}{2}, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 77.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\sqrt{1 - {\left(\frac{M\_m \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \leq 4 \cdot 10^{+109}:\\ \;\;\;\;w0 \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(\left(\left(M\_m \cdot M\_m\right) \cdot h\right) \cdot w0\right) \cdot D}{d} \cdot \frac{D}{\ell \cdot d}, -0.125, w0\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D h l d)
 :precision binary64
 (if (<= (sqrt (- 1.0 (* (pow (/ (* M_m D) (* 2.0 d)) 2.0) (/ h l)))) 4e+109)
   (* w0 1.0)
   (fma (* (/ (* (* (* (* M_m M_m) h) w0) D) d) (/ D (* l d))) -0.125 w0)))

C

M_m = fabs(M);
assert(w0 < M_m && M_m < D && D < h && h < l && l < d);
double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if (sqrt((1.0 - (pow(((M_m * D) / (2.0 * d)), 2.0) * (h / l)))) <= 4e+109) {
		tmp = w0 * 1.0;
	} else {
		tmp = fma(((((((M_m * M_m) * h) * w0) * D) / d) * (D / (l * d))), -0.125, w0);
	}
	return tmp;
}

Julia

M_m = abs(M)
w0, M_m, D, h, l, d = sort([w0, M_m, D, h, l, d])
function code(w0, M_m, D, h, l, d)
	tmp = 0.0
	if (sqrt(Float64(1.0 - Float64((Float64(Float64(M_m * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)))) <= 4e+109)
		tmp = Float64(w0 * 1.0);
	else
		tmp = fma(Float64(Float64(Float64(Float64(Float64(Float64(M_m * M_m) * h) * w0) * D) / d) * Float64(D / Float64(l * d))), -0.125, w0);
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d_] := If[LessEqual[N[Sqrt[N[(1.0 - N[(N[Power[N[(N[(M$95$m * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 4e+109], N[(w0 * 1.0), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * h), $MachinePrecision] * w0), $MachinePrecision] * D), $MachinePrecision] / d), $MachinePrecision] * N[(D / N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125 + w0), $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)))) < 3.99999999999999993e109

   1. Initial program 99.9%

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

   3. Taylor expanded in M around 0

      \[\leadsto w0 \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 90.8%

      \[\leadsto w0 \cdot \color{blue}{1} \]

   if 3.99999999999999993e109 < (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 32.6%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 31.6

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

   4. Applied rewrites 31.6%

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

   5. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 27.3

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

   7. Applied rewrites 27.3%

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

   9. Applied rewrites 37.9%

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

Alternative 3: 82.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+19}:\\ \;\;\;\;w0 \cdot \sqrt{\left(-0.25 \cdot \left(D \cdot D\right)\right) \cdot \left(\frac{h \cdot M\_m}{d} \cdot \frac{M\_m}{d \cdot \ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M_m D) (* 2.0 d)) 2.0) (/ h l)) -2e+19)
   (* w0 (sqrt (* (* -0.25 (* D D)) (* (/ (* h M_m) d) (/ M_m (* d l))))))
   (* w0 1.0)))

C

M_m = fabs(M);
assert(w0 < M_m && M_m < D && D < h && h < l && l < d);
double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M_m * D) / (2.0 * d)), 2.0) * (h / l)) <= -2e+19) {
		tmp = w0 * sqrt(((-0.25 * (D * D)) * (((h * M_m) / d) * (M_m / (d * l)))));
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Fortran

M_m =     private
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
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_m, d, h, l, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_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_m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l)) <= (-2d+19)) then
        tmp = w0 * sqrt((((-0.25d0) * (d * d)) * (((h * m_m) / d_1) * (m_m / (d_1 * l)))))
    else
        tmp = w0 * 1.0d0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert w0 < M_m && M_m < D && D < h && h < l && l < d;
public static double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((M_m * D) / (2.0 * d)), 2.0) * (h / l)) <= -2e+19) {
		tmp = w0 * Math.sqrt(((-0.25 * (D * D)) * (((h * M_m) / d) * (M_m / (d * l)))));
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[w0, M_m, D, h, l, d] = sort([w0, M_m, D, h, l, d])
def code(w0, M_m, D, h, l, d):
	tmp = 0
	if (math.pow(((M_m * D) / (2.0 * d)), 2.0) * (h / l)) <= -2e+19:
		tmp = w0 * math.sqrt(((-0.25 * (D * D)) * (((h * M_m) / d) * (M_m / (d * l)))))
	else:
		tmp = w0 * 1.0
	return tmp

Julia

M_m = abs(M)
w0, M_m, D, h, l, d = sort([w0, M_m, D, h, l, d])
function code(w0, M_m, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -2e+19)
		tmp = Float64(w0 * sqrt(Float64(Float64(-0.25 * Float64(D * D)) * Float64(Float64(Float64(h * M_m) / d) * Float64(M_m / Float64(d * l))))));
	else
		tmp = Float64(w0 * 1.0);
	end
	return tmp
end

MATLAB

M_m = abs(M);
w0, M_m, D, h, l, d = num2cell(sort([w0, M_m, D, h, l, d])){:}
function tmp_2 = code(w0, M_m, D, h, l, d)
	tmp = 0.0;
	if (((((M_m * D) / (2.0 * d)) ^ 2.0) * (h / l)) <= -2e+19)
		tmp = w0 * sqrt(((-0.25 * (D * D)) * (((h * M_m) / d) * (M_m / (d * l)))));
	else
		tmp = w0 * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M$95$m * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -2e+19], N[(w0 * N[Sqrt[N[(N[(-0.25 * N[(D * D), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(h * M$95$m), $MachinePrecision] / d), $MachinePrecision] * N[(M$95$m / N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * 1.0), $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)) < -2e19

   1. Initial program 49.7%

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

   3. Taylor expanded in M around 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. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 21.7

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

   5. Applied rewrites 21.7%

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

   7. Applied rewrites 39.7%

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

   if -2e19 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   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. Taylor expanded in M around 0

      \[\leadsto w0 \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 93.6%

      \[\leadsto w0 \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+19}:\\ \;\;\;\;w0 \cdot \sqrt{\left(-0.25 \cdot \left(D \cdot D\right)\right) \cdot \left(\frac{h \cdot M}{d} \cdot \frac{M}{d \cdot \ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+19}:\\ \;\;\;\;w0 \cdot \sqrt{\frac{M\_m \cdot \left(M\_m \cdot \left(\left(-0.25 \cdot \left(D \cdot D\right)\right) \cdot h\right)\right)}{\left(d \cdot d\right) \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M_m D) (* 2.0 d)) 2.0) (/ h l)) -2e+19)
   (* w0 (sqrt (/ (* M_m (* M_m (* (* -0.25 (* D D)) h))) (* (* d d) l))))
   (* w0 1.0)))

C

M_m = fabs(M);
assert(w0 < M_m && M_m < D && D < h && h < l && l < d);
double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M_m * D) / (2.0 * d)), 2.0) * (h / l)) <= -2e+19) {
		tmp = w0 * sqrt(((M_m * (M_m * ((-0.25 * (D * D)) * h))) / ((d * d) * l)));
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Fortran

M_m =     private
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
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_m, d, h, l, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_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_m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l)) <= (-2d+19)) then
        tmp = w0 * sqrt(((m_m * (m_m * (((-0.25d0) * (d * d)) * h))) / ((d_1 * d_1) * l)))
    else
        tmp = w0 * 1.0d0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert w0 < M_m && M_m < D && D < h && h < l && l < d;
public static double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((M_m * D) / (2.0 * d)), 2.0) * (h / l)) <= -2e+19) {
		tmp = w0 * Math.sqrt(((M_m * (M_m * ((-0.25 * (D * D)) * h))) / ((d * d) * l)));
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[w0, M_m, D, h, l, d] = sort([w0, M_m, D, h, l, d])
def code(w0, M_m, D, h, l, d):
	tmp = 0
	if (math.pow(((M_m * D) / (2.0 * d)), 2.0) * (h / l)) <= -2e+19:
		tmp = w0 * math.sqrt(((M_m * (M_m * ((-0.25 * (D * D)) * h))) / ((d * d) * l)))
	else:
		tmp = w0 * 1.0
	return tmp

Julia

M_m = abs(M)
w0, M_m, D, h, l, d = sort([w0, M_m, D, h, l, d])
function code(w0, M_m, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -2e+19)
		tmp = Float64(w0 * sqrt(Float64(Float64(M_m * Float64(M_m * Float64(Float64(-0.25 * Float64(D * D)) * h))) / Float64(Float64(d * d) * l))));
	else
		tmp = Float64(w0 * 1.0);
	end
	return tmp
end

MATLAB

M_m = abs(M);
w0, M_m, D, h, l, d = num2cell(sort([w0, M_m, D, h, l, d])){:}
function tmp_2 = code(w0, M_m, D, h, l, d)
	tmp = 0.0;
	if (((((M_m * D) / (2.0 * d)) ^ 2.0) * (h / l)) <= -2e+19)
		tmp = w0 * sqrt(((M_m * (M_m * ((-0.25 * (D * D)) * h))) / ((d * d) * l)));
	else
		tmp = w0 * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M$95$m * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -2e+19], N[(w0 * N[Sqrt[N[(N[(M$95$m * N[(M$95$m * N[(N[(-0.25 * N[(D * D), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * 1.0), $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)) < -2e19

   1. Initial program 49.7%

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

   3. Taylor expanded in M around 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. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 21.7

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

   5. Applied rewrites 21.7%

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

   7. Applied rewrites 22.9%

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

   9. Applied rewrites 29.2%

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

   if -2e19 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   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. Taylor expanded in M around 0

      \[\leadsto w0 \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 93.6%

      \[\leadsto w0 \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+19}:\\ \;\;\;\;w0 \cdot \sqrt{\frac{M \cdot \left(M \cdot \left(\left(-0.25 \cdot \left(D \cdot D\right)\right) \cdot h\right)\right)}{\left(d \cdot d\right) \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+64}:\\ \;\;\;\;\mathsf{fma}\left(\frac{M\_m \cdot \left(\left(h \cdot M\_m\right) \cdot \left(\left(D \cdot D\right) \cdot w0\right)\right)}{\left(d \cdot d\right) \cdot \ell}, -0.125, w0\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M_m D) (* 2.0 d)) 2.0) (/ h l)) -5e+64)
   (fma (/ (* M_m (* (* h M_m) (* (* D D) w0))) (* (* d d) l)) -0.125 w0)
   (* w0 1.0)))

C

M_m = fabs(M);
assert(w0 < M_m && M_m < D && D < h && h < l && l < d);
double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M_m * D) / (2.0 * d)), 2.0) * (h / l)) <= -5e+64) {
		tmp = fma(((M_m * ((h * M_m) * ((D * D) * w0))) / ((d * d) * l)), -0.125, w0);
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Julia

M_m = abs(M)
w0, M_m, D, h, l, d = sort([w0, M_m, D, h, l, d])
function code(w0, M_m, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -5e+64)
		tmp = fma(Float64(Float64(M_m * Float64(Float64(h * M_m) * Float64(Float64(D * D) * w0))) / Float64(Float64(d * d) * l)), -0.125, w0);
	else
		tmp = Float64(w0 * 1.0);
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M$95$m * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -5e+64], N[(N[(N[(M$95$m * N[(N[(h * M$95$m), $MachinePrecision] * N[(N[(D * D), $MachinePrecision] * w0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * -0.125 + w0), $MachinePrecision], N[(w0 * 1.0), $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)) < -5e64

   1. Initial program 47.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 46.1

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

   4. Applied rewrites 46.1%

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

   5. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 22.6

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

   7. Applied rewrites 22.6%

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

   9. Applied rewrites 26.4%

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

   if -5e64 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 88.5%

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

   3. Taylor expanded in M around 0

      \[\leadsto w0 \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 91.7%

      \[\leadsto w0 \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 78.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+64}:\\ \;\;\;\;\frac{M\_m \cdot \left(\left(h \cdot M\_m\right) \cdot \left(\left(D \cdot D\right) \cdot w0\right)\right)}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M_m D) (* 2.0 d)) 2.0) (/ h l)) -5e+64)
   (* (/ (* M_m (* (* h M_m) (* (* D D) w0))) (* (* d d) l)) -0.125)
   (* w0 1.0)))

C

M_m = fabs(M);
assert(w0 < M_m && M_m < D && D < h && h < l && l < d);
double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M_m * D) / (2.0 * d)), 2.0) * (h / l)) <= -5e+64) {
		tmp = ((M_m * ((h * M_m) * ((D * D) * w0))) / ((d * d) * l)) * -0.125;
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Fortran

M_m =     private
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
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_m, d, h, l, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_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_m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l)) <= (-5d+64)) then
        tmp = ((m_m * ((h * m_m) * ((d * d) * w0))) / ((d_1 * d_1) * l)) * (-0.125d0)
    else
        tmp = w0 * 1.0d0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert w0 < M_m && M_m < D && D < h && h < l && l < d;
public static double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((M_m * D) / (2.0 * d)), 2.0) * (h / l)) <= -5e+64) {
		tmp = ((M_m * ((h * M_m) * ((D * D) * w0))) / ((d * d) * l)) * -0.125;
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[w0, M_m, D, h, l, d] = sort([w0, M_m, D, h, l, d])
def code(w0, M_m, D, h, l, d):
	tmp = 0
	if (math.pow(((M_m * D) / (2.0 * d)), 2.0) * (h / l)) <= -5e+64:
		tmp = ((M_m * ((h * M_m) * ((D * D) * w0))) / ((d * d) * l)) * -0.125
	else:
		tmp = w0 * 1.0
	return tmp

Julia

M_m = abs(M)
w0, M_m, D, h, l, d = sort([w0, M_m, D, h, l, d])
function code(w0, M_m, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -5e+64)
		tmp = Float64(Float64(Float64(M_m * Float64(Float64(h * M_m) * Float64(Float64(D * D) * w0))) / Float64(Float64(d * d) * l)) * -0.125);
	else
		tmp = Float64(w0 * 1.0);
	end
	return tmp
end

MATLAB

M_m = abs(M);
w0, M_m, D, h, l, d = num2cell(sort([w0, M_m, D, h, l, d])){:}
function tmp_2 = code(w0, M_m, D, h, l, d)
	tmp = 0.0;
	if (((((M_m * D) / (2.0 * d)) ^ 2.0) * (h / l)) <= -5e+64)
		tmp = ((M_m * ((h * M_m) * ((D * D) * w0))) / ((d * d) * l)) * -0.125;
	else
		tmp = w0 * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M$95$m * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -5e+64], N[(N[(N[(M$95$m * N[(N[(h * M$95$m), $MachinePrecision] * N[(N[(D * D), $MachinePrecision] * w0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision], N[(w0 * 1.0), $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)) < -5e64

   1. Initial program 47.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 46.1

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

   4. Applied rewrites 46.1%

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

   5. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 22.6

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

   7. Applied rewrites 22.6%

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

   8. Taylor expanded in M around inf

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

   10. Applied rewrites 21.6%

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

   12. Applied rewrites 25.8%

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

   if -5e64 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 88.5%

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

   3. Taylor expanded in M around 0

      \[\leadsto w0 \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 91.7%

      \[\leadsto w0 \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 77.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -1 \cdot 10^{+219}:\\ \;\;\;\;\left(\left(D \cdot D\right) \cdot \frac{\left(\left(M\_m \cdot M\_m\right) \cdot h\right) \cdot w0}{\left(d \cdot d\right) \cdot \ell}\right) \cdot -0.125\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M_m D) (* 2.0 d)) 2.0) (/ h l)) -1e+219)
   (* (* (* D D) (/ (* (* (* M_m M_m) h) w0) (* (* d d) l))) -0.125)
   (* w0 1.0)))

C

M_m = fabs(M);
assert(w0 < M_m && M_m < D && D < h && h < l && l < d);
double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M_m * D) / (2.0 * d)), 2.0) * (h / l)) <= -1e+219) {
		tmp = ((D * D) * ((((M_m * M_m) * h) * w0) / ((d * d) * l))) * -0.125;
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Fortran

M_m =     private
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
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_m, d, h, l, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_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_m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l)) <= (-1d+219)) then
        tmp = ((d * d) * ((((m_m * m_m) * h) * w0) / ((d_1 * d_1) * l))) * (-0.125d0)
    else
        tmp = w0 * 1.0d0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert w0 < M_m && M_m < D && D < h && h < l && l < d;
public static double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((M_m * D) / (2.0 * d)), 2.0) * (h / l)) <= -1e+219) {
		tmp = ((D * D) * ((((M_m * M_m) * h) * w0) / ((d * d) * l))) * -0.125;
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[w0, M_m, D, h, l, d] = sort([w0, M_m, D, h, l, d])
def code(w0, M_m, D, h, l, d):
	tmp = 0
	if (math.pow(((M_m * D) / (2.0 * d)), 2.0) * (h / l)) <= -1e+219:
		tmp = ((D * D) * ((((M_m * M_m) * h) * w0) / ((d * d) * l))) * -0.125
	else:
		tmp = w0 * 1.0
	return tmp

Julia

M_m = abs(M)
w0, M_m, D, h, l, d = sort([w0, M_m, D, h, l, d])
function code(w0, M_m, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -1e+219)
		tmp = Float64(Float64(Float64(D * D) * Float64(Float64(Float64(Float64(M_m * M_m) * h) * w0) / Float64(Float64(d * d) * l))) * -0.125);
	else
		tmp = Float64(w0 * 1.0);
	end
	return tmp
end

MATLAB

M_m = abs(M);
w0, M_m, D, h, l, d = num2cell(sort([w0, M_m, D, h, l, d])){:}
function tmp_2 = code(w0, M_m, D, h, l, d)
	tmp = 0.0;
	if (((((M_m * D) / (2.0 * d)) ^ 2.0) * (h / l)) <= -1e+219)
		tmp = ((D * D) * ((((M_m * M_m) * h) * w0) / ((d * d) * l))) * -0.125;
	else
		tmp = w0 * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M$95$m * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -1e+219], N[(N[(N[(D * D), $MachinePrecision] * N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * h), $MachinePrecision] * w0), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision], N[(w0 * 1.0), $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)) < -9.99999999999999965e218

   1. Initial program 41.6%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 40.3

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

   4. Applied rewrites 40.3%

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

   5. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 24.9

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

   7. Applied rewrites 24.9%

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

   8. Taylor expanded in M around inf

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

   10. Applied rewrites 23.9%

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

   12. Applied rewrites 23.9%

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

   if -9.99999999999999965e218 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 89.0%

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

   3. Taylor expanded in M around 0

      \[\leadsto w0 \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 88.0%

      \[\leadsto w0 \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 82.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 10^{-198}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{\left(\left(\frac{h}{\ell} \cdot D\right) \cdot \frac{\frac{M\_m}{d}}{2}\right) \cdot \left(M\_m \cdot D\right)}{-2 \cdot d}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{\frac{-M\_m}{d} \cdot \left(\left(D \cdot M\_m\right) \cdot h\right)}{2 \cdot \ell}, \frac{\frac{D}{d}}{2}, 1\right)}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D h l d)
 :precision binary64
 (if (<= d 1e-198)
   (*
    w0
    (sqrt
     (+ 1.0 (/ (* (* (* (/ h l) D) (/ (/ M_m d) 2.0)) (* M_m D)) (* -2.0 d)))))
   (*
    w0
    (sqrt
     (fma
      (/ (* (/ (- M_m) d) (* (* D M_m) h)) (* 2.0 l))
      (/ (/ D d) 2.0)
      1.0)))))

C

M_m = fabs(M);
assert(w0 < M_m && M_m < D && D < h && h < l && l < d);
double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if (d <= 1e-198) {
		tmp = w0 * sqrt((1.0 + (((((h / l) * D) * ((M_m / d) / 2.0)) * (M_m * D)) / (-2.0 * d))));
	} else {
		tmp = w0 * sqrt(fma((((-M_m / d) * ((D * M_m) * h)) / (2.0 * l)), ((D / d) / 2.0), 1.0));
	}
	return tmp;
}

Julia

M_m = abs(M)
w0, M_m, D, h, l, d = sort([w0, M_m, D, h, l, d])
function code(w0, M_m, D, h, l, d)
	tmp = 0.0
	if (d <= 1e-198)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(Float64(Float64(Float64(h / l) * D) * Float64(Float64(M_m / d) / 2.0)) * Float64(M_m * D)) / Float64(-2.0 * d)))));
	else
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(Float64(Float64(-M_m) / d) * Float64(Float64(D * M_m) * h)) / Float64(2.0 * l)), Float64(Float64(D / d) / 2.0), 1.0)));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d_] := If[LessEqual[d, 1e-198], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(N[(N[(N[(h / l), $MachinePrecision] * D), $MachinePrecision] * N[(N[(M$95$m / d), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] * N[(M$95$m * D), $MachinePrecision]), $MachinePrecision] / N[(-2.0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(N[(N[(N[((-M$95$m) / d), $MachinePrecision] * N[(N[(D * M$95$m), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] / N[(2.0 * l), $MachinePrecision]), $MachinePrecision] * N[(N[(D / d), $MachinePrecision] / 2.0), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 9.9999999999999991e-199

   1. Initial program 71.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. lift-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. associate-*r/ N/A

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

      9. lift-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. flip-+ N/A

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

      12. distribute-neg-frac N/A

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

      13. +-inverses N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\frac{h}{\ell} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\mathsf{neg}\left(M \cdot D\right)\right)}{\frac{\mathsf{neg}\left(\color{blue}{0}\right)}{d - d}}} \]

      14. metadata-eval N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\frac{h}{\ell} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\mathsf{neg}\left(M \cdot D\right)\right)}{\frac{\color{blue}{0}}{d - d}}} \]

      15. +-inverses N/A

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

      16. flip-+ N/A

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

      17. count-2-rev N/A

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

      18. lift-*.f64 N/A

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

      19. lower-/.f64 N/A

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

   4. Applied rewrites 69.1%

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

   if 9.9999999999999991e-199 < d

   1. Initial program 79.7%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

   4. Applied rewrites 76.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 79.4

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

   6. Applied rewrites 79.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 81.7

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

   8. Applied rewrites 81.7%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 10^{-198}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{\left(\left(\frac{h}{\ell} \cdot D\right) \cdot \frac{\frac{M}{d}}{2}\right) \cdot \left(M \cdot D\right)}{-2 \cdot d}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{\frac{-M}{d} \cdot \left(\left(D \cdot M\right) \cdot h\right)}{2 \cdot \ell}, \frac{\frac{D}{d}}{2}, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 80.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 9.5 \cdot 10^{-179}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\left(\frac{M\_m}{d} \cdot h\right) \cdot \frac{\frac{D}{\ell}}{-2}, D \cdot \frac{M\_m}{d}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{h \cdot \left(D \cdot \left(\frac{M\_m}{d} \cdot M\_m\right)\right)}{-2 \cdot \ell}, \frac{\frac{D}{d}}{2}, 1\right)}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D h l d)
 :precision binary64
 (if (<= d 9.5e-179)
   (* w0 (sqrt (fma (* (* (/ M_m d) h) (/ (/ D l) -2.0)) (* D (/ M_m d)) 1.0)))
   (*
    w0
    (sqrt
     (fma (/ (* h (* D (* (/ M_m d) M_m))) (* -2.0 l)) (/ (/ D d) 2.0) 1.0)))))

C

M_m = fabs(M);
assert(w0 < M_m && M_m < D && D < h && h < l && l < d);
double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if (d <= 9.5e-179) {
		tmp = w0 * sqrt(fma((((M_m / d) * h) * ((D / l) / -2.0)), (D * (M_m / d)), 1.0));
	} else {
		tmp = w0 * sqrt(fma(((h * (D * ((M_m / d) * M_m))) / (-2.0 * l)), ((D / d) / 2.0), 1.0));
	}
	return tmp;
}

Julia

M_m = abs(M)
w0, M_m, D, h, l, d = sort([w0, M_m, D, h, l, d])
function code(w0, M_m, D, h, l, d)
	tmp = 0.0
	if (d <= 9.5e-179)
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(Float64(M_m / d) * h) * Float64(Float64(D / l) / -2.0)), Float64(D * Float64(M_m / d)), 1.0)));
	else
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(h * Float64(D * Float64(Float64(M_m / d) * M_m))) / Float64(-2.0 * l)), Float64(Float64(D / d) / 2.0), 1.0)));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d_] := If[LessEqual[d, 9.5e-179], N[(w0 * N[Sqrt[N[(N[(N[(N[(M$95$m / d), $MachinePrecision] * h), $MachinePrecision] * N[(N[(D / l), $MachinePrecision] / -2.0), $MachinePrecision]), $MachinePrecision] * N[(D * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(N[(N[(h * N[(D * N[(N[(M$95$m / d), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-2.0 * l), $MachinePrecision]), $MachinePrecision] * N[(N[(D / d), $MachinePrecision] / 2.0), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 9.50000000000000037e-179

   1. Initial program 71.0%

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

   4. Applied rewrites 59.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 66.6

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

   6. Applied rewrites 66.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. frac-times N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

   8. Applied rewrites 60.4%

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

   10. Applied rewrites 67.9%

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

   if 9.50000000000000037e-179 < d

   1. Initial program 80.8%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

   4. Applied rewrites 77.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 80.6

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

   6. Applied rewrites 80.6%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lower-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. distribute-rgt-neg-out N/A

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

      7. remove-double-neg N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. distribute-lft-neg-in N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval 80.6

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

   8. Applied rewrites 80.6%

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

Alternative 10: 84.0% accurate, 1.8× speedup

Math

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

FPCore

M_m = (fabs.f64 M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D h l d)
 :precision binary64
 (*
  w0
  (sqrt
   (fma (/ (* (/ (- M_m) d) (* (* D M_m) h)) (* 2.0 l)) (/ (/ D d) 2.0) 1.0))))

C

M_m = fabs(M);
assert(w0 < M_m && M_m < D && D < h && h < l && l < d);
double code(double w0, double M_m, double D, double h, double l, double d) {
	return w0 * sqrt(fma((((-M_m / d) * ((D * M_m) * h)) / (2.0 * l)), ((D / d) / 2.0), 1.0));
}

Julia

M_m = abs(M)
w0, M_m, D, h, l, d = sort([w0, M_m, D, h, l, d])
function code(w0, M_m, D, h, l, d)
	return Float64(w0 * sqrt(fma(Float64(Float64(Float64(Float64(-M_m) / d) * Float64(Float64(D * M_m) * h)) / Float64(2.0 * l)), Float64(Float64(D / d) / 2.0), 1.0)))
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d_] := N[(w0 * N[Sqrt[N[(N[(N[(N[((-M$95$m) / d), $MachinePrecision] * N[(N[(D * M$95$m), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] / N[(2.0 * l), $MachinePrecision]), $MachinePrecision] * N[(N[(D / d), $MachinePrecision] / 2.0), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.5%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

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

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

   4. +-commutative N/A

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

4. Applied rewrites 70.3%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. associate-*l/ N/A

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

   6. lift-/.f64 N/A

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

   7. frac-times N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-*.f64 75.2

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

6. Applied rewrites 75.2%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. associate-*l* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 77.4

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

8. Applied rewrites 77.4%

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

9. Final simplification 77.4%

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

Alternative 11: 79.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 1.9 \cdot 10^{-175}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\left(\frac{M\_m}{d} \cdot h\right) \cdot \frac{\frac{D}{\ell}}{-2}, D \cdot \frac{M\_m}{d}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\left(\left(\left(\frac{M\_m}{d} \cdot M\_m\right) \cdot D\right) \cdot \left(-h\right)\right) \cdot \frac{\frac{D}{d}}{\ell \cdot 4} + 1}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D h l d)
 :precision binary64
 (if (<= d 1.9e-175)
   (* w0 (sqrt (fma (* (* (/ M_m d) h) (/ (/ D l) -2.0)) (* D (/ M_m d)) 1.0)))
   (*
    w0
    (sqrt
     (+ (* (* (* (* (/ M_m d) M_m) D) (- h)) (/ (/ D d) (* l 4.0))) 1.0)))))

C

M_m = fabs(M);
assert(w0 < M_m && M_m < D && D < h && h < l && l < d);
double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if (d <= 1.9e-175) {
		tmp = w0 * sqrt(fma((((M_m / d) * h) * ((D / l) / -2.0)), (D * (M_m / d)), 1.0));
	} else {
		tmp = w0 * sqrt(((((((M_m / d) * M_m) * D) * -h) * ((D / d) / (l * 4.0))) + 1.0));
	}
	return tmp;
}

Julia

M_m = abs(M)
w0, M_m, D, h, l, d = sort([w0, M_m, D, h, l, d])
function code(w0, M_m, D, h, l, d)
	tmp = 0.0
	if (d <= 1.9e-175)
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(Float64(M_m / d) * h) * Float64(Float64(D / l) / -2.0)), Float64(D * Float64(M_m / d)), 1.0)));
	else
		tmp = Float64(w0 * sqrt(Float64(Float64(Float64(Float64(Float64(Float64(M_m / d) * M_m) * D) * Float64(-h)) * Float64(Float64(D / d) / Float64(l * 4.0))) + 1.0)));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d_] := If[LessEqual[d, 1.9e-175], N[(w0 * N[Sqrt[N[(N[(N[(N[(M$95$m / d), $MachinePrecision] * h), $MachinePrecision] * N[(N[(D / l), $MachinePrecision] / -2.0), $MachinePrecision]), $MachinePrecision] * N[(D * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(N[(N[(N[(N[(N[(M$95$m / d), $MachinePrecision] * M$95$m), $MachinePrecision] * D), $MachinePrecision] * (-h)), $MachinePrecision] * N[(N[(D / d), $MachinePrecision] / N[(l * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 1.9e-175

   1. Initial program 71.0%

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

   4. Applied rewrites 59.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 66.6

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

   6. Applied rewrites 66.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. frac-times N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

   8. Applied rewrites 60.4%

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

   10. Applied rewrites 67.9%

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

   if 1.9e-175 < d

   1. Initial program 80.8%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

   4. Applied rewrites 77.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 80.6

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

   6. Applied rewrites 80.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 82.9

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

   8. Applied rewrites 82.9%

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

      1. lift-fma.f64 N/A

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

      2. lower-+.f64 N/A

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

   10. Applied rewrites 80.5%

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

Alternative 12: 78.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 1.9 \cdot 10^{-175}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\left(\frac{D}{\ell} \cdot \frac{h}{-2}\right) \cdot \frac{M\_m}{d}, D \cdot \frac{M\_m}{d}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\left(\left(\left(\frac{M\_m}{d} \cdot M\_m\right) \cdot D\right) \cdot \left(-h\right)\right) \cdot \frac{\frac{D}{d}}{\ell \cdot 4} + 1}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D h l d)
 :precision binary64
 (if (<= d 1.9e-175)
   (* w0 (sqrt (fma (* (* (/ D l) (/ h -2.0)) (/ M_m d)) (* D (/ M_m d)) 1.0)))
   (*
    w0
    (sqrt
     (+ (* (* (* (* (/ M_m d) M_m) D) (- h)) (/ (/ D d) (* l 4.0))) 1.0)))))

C

M_m = fabs(M);
assert(w0 < M_m && M_m < D && D < h && h < l && l < d);
double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if (d <= 1.9e-175) {
		tmp = w0 * sqrt(fma((((D / l) * (h / -2.0)) * (M_m / d)), (D * (M_m / d)), 1.0));
	} else {
		tmp = w0 * sqrt(((((((M_m / d) * M_m) * D) * -h) * ((D / d) / (l * 4.0))) + 1.0));
	}
	return tmp;
}

Julia

M_m = abs(M)
w0, M_m, D, h, l, d = sort([w0, M_m, D, h, l, d])
function code(w0, M_m, D, h, l, d)
	tmp = 0.0
	if (d <= 1.9e-175)
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(Float64(D / l) * Float64(h / -2.0)) * Float64(M_m / d)), Float64(D * Float64(M_m / d)), 1.0)));
	else
		tmp = Float64(w0 * sqrt(Float64(Float64(Float64(Float64(Float64(Float64(M_m / d) * M_m) * D) * Float64(-h)) * Float64(Float64(D / d) / Float64(l * 4.0))) + 1.0)));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d_] := If[LessEqual[d, 1.9e-175], N[(w0 * N[Sqrt[N[(N[(N[(N[(D / l), $MachinePrecision] * N[(h / -2.0), $MachinePrecision]), $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] * N[(D * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(N[(N[(N[(N[(N[(M$95$m / d), $MachinePrecision] * M$95$m), $MachinePrecision] * D), $MachinePrecision] * (-h)), $MachinePrecision] * N[(N[(D / d), $MachinePrecision] / N[(l * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 1.9e-175

   1. Initial program 71.0%

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

   4. Applied rewrites 59.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 66.6

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

   6. Applied rewrites 66.6%

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

      1. lift-fma.f64 N/A

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

   8. Applied rewrites 66.2%

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

   if 1.9e-175 < d

   1. Initial program 80.8%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

   4. Applied rewrites 77.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 80.6

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

   6. Applied rewrites 80.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 82.9

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

   8. Applied rewrites 82.9%

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

      1. lift-fma.f64 N/A

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

      2. lower-+.f64 N/A

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

   10. Applied rewrites 80.5%

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

Alternative 13: 74.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \cdot D \leq 2 \cdot 10^{-152}:\\ \;\;\;\;w0 \cdot 1\\ \mathbf{elif}\;M\_m \cdot D \leq 10^{+132}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{\left(D \cdot M\_m\right) \cdot \left(D \cdot M\_m\right)}{\left(d \cdot d\right) \cdot \ell} \cdot -0.25, h, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(\left(\left(M\_m \cdot M\_m\right) \cdot h\right) \cdot w0\right) \cdot D}{d} \cdot \frac{D}{\ell \cdot d}, -0.125, w0\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D h l d)
 :precision binary64
 (if (<= (* M_m D) 2e-152)
   (* w0 1.0)
   (if (<= (* M_m D) 1e+132)
     (*
      w0
      (sqrt (fma (* (/ (* (* D M_m) (* D M_m)) (* (* d d) l)) -0.25) h 1.0)))
     (fma (* (/ (* (* (* (* M_m M_m) h) w0) D) d) (/ D (* l d))) -0.125 w0))))

C

M_m = fabs(M);
assert(w0 < M_m && M_m < D && D < h && h < l && l < d);
double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if ((M_m * D) <= 2e-152) {
		tmp = w0 * 1.0;
	} else if ((M_m * D) <= 1e+132) {
		tmp = w0 * sqrt(fma(((((D * M_m) * (D * M_m)) / ((d * d) * l)) * -0.25), h, 1.0));
	} else {
		tmp = fma(((((((M_m * M_m) * h) * w0) * D) / d) * (D / (l * d))), -0.125, w0);
	}
	return tmp;
}

Julia

M_m = abs(M)
w0, M_m, D, h, l, d = sort([w0, M_m, D, h, l, d])
function code(w0, M_m, D, h, l, d)
	tmp = 0.0
	if (Float64(M_m * D) <= 2e-152)
		tmp = Float64(w0 * 1.0);
	elseif (Float64(M_m * D) <= 1e+132)
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(Float64(Float64(D * M_m) * Float64(D * M_m)) / Float64(Float64(d * d) * l)) * -0.25), h, 1.0)));
	else
		tmp = fma(Float64(Float64(Float64(Float64(Float64(Float64(M_m * M_m) * h) * w0) * D) / d) * Float64(D / Float64(l * d))), -0.125, w0);
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d_] := If[LessEqual[N[(M$95$m * D), $MachinePrecision], 2e-152], N[(w0 * 1.0), $MachinePrecision], If[LessEqual[N[(M$95$m * D), $MachinePrecision], 1e+132], N[(w0 * N[Sqrt[N[(N[(N[(N[(N[(D * M$95$m), $MachinePrecision] * N[(D * M$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision] * h + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * h), $MachinePrecision] * w0), $MachinePrecision] * D), $MachinePrecision] / d), $MachinePrecision] * N[(D / N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125 + w0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 M D) < 2.00000000000000013e-152

   1. Initial program 77.1%

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

   3. Taylor expanded in M around 0

      \[\leadsto w0 \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 69.8%

      \[\leadsto w0 \cdot \color{blue}{1} \]

   if 2.00000000000000013e-152 < (*.f64 M D) < 9.99999999999999991e131

   1. Initial program 73.6%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 72.0

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

   4. Applied rewrites 72.0%

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

   5. Taylor expanded in h around inf

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. lft-mult-inverse N/A

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

      6. lower-fma.f64 N/A

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

   7. Applied rewrites 77.8%

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

   if 9.99999999999999991e131 < (*.f64 M D)

   1. Initial program 68.2%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 68.2

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

   4. Applied rewrites 68.2%

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

   5. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 24.9

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

   7. Applied rewrites 24.9%

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

   9. Applied rewrites 44.3%

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

Alternative 14: 76.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 1.45 \cdot 10^{-243}:\\ \;\;\;\;w0 \cdot 1\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\left(\left(\left(\frac{M\_m}{d} \cdot M\_m\right) \cdot D\right) \cdot \left(-h\right)\right) \cdot \frac{\frac{D}{d}}{\ell \cdot 4} + 1}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D h l d)
 :precision binary64
 (if (<= M_m 1.45e-243)
   (* w0 1.0)
   (*
    w0
    (sqrt
     (+ (* (* (* (* (/ M_m d) M_m) D) (- h)) (/ (/ D d) (* l 4.0))) 1.0)))))

C

M_m = fabs(M);
assert(w0 < M_m && M_m < D && D < h && h < l && l < d);
double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if (M_m <= 1.45e-243) {
		tmp = w0 * 1.0;
	} else {
		tmp = w0 * sqrt(((((((M_m / d) * M_m) * D) * -h) * ((D / d) / (l * 4.0))) + 1.0));
	}
	return tmp;
}

Fortran

M_m =     private
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
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_m, d, h, l, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_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_m <= 1.45d-243) then
        tmp = w0 * 1.0d0
    else
        tmp = w0 * sqrt(((((((m_m / d_1) * m_m) * d) * -h) * ((d / d_1) / (l * 4.0d0))) + 1.0d0))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert w0 < M_m && M_m < D && D < h && h < l && l < d;
public static double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if (M_m <= 1.45e-243) {
		tmp = w0 * 1.0;
	} else {
		tmp = w0 * Math.sqrt(((((((M_m / d) * M_m) * D) * -h) * ((D / d) / (l * 4.0))) + 1.0));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[w0, M_m, D, h, l, d] = sort([w0, M_m, D, h, l, d])
def code(w0, M_m, D, h, l, d):
	tmp = 0
	if M_m <= 1.45e-243:
		tmp = w0 * 1.0
	else:
		tmp = w0 * math.sqrt(((((((M_m / d) * M_m) * D) * -h) * ((D / d) / (l * 4.0))) + 1.0))
	return tmp

Julia

M_m = abs(M)
w0, M_m, D, h, l, d = sort([w0, M_m, D, h, l, d])
function code(w0, M_m, D, h, l, d)
	tmp = 0.0
	if (M_m <= 1.45e-243)
		tmp = Float64(w0 * 1.0);
	else
		tmp = Float64(w0 * sqrt(Float64(Float64(Float64(Float64(Float64(Float64(M_m / d) * M_m) * D) * Float64(-h)) * Float64(Float64(D / d) / Float64(l * 4.0))) + 1.0)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
w0, M_m, D, h, l, d = num2cell(sort([w0, M_m, D, h, l, d])){:}
function tmp_2 = code(w0, M_m, D, h, l, d)
	tmp = 0.0;
	if (M_m <= 1.45e-243)
		tmp = w0 * 1.0;
	else
		tmp = w0 * sqrt(((((((M_m / d) * M_m) * D) * -h) * ((D / d) / (l * 4.0))) + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d_] := If[LessEqual[M$95$m, 1.45e-243], N[(w0 * 1.0), $MachinePrecision], N[(w0 * N[Sqrt[N[(N[(N[(N[(N[(N[(M$95$m / d), $MachinePrecision] * M$95$m), $MachinePrecision] * D), $MachinePrecision] * (-h)), $MachinePrecision] * N[(N[(D / d), $MachinePrecision] / N[(l * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.44999999999999988e-243

   1. Initial program 75.4%

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

   3. Taylor expanded in M around 0

      \[\leadsto w0 \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 68.0%

      \[\leadsto w0 \cdot \color{blue}{1} \]

   if 1.44999999999999988e-243 < M

   1. Initial program 75.5%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

   4. Applied rewrites 70.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 74.9

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

   6. Applied rewrites 74.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 76.6

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

   8. Applied rewrites 76.6%

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

      1. lift-fma.f64 N/A

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

      2. lower-+.f64 N/A

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

   10. Applied rewrites 74.9%

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

Alternative 15: 76.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 1.45 \cdot 10^{-243}:\\ \;\;\;\;w0 \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\left(\frac{M\_m}{d} \cdot M\_m\right) \cdot D\right) \cdot \left(-h\right), \frac{\frac{D}{d}}{\ell \cdot 4}, 1\right)} \cdot w0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D h l d)
 :precision binary64
 (if (<= M_m 1.45e-243)
   (* w0 1.0)
   (*
    (sqrt (fma (* (* (* (/ M_m d) M_m) D) (- h)) (/ (/ D d) (* l 4.0)) 1.0))
    w0)))

C

M_m = fabs(M);
assert(w0 < M_m && M_m < D && D < h && h < l && l < d);
double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if (M_m <= 1.45e-243) {
		tmp = w0 * 1.0;
	} else {
		tmp = sqrt(fma(((((M_m / d) * M_m) * D) * -h), ((D / d) / (l * 4.0)), 1.0)) * w0;
	}
	return tmp;
}

Julia

M_m = abs(M)
w0, M_m, D, h, l, d = sort([w0, M_m, D, h, l, d])
function code(w0, M_m, D, h, l, d)
	tmp = 0.0
	if (M_m <= 1.45e-243)
		tmp = Float64(w0 * 1.0);
	else
		tmp = Float64(sqrt(fma(Float64(Float64(Float64(Float64(M_m / d) * M_m) * D) * Float64(-h)), Float64(Float64(D / d) / Float64(l * 4.0)), 1.0)) * w0);
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d_] := If[LessEqual[M$95$m, 1.45e-243], N[(w0 * 1.0), $MachinePrecision], N[(N[Sqrt[N[(N[(N[(N[(N[(M$95$m / d), $MachinePrecision] * M$95$m), $MachinePrecision] * D), $MachinePrecision] * (-h)), $MachinePrecision] * N[(N[(D / d), $MachinePrecision] / N[(l * 4.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.44999999999999988e-243

   1. Initial program 75.4%

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

   3. Taylor expanded in M around 0

      \[\leadsto w0 \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 68.0%

      \[\leadsto w0 \cdot \color{blue}{1} \]

   if 1.44999999999999988e-243 < M

   1. Initial program 75.5%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

   4. Applied rewrites 70.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 74.9

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

   6. Applied rewrites 74.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 76.6

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

   8. Applied rewrites 76.6%

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

   10. Applied rewrites 74.9%

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

Alternative 16: 68.2% accurate, 26.2× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ w0 \cdot 1 \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D h l d) :precision binary64 (* w0 1.0))

C

M_m = fabs(M);
assert(w0 < M_m && M_m < D && D < h && h < l && l < d);
double code(double w0, double M_m, double D, double h, double l, double d) {
	return w0 * 1.0;
}

Fortran

M_m =     private
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
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_m, d, h, l, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    code = w0 * 1.0d0
end function

Java

M_m = Math.abs(M);
assert w0 < M_m && M_m < D && D < h && h < l && l < d;
public static double code(double w0, double M_m, double D, double h, double l, double d) {
	return w0 * 1.0;
}

Python

M_m = math.fabs(M)
[w0, M_m, D, h, l, d] = sort([w0, M_m, D, h, l, d])
def code(w0, M_m, D, h, l, d):
	return w0 * 1.0

Julia

M_m = abs(M)
w0, M_m, D, h, l, d = sort([w0, M_m, D, h, l, d])
function code(w0, M_m, D, h, l, d)
	return Float64(w0 * 1.0)
end

MATLAB

M_m = abs(M);
w0, M_m, D, h, l, d = num2cell(sort([w0, M_m, D, h, l, d])){:}
function tmp = code(w0, M_m, D, h, l, d)
	tmp = w0 * 1.0;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d_] := N[(w0 * 1.0), $MachinePrecision]

Derivation

1. Initial program 75.5%

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

3. Taylor expanded in M around 0

   \[\leadsto w0 \cdot \color{blue}{1} \]
4. Step-by-step derivation

5. Applied rewrites 64.4%

   \[\leadsto w0 \cdot \color{blue}{1} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024346 
(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))))))