Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.4% → 87.1%

Time: 5.0s

Alternatives: 7

Speedup: 0.7×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 81.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 87.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \leq 5 \cdot 10^{+139}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{d + d}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(-0.25, \frac{\left(M \cdot D\right) \cdot \left(\left(M \cdot D\right) \cdot h\right)}{d \cdot \left(d \cdot \ell\right)}, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (sqrt (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)))) 5e+139)
   (* w0 (sqrt (- 1.0 (* (pow (/ (* M D) (+ d d)) 2.0) (/ h l)))))
   (* w0 (sqrt (fma -0.25 (/ (* (* M D) (* (* M D) h)) (* d (* d l))) 1.0)))))

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)))) <= 5e+139) {
		tmp = w0 * sqrt((1.0 - (pow(((M * D) / (d + d)), 2.0) * (h / l))));
	} else {
		tmp = w0 * sqrt(fma(-0.25, (((M * D) * ((M * D) * h)) / (d * (d * l))), 1.0));
	}
	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)))) <= 5e+139)
		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(fma(-0.25, Float64(Float64(Float64(M * D) * Float64(Float64(M * D) * h)) / Float64(d * Float64(d * l))), 1.0)));
	end
	return 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], 5e+139], 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[(-0.25 * N[(N[(N[(M * D), $MachinePrecision] * N[(N[(M * D), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] / N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)))) < 5.0000000000000003e139

   1. Initial program 99.8%

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

      1. lift-*.f64 N/A

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

      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 99.8

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

   3. Applied rewrites 99.8%

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

   if 5.0000000000000003e139 < (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 42.9%

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

   2. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. pow-prod-down N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 54.6

          \[\leadsto w0 \cdot \sqrt{\mathsf{fma}\left(-0.25, \frac{{\left(D \cdot M\right)}^{2} \cdot h}{\left(d \cdot d\right) \cdot \ell}, 1\right)} \]

   4. Applied rewrites 54.6%

      \[\leadsto w0 \cdot \sqrt{\color{blue}{\mathsf{fma}\left(-0.25, \frac{{\left(D \cdot M\right)}^{2} \cdot h}{\left(d \cdot d\right) \cdot \ell}, 1\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 58.4

         \[\leadsto w0 \cdot \sqrt{\mathsf{fma}\left(-0.25, \frac{{\left(D \cdot M\right)}^{2} \cdot h}{d \cdot \left(d \cdot \color{blue}{\ell}\right)}, 1\right)} \]

   6. Applied rewrites 58.4%

      \[\leadsto w0 \cdot \sqrt{\mathsf{fma}\left(-0.25, \frac{{\left(D \cdot M\right)}^{2} \cdot h}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}, 1\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 58.4

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

   8. Applied rewrites 58.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 60.5

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

   10. Applied rewrites 60.5%

       \[\leadsto w0 \cdot \sqrt{\mathsf{fma}\left(-0.25, \frac{\left(M \cdot D\right) \cdot \left(\left(M \cdot D\right) \cdot h\right)}{\color{blue}{d} \cdot \left(d \cdot \ell\right)}, 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 83.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \leq 1.1:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(-0.25, \frac{\left(M \cdot D\right) \cdot \left(\left(M \cdot D\right) \cdot h\right)}{d \cdot \left(d \cdot \ell\right)}, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (sqrt (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)))) 1.1)
   w0
   (* w0 (sqrt (fma -0.25 (/ (* (* M D) (* (* M D) h)) (* d (* d l))) 1.0)))))

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)))) <= 1.1) {
		tmp = w0;
	} else {
		tmp = w0 * sqrt(fma(-0.25, (((M * D) * ((M * D) * h)) / (d * (d * l))), 1.0));
	}
	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)))) <= 1.1)
		tmp = w0;
	else
		tmp = Float64(w0 * sqrt(fma(-0.25, Float64(Float64(Float64(M * D) * Float64(Float64(M * D) * h)) / Float64(d * Float64(d * l))), 1.0)));
	end
	return 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], 1.1], w0, N[(w0 * N[Sqrt[N[(-0.25 * N[(N[(N[(M * D), $MachinePrecision] * N[(N[(M * D), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] / N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)))) < 1.1000000000000001

   1. Initial program 99.9%

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

   2. Taylor expanded in M around 0

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

   4. Applied rewrites 99.3%

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

   if 1.1000000000000001 < (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 52.4%

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

   2. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. pow-prod-down N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 52.7

          \[\leadsto w0 \cdot \sqrt{\mathsf{fma}\left(-0.25, \frac{{\left(D \cdot M\right)}^{2} \cdot h}{\left(d \cdot d\right) \cdot \ell}, 1\right)} \]

   4. Applied rewrites 52.7%

      \[\leadsto w0 \cdot \sqrt{\color{blue}{\mathsf{fma}\left(-0.25, \frac{{\left(D \cdot M\right)}^{2} \cdot h}{\left(d \cdot d\right) \cdot \ell}, 1\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 57.1

         \[\leadsto w0 \cdot \sqrt{\mathsf{fma}\left(-0.25, \frac{{\left(D \cdot M\right)}^{2} \cdot h}{d \cdot \left(d \cdot \color{blue}{\ell}\right)}, 1\right)} \]

   6. Applied rewrites 57.1%

      \[\leadsto w0 \cdot \sqrt{\mathsf{fma}\left(-0.25, \frac{{\left(D \cdot M\right)}^{2} \cdot h}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}, 1\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 57.1

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

   8. Applied rewrites 57.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 59.8

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

   10. Applied rewrites 59.8%

       \[\leadsto w0 \cdot \sqrt{\mathsf{fma}\left(-0.25, \frac{\left(M \cdot D\right) \cdot \left(\left(M \cdot D\right) \cdot h\right)}{\color{blue}{d} \cdot \left(d \cdot \ell\right)}, 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 86.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq 10^{+231}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\left(\left(M \cdot \frac{D}{2 \cdot d}\right) \cdot \left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right)\right) \cdot h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(-0.25, \frac{\left(M \cdot D\right) \cdot \left(\left(M \cdot D\right) \cdot h\right)}{d \cdot \left(d \cdot \ell\right)}, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (pow (/ (* M D) (* 2.0 d)) 2.0) 1e+231)
   (*
    w0
    (sqrt (- 1.0 (/ (* (* (* M (/ D (* 2.0 d))) (* (/ D d) (* 0.5 M))) h) l))))
   (* w0 (sqrt (fma -0.25 (/ (* (* M D) (* (* M D) h)) (* d (* d l))) 1.0)))))

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) <= 1e+231) {
		tmp = w0 * sqrt((1.0 - ((((M * (D / (2.0 * d))) * ((D / d) * (0.5 * M))) * h) / l)));
	} else {
		tmp = w0 * sqrt(fma(-0.25, (((M * D) * ((M * D) * h)) / (d * (d * l))), 1.0));
	}
	return tmp;
}

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if ((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) <= 1e+231)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(Float64(M * Float64(D / Float64(2.0 * d))) * Float64(Float64(D / d) * Float64(0.5 * M))) * h) / l))));
	else
		tmp = Float64(w0 * sqrt(fma(-0.25, Float64(Float64(Float64(M * D) * Float64(Float64(M * D) * h)) / Float64(d * Float64(d * l))), 1.0)));
	end
	return tmp
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], 1e+231], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(N[(M * N[(D / N[(2.0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(D / d), $MachinePrecision] * N[(0.5 * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(-0.25 * N[(N[(N[(M * D), $MachinePrecision] * N[(N[(M * D), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] / N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) < 1.0000000000000001e231

   1. Initial program 90.6%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 95.6

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

   3. Applied rewrites 95.6%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-/.f64 95.6

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

   5. Applied rewrites 95.6%

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

   6. Taylor expanded in M around 0

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

      1. lower-*.f64 95.6

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

   8. Applied rewrites 95.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 95.5

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

   10. Applied rewrites 95.5%

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

   if 1.0000000000000001e231 < (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))

   1. Initial program 54.6%

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

   2. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. pow-prod-down N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 53.0

          \[\leadsto w0 \cdot \sqrt{\mathsf{fma}\left(-0.25, \frac{{\left(D \cdot M\right)}^{2} \cdot h}{\left(d \cdot d\right) \cdot \ell}, 1\right)} \]

   4. Applied rewrites 53.0%

      \[\leadsto w0 \cdot \sqrt{\color{blue}{\mathsf{fma}\left(-0.25, \frac{{\left(D \cdot M\right)}^{2} \cdot h}{\left(d \cdot d\right) \cdot \ell}, 1\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 57.8

         \[\leadsto w0 \cdot \sqrt{\mathsf{fma}\left(-0.25, \frac{{\left(D \cdot M\right)}^{2} \cdot h}{d \cdot \left(d \cdot \color{blue}{\ell}\right)}, 1\right)} \]

   6. Applied rewrites 57.8%

      \[\leadsto w0 \cdot \sqrt{\mathsf{fma}\left(-0.25, \frac{{\left(D \cdot M\right)}^{2} \cdot h}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}, 1\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 57.8

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

   8. Applied rewrites 57.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 60.5

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

   10. Applied rewrites 60.5%

       \[\leadsto w0 \cdot \sqrt{\mathsf{fma}\left(-0.25, \frac{\left(M \cdot D\right) \cdot \left(\left(M \cdot D\right) \cdot h\right)}{\color{blue}{d} \cdot \left(d \cdot \ell\right)}, 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 79.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -1 \cdot 10^{+245}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(-0.125, D \cdot \left(\left(\left(\frac{h}{\left(d \cdot d\right) \cdot \ell} \cdot M\right) \cdot M\right) \cdot D\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -1e+245)
   (* w0 (fma -0.125 (* D (* (* (* (/ h (* (* d d) l)) M) M) D)) 1.0))
   w0))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -1e+245) {
		tmp = w0 * fma(-0.125, (D * ((((h / ((d * d) * l)) * M) * M) * D)), 1.0);
	} else {
		tmp = w0;
	}
	return tmp;
}

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -1e+245)
		tmp = Float64(w0 * fma(-0.125, Float64(D * Float64(Float64(Float64(Float64(h / Float64(Float64(d * d) * l)) * M) * M) * D)), 1.0));
	else
		tmp = w0;
	end
	return tmp
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -1e+245], N[(w0 * N[(-0.125 * N[(D * N[(N[(N[(N[(h / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * M), $MachinePrecision] * M), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -1.00000000000000004e245

   1. Initial program 57.5%

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

   2. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. pow-prod-down N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 50.0

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

   4. Applied rewrites 50.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow-prod-down N/A

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

      8. associate-*r* N/A

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

      9. pow2 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 N/A

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

      21. lift-*.f64 44.9

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

   6. Applied rewrites 44.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 48.3

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

   8. Applied rewrites 48.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. pow2 N/A

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

      10. associate-*l/ N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. pow2 N/A

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

      14. lower-*.f64 N/A

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

   10. Applied rewrites 51.8%

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

   if -1.00000000000000004e245 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 89.3%

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

   2. Taylor expanded in M around 0

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

   4. Applied rewrites 88.7%

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

Alternative 5: 70.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \cdot D \leq 2.5 \cdot 10^{-98}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(-0.25, D \cdot \left(D \cdot \left(\frac{h}{\left(d \cdot d\right) \cdot \ell} \cdot \left(M \cdot M\right)\right)\right), 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* M D) 2.5e-98)
   w0
   (* w0 (sqrt (fma -0.25 (* D (* D (* (/ h (* (* d d) l)) (* M M)))) 1.0)))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((M * D) <= 2.5e-98) {
		tmp = w0;
	} else {
		tmp = w0 * sqrt(fma(-0.25, (D * (D * ((h / ((d * d) * l)) * (M * M)))), 1.0));
	}
	return tmp;
}

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64(M * D) <= 2.5e-98)
		tmp = w0;
	else
		tmp = Float64(w0 * sqrt(fma(-0.25, Float64(D * Float64(D * Float64(Float64(h / Float64(Float64(d * d) * l)) * Float64(M * M)))), 1.0)));
	end
	return tmp
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(M * D), $MachinePrecision], 2.5e-98], w0, N[(w0 * N[Sqrt[N[(-0.25 * N[(D * N[(D * N[(N[(h / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M D) < 2.50000000000000009e-98

   1. Initial program 83.8%

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

   2. Taylor expanded in M around 0

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

   4. Applied rewrites 74.6%

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

   if 2.50000000000000009e-98 < (*.f64 M D)

   1. Initial program 75.1%

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

   2. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. pow-prod-down N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 65.4

          \[\leadsto w0 \cdot \sqrt{\mathsf{fma}\left(-0.25, \frac{{\left(D \cdot M\right)}^{2} \cdot h}{\left(d \cdot d\right) \cdot \ell}, 1\right)} \]

   4. Applied rewrites 65.4%

      \[\leadsto w0 \cdot \sqrt{\color{blue}{\mathsf{fma}\left(-0.25, \frac{{\left(D \cdot M\right)}^{2} \cdot h}{\left(d \cdot d\right) \cdot \ell}, 1\right)}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow-prod-down N/A

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

      8. associate-*r* N/A

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

      9. pow2 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 N/A

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

      21. lift-*.f64 44.7

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

   6. Applied rewrites 44.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 58.7

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

   8. Applied rewrites 58.7%

      \[\leadsto w0 \cdot \sqrt{\mathsf{fma}\left(-0.25, D \cdot \color{blue}{\left(D \cdot \left(\frac{h}{\left(d \cdot d\right) \cdot \ell} \cdot \left(M \cdot M\right)\right)\right)}, 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 69.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \cdot D \leq 2.5 \cdot 10^{-98}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(-0.125, D \cdot \left(D \cdot \left(\frac{h}{\left(d \cdot d\right) \cdot \ell} \cdot \left(M \cdot M\right)\right)\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* M D) 2.5e-98)
   w0
   (* w0 (fma -0.125 (* D (* D (* (/ h (* (* d d) l)) (* M M)))) 1.0))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((M * D) <= 2.5e-98) {
		tmp = w0;
	} else {
		tmp = w0 * fma(-0.125, (D * (D * ((h / ((d * d) * l)) * (M * M)))), 1.0);
	}
	return tmp;
}

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64(M * D) <= 2.5e-98)
		tmp = w0;
	else
		tmp = Float64(w0 * fma(-0.125, Float64(D * Float64(D * Float64(Float64(h / Float64(Float64(d * d) * l)) * Float64(M * M)))), 1.0));
	end
	return tmp
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(M * D), $MachinePrecision], 2.5e-98], w0, N[(w0 * N[(-0.125 * N[(D * N[(D * N[(N[(h / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M D) < 2.50000000000000009e-98

   1. Initial program 83.8%

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

   2. Taylor expanded in M around 0

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

   4. Applied rewrites 74.6%

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

   if 2.50000000000000009e-98 < (*.f64 M D)

   1. Initial program 75.1%

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

   2. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. pow-prod-down N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 60.9

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

   4. Applied rewrites 60.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow-prod-down N/A

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

      8. associate-*r* N/A

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

      9. pow2 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 N/A

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

      21. lift-*.f64 42.6

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

   6. Applied rewrites 42.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 56.3

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

   8. Applied rewrites 56.3%

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

Alternative 7: 67.8% 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 81.4%

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

2. Taylor expanded in M around 0

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

4. Applied rewrites 67.8%

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

Reproduce

herbie shell --seed 2025097 
(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))))))