Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.8% → 87.0%

Time: 6.8s

Alternatives: 14

Speedup: 0.8×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 80.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 87.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \leq 5 \cdot 10^{+267}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{d + d}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right) \cdot \frac{0.5 \cdot \left(\left(h \cdot M\right) \cdot D\right)}{\ell \cdot d}}\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<=
      (* w0 (sqrt (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)))))
      5e+267)
   (* w0 (sqrt (- 1.0 (* (pow (/ (* M D) (+ d d)) 2.0) (/ h l)))))
   (*
    w0
    (sqrt
     (- 1.0 (* (* (/ D d) (* 0.5 M)) (/ (* 0.5 (* (* h M) D)) (* l d))))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((w0 * sqrt((1.0 - (pow(((M * D) / (2.0 * d)), 2.0) * (h / l))))) <= 5e+267) {
		tmp = w0 * sqrt((1.0 - (pow(((M * D) / (d + d)), 2.0) * (h / l))));
	} else {
		tmp = w0 * sqrt((1.0 - (((D / d) * (0.5 * M)) * ((0.5 * ((h * M) * D)) / (l * d)))));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(w0, m, d, h, l, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if ((w0 * sqrt((1.0d0 - ((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l))))) <= 5d+267) then
        tmp = w0 * sqrt((1.0d0 - ((((m * d) / (d_1 + d_1)) ** 2.0d0) * (h / l))))
    else
        tmp = w0 * sqrt((1.0d0 - (((d / d_1) * (0.5d0 * m)) * ((0.5d0 * ((h * m) * d)) / (l * d_1)))))
    end if
    code = tmp
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((w0 * Math.sqrt((1.0 - (Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l))))) <= 5e+267) {
		tmp = w0 * Math.sqrt((1.0 - (Math.pow(((M * D) / (d + d)), 2.0) * (h / l))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (((D / d) * (0.5 * M)) * ((0.5 * ((h * M) * D)) / (l * d)))));
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if (w0 * math.sqrt((1.0 - (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l))))) <= 5e+267:
		tmp = w0 * math.sqrt((1.0 - (math.pow(((M * D) / (d + d)), 2.0) * (h / l))))
	else:
		tmp = w0 * math.sqrt((1.0 - (((D / d) * (0.5 * M)) * ((0.5 * ((h * M) * D)) / (l * d)))))
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64(w0 * sqrt(Float64(1.0 - Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l))))) <= 5e+267)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64((Float64(Float64(M * D) / Float64(d + d)) ^ 2.0) * Float64(h / l)))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(D / d) * Float64(0.5 * M)) * Float64(Float64(0.5 * Float64(Float64(h * M) * D)) / Float64(l * d))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if ((w0 * sqrt((1.0 - ((((M * D) / (2.0 * d)) ^ 2.0) * (h / l))))) <= 5e+267)
		tmp = w0 * sqrt((1.0 - ((((M * D) / (d + d)) ^ 2.0) * (h / l))));
	else
		tmp = w0 * sqrt((1.0 - (((D / d) * (0.5 * M)) * ((0.5 * ((h * M) * D)) / (l * d)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[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], 5e+267], N[(w0 * N[Sqrt[N[(1.0 - N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(d + d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(D / d), $MachinePrecision] * N[(0.5 * M), $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 * N[(N[(h * M), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 w0 (sqrt.f64 (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))))) < 4.9999999999999999e267

   1. Initial program 96.1%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 96.1

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

   4. Applied rewrites 96.1%

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

   if 4.9999999999999999e267 < (*.f64 w0 (sqrt.f64 (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)))))

   1. Initial program 34.8%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 34.8

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

   4. Applied rewrites 34.8%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. count-2-rev N/A

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

      7. frac-times N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-/.f64 N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. lift-/.f64 N/A

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

      20. lower-*.f64 N/A

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

   6. Applied rewrites 47.8%

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

   7. Taylor expanded in M around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 69.9

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

   9. Applied rewrites 69.9%

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

   10. Taylor expanded in M around 0

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

       1. lower-*.f64 69.9

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

   12. Applied rewrites 69.9%

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

Alternative 2: 86.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \leq 5 \cdot 10^{+267}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)\right) \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right) \cdot \frac{0.5 \cdot \left(\left(h \cdot M\right) \cdot D\right)}{\ell \cdot d}}\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<=
      (* w0 (sqrt (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)))))
      5e+267)
   (*
    w0
    (sqrt (- 1.0 (* (* (* (/ M 2.0) (/ D d)) (* (* 0.5 M) (/ D d))) (/ h l)))))
   (*
    w0
    (sqrt
     (- 1.0 (* (* (/ D d) (* 0.5 M)) (/ (* 0.5 (* (* h M) D)) (* l d))))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((w0 * sqrt((1.0 - (pow(((M * D) / (2.0 * d)), 2.0) * (h / l))))) <= 5e+267) {
		tmp = w0 * sqrt((1.0 - ((((M / 2.0) * (D / d)) * ((0.5 * M) * (D / d))) * (h / l))));
	} else {
		tmp = w0 * sqrt((1.0 - (((D / d) * (0.5 * M)) * ((0.5 * ((h * M) * D)) / (l * d)))));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(w0, m, d, h, l, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if ((w0 * sqrt((1.0d0 - ((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l))))) <= 5d+267) then
        tmp = w0 * sqrt((1.0d0 - ((((m / 2.0d0) * (d / d_1)) * ((0.5d0 * m) * (d / d_1))) * (h / l))))
    else
        tmp = w0 * sqrt((1.0d0 - (((d / d_1) * (0.5d0 * m)) * ((0.5d0 * ((h * m) * d)) / (l * d_1)))))
    end if
    code = tmp
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((w0 * Math.sqrt((1.0 - (Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l))))) <= 5e+267) {
		tmp = w0 * Math.sqrt((1.0 - ((((M / 2.0) * (D / d)) * ((0.5 * M) * (D / d))) * (h / l))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (((D / d) * (0.5 * M)) * ((0.5 * ((h * M) * D)) / (l * d)))));
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if (w0 * math.sqrt((1.0 - (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l))))) <= 5e+267:
		tmp = w0 * math.sqrt((1.0 - ((((M / 2.0) * (D / d)) * ((0.5 * M) * (D / d))) * (h / l))))
	else:
		tmp = w0 * math.sqrt((1.0 - (((D / d) * (0.5 * M)) * ((0.5 * ((h * M) * D)) / (l * d)))))
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64(w0 * sqrt(Float64(1.0 - Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l))))) <= 5e+267)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(Float64(M / 2.0) * Float64(D / d)) * Float64(Float64(0.5 * M) * Float64(D / d))) * Float64(h / l)))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(D / d) * Float64(0.5 * M)) * Float64(Float64(0.5 * Float64(Float64(h * M) * D)) / Float64(l * d))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if ((w0 * sqrt((1.0 - ((((M * D) / (2.0 * d)) ^ 2.0) * (h / l))))) <= 5e+267)
		tmp = w0 * sqrt((1.0 - ((((M / 2.0) * (D / d)) * ((0.5 * M) * (D / d))) * (h / l))));
	else
		tmp = w0 * sqrt((1.0 - (((D / d) * (0.5 * M)) * ((0.5 * ((h * M) * D)) / (l * d)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[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], 5e+267], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(N[(M / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 * M), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(D / d), $MachinePrecision] * N[(0.5 * M), $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 * N[(N[(h * M), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 w0 (sqrt.f64 (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))))) < 4.9999999999999999e267

   1. Initial program 96.1%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 95.1

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

   4. Applied rewrites 95.1%

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

   5. Taylor expanded in M around 0

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

      1. lower-*.f64 95.1

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

   7. Applied rewrites 95.1%

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

   if 4.9999999999999999e267 < (*.f64 w0 (sqrt.f64 (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)))))

   1. Initial program 34.8%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 34.8

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

   4. Applied rewrites 34.8%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. count-2-rev N/A

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

      7. frac-times N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-/.f64 N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. lift-/.f64 N/A

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

      20. lower-*.f64 N/A

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

   6. Applied rewrites 47.8%

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

   7. Taylor expanded in M around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 69.9

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

   9. Applied rewrites 69.9%

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

   10. Taylor expanded in M around 0

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

       1. lower-*.f64 69.9

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

   12. Applied rewrites 69.9%

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

Alternative 3: 87.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{D}{d} \cdot \frac{M}{2}\\ \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -1 \cdot 10^{-9}:\\ \;\;\;\;w0 \cdot \sqrt{1 - t\_0 \cdot \left(t\_0 \cdot \frac{h}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (* (/ D d) (/ M 2.0))))
   (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -1e-9)
     (* w0 (sqrt (- 1.0 (* t_0 (* t_0 (/ h l))))))
     w0)))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = (D / d) * (M / 2.0);
	double tmp;
	if ((pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -1e-9) {
		tmp = w0 * sqrt((1.0 - (t_0 * (t_0 * (h / l)))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(w0, m, d, h, l, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (d / d_1) * (m / 2.0d0)
    if (((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l)) <= (-1d-9)) then
        tmp = w0 * sqrt((1.0d0 - (t_0 * (t_0 * (h / l)))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = (D / d) * (M / 2.0);
	double tmp;
	if ((Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -1e-9) {
		tmp = w0 * Math.sqrt((1.0 - (t_0 * (t_0 * (h / l)))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	t_0 = (D / d) * (M / 2.0)
	tmp = 0
	if (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -1e-9:
		tmp = w0 * math.sqrt((1.0 - (t_0 * (t_0 * (h / l)))))
	else:
		tmp = w0
	return tmp

Julia

function code(w0, M, D, h, l, d)
	t_0 = Float64(Float64(D / d) * Float64(M / 2.0))
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -1e-9)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(t_0 * Float64(t_0 * Float64(h / l))))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	t_0 = (D / d) * (M / 2.0);
	tmp = 0.0;
	if (((((M * D) / (2.0 * d)) ^ 2.0) * (h / l)) <= -1e-9)
		tmp = w0 * sqrt((1.0 - (t_0 * (t_0 * (h / l)))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := Block[{t$95$0 = N[(N[(D / d), $MachinePrecision] * N[(M / 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -1e-9], N[(w0 * N[Sqrt[N[(1.0 - N[(t$95$0 * N[(t$95$0 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -1.00000000000000006e-9

   1. Initial program 66.5%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 66.5

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

   4. Applied rewrites 66.5%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. count-2-rev N/A

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

      7. frac-times N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-/.f64 N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. lift-/.f64 N/A

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

      20. lower-*.f64 N/A

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

   6. Applied rewrites 70.3%

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

   if -1.00000000000000006e-9 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 87.2%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 96.0%

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

Alternative 4: 85.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 100.0%

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

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

   1. Initial program 49.1%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 49.1

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

   4. Applied rewrites 49.1%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. count-2-rev N/A

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

      7. frac-times N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-/.f64 N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. lift-/.f64 N/A

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

      20. lower-*.f64 N/A

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

   6. Applied rewrites 57.6%

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

   7. Taylor expanded in M around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 62.2

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

   9. Applied rewrites 62.2%

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

   10. Taylor expanded in M around 0

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

       1. lower-*.f64 62.2

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

   12. Applied rewrites 62.2%

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

Alternative 5: 86.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{M \cdot D}{2 \cdot d}\\ \mathbf{if}\;{t\_0}^{2} \cdot \frac{h}{\ell} \leq -1 \cdot 10^{-9}:\\ \;\;\;\;w0 \cdot \sqrt{1 - t\_0 \cdot \left(\left(\frac{D}{d} \cdot \frac{M}{2}\right) \cdot \frac{h}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (/ (* M D) (* 2.0 d))))
   (if (<= (* (pow t_0 2.0) (/ h l)) -1e-9)
     (* w0 (sqrt (- 1.0 (* t_0 (* (* (/ D d) (/ M 2.0)) (/ h l))))))
     w0)))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = (M * D) / (2.0 * d);
	double tmp;
	if ((pow(t_0, 2.0) * (h / l)) <= -1e-9) {
		tmp = w0 * sqrt((1.0 - (t_0 * (((D / d) * (M / 2.0)) * (h / l)))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(w0, m, d, h, l, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (m * d) / (2.0d0 * d_1)
    if (((t_0 ** 2.0d0) * (h / l)) <= (-1d-9)) then
        tmp = w0 * sqrt((1.0d0 - (t_0 * (((d / d_1) * (m / 2.0d0)) * (h / l)))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = (M * D) / (2.0 * d);
	double tmp;
	if ((Math.pow(t_0, 2.0) * (h / l)) <= -1e-9) {
		tmp = w0 * Math.sqrt((1.0 - (t_0 * (((D / d) * (M / 2.0)) * (h / l)))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	t_0 = (M * D) / (2.0 * d)
	tmp = 0
	if (math.pow(t_0, 2.0) * (h / l)) <= -1e-9:
		tmp = w0 * math.sqrt((1.0 - (t_0 * (((D / d) * (M / 2.0)) * (h / l)))))
	else:
		tmp = w0
	return tmp

Julia

function code(w0, M, D, h, l, d)
	t_0 = Float64(Float64(M * D) / Float64(2.0 * d))
	tmp = 0.0
	if (Float64((t_0 ^ 2.0) * Float64(h / l)) <= -1e-9)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(t_0 * Float64(Float64(Float64(D / d) * Float64(M / 2.0)) * Float64(h / l))))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	t_0 = (M * D) / (2.0 * d);
	tmp = 0.0;
	if (((t_0 ^ 2.0) * (h / l)) <= -1e-9)
		tmp = w0 * sqrt((1.0 - (t_0 * (((D / d) * (M / 2.0)) * (h / l)))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := Block[{t$95$0 = N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Power[t$95$0, 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -1e-9], N[(w0 * N[Sqrt[N[(1.0 - N[(t$95$0 * N[(N[(N[(D / d), $MachinePrecision] * N[(M / 2.0), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -1.00000000000000006e-9

   1. Initial program 66.5%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 66.5

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

   4. Applied rewrites 66.5%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. count-2-rev N/A

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

      7. frac-times N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-/.f64 N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. lift-/.f64 N/A

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

      20. lower-*.f64 N/A

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

   6. Applied rewrites 70.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. *-commutative N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 68.9

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

   8. Applied rewrites 68.9%

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

   if -1.00000000000000006e-9 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 87.2%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 96.0%

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

Alternative 6: 81.9% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -5e+19)
   (* w0 (sqrt (* -0.25 (* (* D D) (* (* (/ M d) (/ M d)) (/ h l))))))
   w0))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -5e+19) {
		tmp = w0 * sqrt((-0.25 * ((D * D) * (((M / d) * (M / d)) * (h / l)))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(w0, m, d, h, l, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l)) <= (-5d+19)) then
        tmp = w0 * sqrt(((-0.25d0) * ((d * d) * (((m / d_1) * (m / d_1)) * (h / l)))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -5e+19) {
		tmp = w0 * Math.sqrt((-0.25 * ((D * D) * (((M / d) * (M / d)) * (h / l)))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -5e+19:
		tmp = w0 * math.sqrt((-0.25 * ((D * D) * (((M / d) * (M / d)) * (h / l)))))
	else:
		tmp = w0
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -5e+19)
		tmp = Float64(w0 * sqrt(Float64(-0.25 * Float64(Float64(D * D) * Float64(Float64(Float64(M / d) * Float64(M / d)) * Float64(h / l))))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (((((M * D) / (2.0 * d)) ^ 2.0) * (h / l)) <= -5e+19)
		tmp = w0 * sqrt((-0.25 * ((D * D) * (((M / d) * (M / d)) * (h / l)))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -5e+19], N[(w0 * N[Sqrt[N[(-0.25 * N[(N[(D * D), $MachinePrecision] * N[(N[(N[(M / d), $MachinePrecision] * N[(M / d), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -5e19

   1. Initial program 65.5%

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

   3. Taylor expanded in M around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. pow-prod-down N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 44.3

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

   5. Applied rewrites 44.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow-prod-down N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 34.9

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

   7. Applied rewrites 34.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 36.6

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

   9. Applied rewrites 36.6%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. pow2 N/A

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

       4. lower-/.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. pow2 N/A

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

       8. *-commutative N/A

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

       9. lift-*.f64 N/A

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

       10. lift-*.f64 N/A

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

       11. *-commutative N/A

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

       12. associate-*l* N/A

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

       13. pow2 N/A

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

       14. associate-/l* N/A

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

       15. lower-*.f64 N/A

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

       16. pow2 N/A

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

       17. lift-*.f64 N/A

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

       18. times-frac N/A

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

       19. pow2 N/A

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

       20. pow2 N/A

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

   11. Applied rewrites 50.3%

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

   if -5e19 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 87.4%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 95.6%

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

Alternative 7: 81.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+25}:\\ \;\;\;\;w0 \cdot \sqrt{-0.25 \cdot \frac{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right) \cdot h}{\left(d \cdot d\right) \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -2e+25)
   (* w0 (sqrt (* -0.25 (/ (* (* (* D M) (* D M)) h) (* (* d d) l)))))
   w0))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -2e+25) {
		tmp = w0 * sqrt((-0.25 * ((((D * M) * (D * M)) * h) / ((d * d) * l))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(w0, m, d, h, l, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l)) <= (-2d+25)) then
        tmp = w0 * sqrt(((-0.25d0) * ((((d * m) * (d * m)) * h) / ((d_1 * d_1) * l))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -2e+25) {
		tmp = w0 * Math.sqrt((-0.25 * ((((D * M) * (D * M)) * h) / ((d * d) * l))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -2e+25:
		tmp = w0 * math.sqrt((-0.25 * ((((D * M) * (D * M)) * h) / ((d * d) * l))))
	else:
		tmp = w0
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -2e+25)
		tmp = Float64(w0 * sqrt(Float64(-0.25 * Float64(Float64(Float64(Float64(D * M) * Float64(D * M)) * h) / Float64(Float64(d * d) * l)))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (((((M * D) / (2.0 * d)) ^ 2.0) * (h / l)) <= -2e+25)
		tmp = w0 * sqrt((-0.25 * ((((D * M) * (D * M)) * h) / ((d * d) * l))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -2e+25], N[(w0 * N[Sqrt[N[(-0.25 * N[(N[(N[(N[(D * M), $MachinePrecision] * N[(D * M), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -2.00000000000000018e25

   1. Initial program 64.4%

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

   3. Taylor expanded in M around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. pow-prod-down N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 45.7

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

   5. Applied rewrites 45.7%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 45.7

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

   7. Applied rewrites 45.7%

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

   if -2.00000000000000018e25 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 87.5%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 94.7%

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

Alternative 8: 78.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+25}:\\ \;\;\;\;w0 \cdot \sqrt{-0.25 \cdot \left(\left(\left(h \cdot M\right) \cdot M\right) \cdot \frac{D \cdot D}{\left(d \cdot d\right) \cdot \ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -2e+25)
   (* w0 (sqrt (* -0.25 (* (* (* h M) M) (/ (* D D) (* (* d d) l))))))
   w0))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -2e+25) {
		tmp = w0 * sqrt((-0.25 * (((h * M) * M) * ((D * D) / ((d * d) * l)))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(w0, m, d, h, l, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l)) <= (-2d+25)) then
        tmp = w0 * sqrt(((-0.25d0) * (((h * m) * m) * ((d * d) / ((d_1 * d_1) * l)))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -2e+25) {
		tmp = w0 * Math.sqrt((-0.25 * (((h * M) * M) * ((D * D) / ((d * d) * l)))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -2e+25:
		tmp = w0 * math.sqrt((-0.25 * (((h * M) * M) * ((D * D) / ((d * d) * l)))))
	else:
		tmp = w0
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -2e+25)
		tmp = Float64(w0 * sqrt(Float64(-0.25 * Float64(Float64(Float64(h * M) * M) * Float64(Float64(D * D) / Float64(Float64(d * d) * l))))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (((((M * D) / (2.0 * d)) ^ 2.0) * (h / l)) <= -2e+25)
		tmp = w0 * sqrt((-0.25 * (((h * M) * M) * ((D * D) / ((d * d) * l)))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -2e+25], N[(w0 * N[Sqrt[N[(-0.25 * N[(N[(N[(h * M), $MachinePrecision] * M), $MachinePrecision] * N[(N[(D * D), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -2.00000000000000018e25

   1. Initial program 64.4%

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

   3. Taylor expanded in M around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. pow-prod-down N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 45.7

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

   5. Applied rewrites 45.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow-prod-down N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 36.0

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

   7. Applied rewrites 36.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 37.7

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

   9. Applied rewrites 37.7%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. pow2 N/A

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

       5. associate-/l* N/A

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. *-commutative N/A

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

       9. associate-*l* N/A

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

       10. lift-*.f64 N/A

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

       11. lift-*.f64 N/A

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

       12. lower-*.f64 N/A

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

       13. lift-*.f64 N/A

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

       14. lift-*.f64 N/A

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

       15. associate-*l* N/A

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

       16. *-commutative N/A

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

       17. lift-*.f64 N/A

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

       18. *-commutative N/A

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

       19. lower-*.f64 N/A

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

       20. lift-*.f64 N/A

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

       21. lift-*.f64 N/A

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

       22. pow2 N/A

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

       23. lower-/.f64 N/A

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

       24. pow2 N/A

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

       25. lift-*.f64 N/A

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

       26. pow2 N/A

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

   11. Applied rewrites 40.8%

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

   if -2.00000000000000018e25 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 87.5%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 94.7%

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

Alternative 9: 78.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+25}:\\ \;\;\;\;w0 \cdot \sqrt{-0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{h}{\left(d \cdot d\right) \cdot \ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -2e+25)
   (* w0 (sqrt (* -0.25 (* (* D D) (* (* M M) (/ h (* (* d d) l)))))))
   w0))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -2e+25) {
		tmp = w0 * sqrt((-0.25 * ((D * D) * ((M * M) * (h / ((d * d) * l))))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(w0, m, d, h, l, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l)) <= (-2d+25)) then
        tmp = w0 * sqrt(((-0.25d0) * ((d * d) * ((m * m) * (h / ((d_1 * d_1) * l))))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -2e+25) {
		tmp = w0 * Math.sqrt((-0.25 * ((D * D) * ((M * M) * (h / ((d * d) * l))))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -2e+25:
		tmp = w0 * math.sqrt((-0.25 * ((D * D) * ((M * M) * (h / ((d * d) * l))))))
	else:
		tmp = w0
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -2e+25)
		tmp = Float64(w0 * sqrt(Float64(-0.25 * Float64(Float64(D * D) * Float64(Float64(M * M) * Float64(h / Float64(Float64(d * d) * l)))))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (((((M * D) / (2.0 * d)) ^ 2.0) * (h / l)) <= -2e+25)
		tmp = w0 * sqrt((-0.25 * ((D * D) * ((M * M) * (h / ((d * d) * l))))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -2e+25], N[(w0 * N[Sqrt[N[(-0.25 * N[(N[(D * D), $MachinePrecision] * N[(N[(M * M), $MachinePrecision] * N[(h / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -2.00000000000000018e25

   1. Initial program 64.4%

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

   3. Taylor expanded in M around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. pow-prod-down N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 45.7

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

   5. Applied rewrites 45.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow-prod-down N/A

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

      9. associate-*r* N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 N/A

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

      21. lift-*.f64 40.7

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

   7. Applied rewrites 40.7%

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

   if -2.00000000000000018e25 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 87.5%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 94.7%

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

Alternative 10: 78.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -1 \cdot 10^{+272}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \left(h \cdot w0\right)}{d \cdot \left(d \cdot \ell\right)}, -0.125, w0\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -1e+272)
   (fma (/ (* (* (* M D) (* M D)) (* h w0)) (* d (* d l))) -0.125 w0)
   w0))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -1e+272) {
		tmp = fma(((((M * D) * (M * D)) * (h * w0)) / (d * (d * l))), -0.125, w0);
	} else {
		tmp = w0;
	}
	return tmp;
}

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -1e+272)
		tmp = fma(Float64(Float64(Float64(Float64(M * D) * Float64(M * D)) * Float64(h * w0)) / Float64(d * Float64(d * l))), -0.125, w0);
	else
		tmp = w0;
	end
	return tmp
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -1e+272], N[(N[(N[(N[(N[(M * D), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] * N[(h * w0), $MachinePrecision]), $MachinePrecision] / N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125 + w0), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -1.0000000000000001e272

   1. Initial program 54.5%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. pow-prod-down N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 48.1

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

   5. Applied rewrites 48.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 50.2

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

   7. Applied rewrites 50.2%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 50.2

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

   9. Applied rewrites 50.2%

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

   if -1.0000000000000001e272 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 88.3%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 88.8%

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

Alternative 11: 78.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -1 \cdot 10^{+272}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right) \cdot \left(h \cdot w0\right)}{\left(d \cdot d\right) \cdot \ell}, -0.125, w0\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -1e+272)
   (fma (/ (* (* (* D M) (* D M)) (* h w0)) (* (* d d) l)) -0.125 w0)
   w0))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -1e+272) {
		tmp = fma(((((D * M) * (D * M)) * (h * w0)) / ((d * d) * l)), -0.125, w0);
	} else {
		tmp = w0;
	}
	return tmp;
}

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -1e+272)
		tmp = fma(Float64(Float64(Float64(Float64(D * M) * Float64(D * M)) * Float64(h * w0)) / Float64(Float64(d * d) * l)), -0.125, w0);
	else
		tmp = w0;
	end
	return tmp
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -1e+272], N[(N[(N[(N[(N[(D * M), $MachinePrecision] * N[(D * M), $MachinePrecision]), $MachinePrecision] * N[(h * w0), $MachinePrecision]), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * -0.125 + w0), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -1.0000000000000001e272

   1. Initial program 54.5%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. pow-prod-down N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 48.1

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

   5. Applied rewrites 48.1%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 48.1

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

   7. Applied rewrites 48.1%

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

   if -1.0000000000000001e272 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 88.3%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 88.8%

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

Alternative 12: 78.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+103}:\\ \;\;\;\;\mathsf{fma}\left(D \cdot \frac{\left(\left(\left(h \cdot M\right) \cdot M\right) \cdot w0\right) \cdot D}{\left(d \cdot d\right) \cdot \ell}, -0.125, w0\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -2e+103)
   (fma (* D (/ (* (* (* (* h M) M) w0) D) (* (* d d) l))) -0.125 w0)
   w0))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -2e+103) {
		tmp = fma((D * (((((h * M) * M) * w0) * D) / ((d * d) * l))), -0.125, w0);
	} else {
		tmp = w0;
	}
	return tmp;
}

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -2e+103)
		tmp = fma(Float64(D * Float64(Float64(Float64(Float64(Float64(h * M) * M) * w0) * D) / Float64(Float64(d * d) * l))), -0.125, w0);
	else
		tmp = w0;
	end
	return tmp
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -2e+103], N[(N[(D * N[(N[(N[(N[(N[(h * M), $MachinePrecision] * M), $MachinePrecision] * w0), $MachinePrecision] * D), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125 + w0), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -2e103

   1. Initial program 60.8%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. pow-prod-down N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 41.9

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

   5. Applied rewrites 41.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow-prod-down N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. unpow2 N/A

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

      20. lower-*.f64 N/A

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

      21. pow2 N/A

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

      22. lift-*.f64 N/A

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

      23. lift-*.f64 39.6

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

   7. Applied rewrites 39.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 40.5

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lower-/.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 40.3

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

   9. Applied rewrites 40.3%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-/.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. lift-*.f64 N/A

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

       11. lift-*.f64 N/A

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

       12. associate-*r/ N/A

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

       13. lift-*.f64 N/A

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

       14. lift-*.f64 N/A

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

       15. pow2 N/A

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

       16. associate-*r* N/A

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

       17. lift-*.f64 N/A

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

       18. lift-*.f64 N/A

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

       19. pow2 N/A

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

   11. Applied rewrites 42.3%

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

   if -2e103 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 87.9%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 92.1%

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

Alternative 13: 78.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -1 \cdot 10^{+135}:\\ \;\;\;\;\mathsf{fma}\left(D \cdot \left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{h \cdot w0}{\left(d \cdot d\right) \cdot \ell}\right)\right), -0.125, w0\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -1e+135)
   (fma (* D (* D (* (* M M) (/ (* h w0) (* (* d d) l))))) -0.125 w0)
   w0))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -1e+135) {
		tmp = fma((D * (D * ((M * M) * ((h * w0) / ((d * d) * l))))), -0.125, w0);
	} else {
		tmp = w0;
	}
	return tmp;
}

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -1e+135)
		tmp = fma(Float64(D * Float64(D * Float64(Float64(M * M) * Float64(Float64(h * w0) / Float64(Float64(d * d) * l))))), -0.125, w0);
	else
		tmp = w0;
	end
	return tmp
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -1e+135], N[(N[(D * N[(D * N[(N[(M * M), $MachinePrecision] * N[(N[(h * w0), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125 + w0), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -9.99999999999999962e134

   1. Initial program 60.1%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. pow-prod-down N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 42.6

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

   5. Applied rewrites 42.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow-prod-down N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. unpow2 N/A

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

      20. lower-*.f64 N/A

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

      21. pow2 N/A

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

      22. lift-*.f64 N/A

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

      23. lift-*.f64 40.3

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

   7. Applied rewrites 40.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 41.1

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lower-/.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 40.9

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

   9. Applied rewrites 40.9%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-*r/ N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. pow2 N/A

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

       7. associate-*r* N/A

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

       8. lift-*.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. pow2 N/A

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

       11. associate-/l* N/A

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

       12. lower-*.f64 N/A

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

       13. pow2 N/A

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

       14. lift-*.f64 N/A

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

       15. lower-/.f64 N/A

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

       16. lift-*.f64 N/A

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

       17. pow2 N/A

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

       18. lift-*.f64 N/A

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

       19. lift-*.f64 42.6

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

   11. Applied rewrites 42.6%

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

   if -9.99999999999999962e134 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 87.9%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 91.7%

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

Alternative 14: 67.6% accurate, 157.0× speedup

Math

\[\begin{array}{l} \\ w0 \end{array} \]

FPCore

(FPCore (w0 M D h l d) :precision binary64 w0)

C

double code(double w0, double M, double D, double h, double l, double d) {
	return w0;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(w0, m, d, h, l, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    code = w0
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	return w0;
}

Python

def code(w0, M, D, h, l, d):
	return w0

Julia

function code(w0, M, D, h, l, d)
	return w0
end

MATLAB

function tmp = code(w0, M, D, h, l, d)
	tmp = w0;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := w0

Derivation

1. Initial program 81.7%

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

3. Taylor expanded in M around 0

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

5. Applied rewrites 72.3%

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

Reproduce

herbie shell --seed 2025040 
(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))))))