Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.3% → 88.9%

Time: 15.9s

Alternatives: 16

Speedup: 2.3×

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

real(8) function code(w0, m, d, h, l, d_1)
    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: 81.3% 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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 88.9% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.4%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

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

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

   6. lift-/.f64 N/A

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

   7. clear-num N/A

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

   8. un-div-inv N/A

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

   9. lift-pow.f64 N/A

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

   10. unpow2 N/A

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

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

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

   12. div-inv N/A

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

   13. times-frac N/A

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

   14. lower-fma.f64 N/A

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

4. Applied rewrites 91.8%

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

5. Final simplification 91.8%

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

Alternative 2: 78.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+172}:\\ \;\;\;\;\left(M \cdot \left(\left(h \cdot \left(D \cdot D\right)\right) \cdot \frac{-0.125}{d}\right)\right) \cdot \left(w0 \cdot \frac{M}{d \cdot \ell}\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* d 2.0)) 2.0) (/ h l)) -5e+172)
   (* (* M (* (* h (* D D)) (/ -0.125 d))) (* w0 (/ M (* d l))))
   (* w0 1.0)))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (d * 2.0)), 2.0) * (h / l)) <= -5e+172) {
		tmp = (M * ((h * (D * D)) * (-0.125 / d))) * (w0 * (M / (d * l)));
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (((((m * d) / (d_1 * 2.0d0)) ** 2.0d0) * (h / l)) <= (-5d+172)) then
        tmp = (m * ((h * (d * d)) * ((-0.125d0) / d_1))) * (w0 * (m / (d_1 * l)))
    else
        tmp = w0 * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((M * D) / (d * 2.0)), 2.0) * (h / l)) <= -5e+172) {
		tmp = (M * ((h * (D * D)) * (-0.125 / d))) * (w0 * (M / (d * l)));
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if (math.pow(((M * D) / (d * 2.0)), 2.0) * (h / l)) <= -5e+172:
		tmp = (M * ((h * (D * D)) * (-0.125 / d))) * (w0 * (M / (d * l)))
	else:
		tmp = w0 * 1.0
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(d * 2.0)) ^ 2.0) * Float64(h / l)) <= -5e+172)
		tmp = Float64(Float64(M * Float64(Float64(h * Float64(D * D)) * Float64(-0.125 / d))) * Float64(w0 * Float64(M / Float64(d * l))));
	else
		tmp = Float64(w0 * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (((((M * D) / (d * 2.0)) ^ 2.0) * (h / l)) <= -5e+172)
		tmp = (M * ((h * (D * D)) * (-0.125 / d))) * (w0 * (M / (d * l)));
	else
		tmp = w0 * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -5e+172], N[(N[(M * N[(N[(h * N[(D * D), $MachinePrecision]), $MachinePrecision] * N[(-0.125 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(w0 * N[(M / N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(w0 * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -5.0000000000000001e172

   1. Initial program 55.2%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 4.8%

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

   6. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   8. Applied rewrites 42.0%

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

   9. Taylor expanded in D around inf

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

   11. Applied rewrites 36.6%

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

   13. Applied rewrites 44.4%

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

   if -5.0000000000000001e172 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 86.4%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 90.2%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+172}:\\ \;\;\;\;\left(M \cdot \left(\left(h \cdot \left(D \cdot D\right)\right) \cdot \frac{-0.125}{d}\right)\right) \cdot \left(w0 \cdot \frac{M}{d \cdot \ell}\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+172}:\\ \;\;\;\;\left(D \cdot D\right) \cdot \left(\left(w0 \cdot \frac{M}{d \cdot \ell}\right) \cdot \left(M \cdot \frac{h \cdot -0.125}{d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* d 2.0)) 2.0) (/ h l)) -5e+172)
   (* (* D D) (* (* w0 (/ M (* d l))) (* M (/ (* h -0.125) d))))
   (* w0 1.0)))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (d * 2.0)), 2.0) * (h / l)) <= -5e+172) {
		tmp = (D * D) * ((w0 * (M / (d * l))) * (M * ((h * -0.125) / d)));
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (((((m * d) / (d_1 * 2.0d0)) ** 2.0d0) * (h / l)) <= (-5d+172)) then
        tmp = (d * d) * ((w0 * (m / (d_1 * l))) * (m * ((h * (-0.125d0)) / d_1)))
    else
        tmp = w0 * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((M * D) / (d * 2.0)), 2.0) * (h / l)) <= -5e+172) {
		tmp = (D * D) * ((w0 * (M / (d * l))) * (M * ((h * -0.125) / d)));
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if (math.pow(((M * D) / (d * 2.0)), 2.0) * (h / l)) <= -5e+172:
		tmp = (D * D) * ((w0 * (M / (d * l))) * (M * ((h * -0.125) / d)))
	else:
		tmp = w0 * 1.0
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(d * 2.0)) ^ 2.0) * Float64(h / l)) <= -5e+172)
		tmp = Float64(Float64(D * D) * Float64(Float64(w0 * Float64(M / Float64(d * l))) * Float64(M * Float64(Float64(h * -0.125) / d))));
	else
		tmp = Float64(w0 * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (((((M * D) / (d * 2.0)) ^ 2.0) * (h / l)) <= -5e+172)
		tmp = (D * D) * ((w0 * (M / (d * l))) * (M * ((h * -0.125) / d)));
	else
		tmp = w0 * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -5e+172], N[(N[(D * D), $MachinePrecision] * N[(N[(w0 * N[(M / N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(M * N[(N[(h * -0.125), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(w0 * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -5.0000000000000001e172

   1. Initial program 55.2%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 4.8%

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

   6. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   8. Applied rewrites 42.0%

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

   9. Taylor expanded in D around inf

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

   11. Applied rewrites 36.6%

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

   13. Applied rewrites 44.2%

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

   if -5.0000000000000001e172 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 86.4%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 90.2%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+172}:\\ \;\;\;\;\left(D \cdot D\right) \cdot \left(\left(w0 \cdot \frac{M}{d \cdot \ell}\right) \cdot \left(M \cdot \frac{h \cdot -0.125}{d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+114}:\\ \;\;\;\;\mathsf{fma}\left(D \cdot D, \frac{-0.125 \cdot \left(M \cdot \left(h \cdot \left(w0 \cdot M\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}, w0\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* d 2.0)) 2.0) (/ h l)) -2e+114)
   (fma (* D D) (/ (* -0.125 (* M (* h (* w0 M)))) (* d (* d l))) w0)
   (* w0 1.0)))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (d * 2.0)), 2.0) * (h / l)) <= -2e+114) {
		tmp = fma((D * D), ((-0.125 * (M * (h * (w0 * M)))) / (d * (d * l))), w0);
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(d * 2.0)) ^ 2.0) * Float64(h / l)) <= -2e+114)
		tmp = fma(Float64(D * D), Float64(Float64(-0.125 * Float64(M * Float64(h * Float64(w0 * M)))) / Float64(d * Float64(d * l))), w0);
	else
		tmp = Float64(w0 * 1.0);
	end
	return tmp
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -2e+114], N[(N[(D * D), $MachinePrecision] * N[(N[(-0.125 * N[(M * N[(h * N[(w0 * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + w0), $MachinePrecision], N[(w0 * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -2e114

   1. Initial program 56.7%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 4.9%

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

   6. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   8. Applied rewrites 40.8%

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

   10. Applied rewrites 42.8%

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

   if -2e114 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 86.2%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 91.1%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+114}:\\ \;\;\;\;\mathsf{fma}\left(D \cdot D, \frac{-0.125 \cdot \left(M \cdot \left(h \cdot \left(w0 \cdot M\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}, w0\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+114}:\\ \;\;\;\;\mathsf{fma}\left(D \cdot D, \frac{-0.125 \cdot \left(h \cdot \left(M \cdot \left(w0 \cdot M\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}, w0\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* d 2.0)) 2.0) (/ h l)) -2e+114)
   (fma (* D D) (/ (* -0.125 (* h (* M (* w0 M)))) (* d (* d l))) w0)
   (* w0 1.0)))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (d * 2.0)), 2.0) * (h / l)) <= -2e+114) {
		tmp = fma((D * D), ((-0.125 * (h * (M * (w0 * M)))) / (d * (d * l))), w0);
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(d * 2.0)) ^ 2.0) * Float64(h / l)) <= -2e+114)
		tmp = fma(Float64(D * D), Float64(Float64(-0.125 * Float64(h * Float64(M * Float64(w0 * M)))) / Float64(d * Float64(d * l))), w0);
	else
		tmp = Float64(w0 * 1.0);
	end
	return tmp
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -2e+114], N[(N[(D * D), $MachinePrecision] * N[(N[(-0.125 * N[(h * N[(M * N[(w0 * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + w0), $MachinePrecision], N[(w0 * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -2e114

   1. Initial program 56.7%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 4.9%

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

   6. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   8. Applied rewrites 40.8%

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

   if -2e114 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 86.2%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 91.1%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+114}:\\ \;\;\;\;\mathsf{fma}\left(D \cdot D, \frac{-0.125 \cdot \left(h \cdot \left(M \cdot \left(w0 \cdot M\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}, w0\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+210}:\\ \;\;\;\;\left(D \cdot D\right) \cdot \left(\left(-0.125 \cdot \left(M \cdot \left(M \cdot h\right)\right)\right) \cdot \frac{w0}{\ell \cdot \left(d \cdot d\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* d 2.0)) 2.0) (/ h l)) -2e+210)
   (* (* D D) (* (* -0.125 (* M (* M h))) (/ w0 (* l (* d d)))))
   (* w0 1.0)))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (d * 2.0)), 2.0) * (h / l)) <= -2e+210) {
		tmp = (D * D) * ((-0.125 * (M * (M * h))) * (w0 / (l * (d * d))));
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (((((m * d) / (d_1 * 2.0d0)) ** 2.0d0) * (h / l)) <= (-2d+210)) then
        tmp = (d * d) * (((-0.125d0) * (m * (m * h))) * (w0 / (l * (d_1 * d_1))))
    else
        tmp = w0 * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((M * D) / (d * 2.0)), 2.0) * (h / l)) <= -2e+210) {
		tmp = (D * D) * ((-0.125 * (M * (M * h))) * (w0 / (l * (d * d))));
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if (math.pow(((M * D) / (d * 2.0)), 2.0) * (h / l)) <= -2e+210:
		tmp = (D * D) * ((-0.125 * (M * (M * h))) * (w0 / (l * (d * d))))
	else:
		tmp = w0 * 1.0
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(d * 2.0)) ^ 2.0) * Float64(h / l)) <= -2e+210)
		tmp = Float64(Float64(D * D) * Float64(Float64(-0.125 * Float64(M * Float64(M * h))) * Float64(w0 / Float64(l * Float64(d * d)))));
	else
		tmp = Float64(w0 * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (((((M * D) / (d * 2.0)) ^ 2.0) * (h / l)) <= -2e+210)
		tmp = (D * D) * ((-0.125 * (M * (M * h))) * (w0 / (l * (d * d))));
	else
		tmp = w0 * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -2e+210], N[(N[(D * D), $MachinePrecision] * N[(N[(-0.125 * N[(M * N[(M * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(w0 / N[(l * N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(w0 * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -1.99999999999999985e210

   1. Initial program 52.7%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 4.7%

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

   6. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   8. Applied rewrites 42.5%

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

   9. Taylor expanded in D around inf

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

   11. Applied rewrites 36.8%

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

   13. Applied rewrites 40.7%

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

   if -1.99999999999999985e210 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 86.6%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 89.0%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+210}:\\ \;\;\;\;\left(D \cdot D\right) \cdot \left(\left(-0.125 \cdot \left(M \cdot \left(M \cdot h\right)\right)\right) \cdot \frac{w0}{\ell \cdot \left(d \cdot d\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \cdot D \leq 2.5 \cdot 10^{-11}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{M}{d} \cdot \frac{D \cdot \left(M \cdot D\right)}{-4 \cdot \left(d \cdot \ell\right)}, h, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{D}{d}, \frac{M \cdot \left(M \cdot D\right)}{d \cdot \ell} \cdot \left(h \cdot -0.25\right), 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* M D) 2.5e-11)
   (* w0 (sqrt (fma (* (/ M d) (/ (* D (* M D)) (* -4.0 (* d l)))) h 1.0)))
   (* w0 (sqrt (fma (/ D d) (* (/ (* M (* M D)) (* d l)) (* h -0.25)) 1.0)))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((M * D) <= 2.5e-11) {
		tmp = w0 * sqrt(fma(((M / d) * ((D * (M * D)) / (-4.0 * (d * l)))), h, 1.0));
	} else {
		tmp = w0 * sqrt(fma((D / d), (((M * (M * D)) / (d * l)) * (h * -0.25)), 1.0));
	}
	return tmp;
}

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64(M * D) <= 2.5e-11)
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(M / d) * Float64(Float64(D * Float64(M * D)) / Float64(-4.0 * Float64(d * l)))), h, 1.0)));
	else
		tmp = Float64(w0 * sqrt(fma(Float64(D / d), Float64(Float64(Float64(M * Float64(M * D)) / Float64(d * l)) * Float64(h * -0.25)), 1.0)));
	end
	return tmp
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(M * D), $MachinePrecision], 2.5e-11], N[(w0 * N[Sqrt[N[(N[(N[(M / d), $MachinePrecision] * N[(N[(D * N[(M * D), $MachinePrecision]), $MachinePrecision] / N[(-4.0 * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * h + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(N[(D / d), $MachinePrecision] * N[(N[(N[(M * N[(M * D), $MachinePrecision]), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision] * N[(h * -0.25), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M D) < 2.50000000000000009e-11

   1. Initial program 78.2%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

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

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

      12. div-inv N/A

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

      13. times-frac N/A

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

      14. lower-fma.f64 N/A

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

   4. Applied rewrites 93.7%

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

   6. Applied rewrites 74.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l* N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. lower-*.f64 N/A

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

      19. metadata-eval N/A

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

      20. lower-/.f64 85.9

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

   8. Applied rewrites 85.9%

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

   if 2.50000000000000009e-11 < (*.f64 M D)

   1. Initial program 83.5%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. swap-sqr N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. metadata-eval 76.9

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

   4. Applied rewrites 76.9%

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

   5. Taylor expanded in h around inf

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lft-mult-inverse N/A

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

      8. lower-fma.f64 N/A

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

   7. Applied rewrites 53.0%

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

   9. Applied rewrites 70.0%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot D \leq 2.5 \cdot 10^{-11}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{M}{d} \cdot \frac{D \cdot \left(M \cdot D\right)}{-4 \cdot \left(d \cdot \ell\right)}, h, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{D}{d}, \frac{M \cdot \left(M \cdot D\right)}{d \cdot \ell} \cdot \left(h \cdot -0.25\right), 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 88.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{M \cdot D}{d}\\ w0 \cdot \sqrt{\mathsf{fma}\left(t\_0 \cdot \frac{t\_0}{\ell \cdot -4}, h, 1\right)} \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (/ (* M D) d)))
   (* w0 (sqrt (fma (* t_0 (/ t_0 (* l -4.0))) h 1.0)))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = (M * D) / d;
	return w0 * sqrt(fma((t_0 * (t_0 / (l * -4.0))), h, 1.0));
}

Julia

function code(w0, M, D, h, l, d)
	t_0 = Float64(Float64(M * D) / d)
	return Float64(w0 * sqrt(fma(Float64(t_0 * Float64(t_0 / Float64(l * -4.0))), h, 1.0)))
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := Block[{t$95$0 = N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision]}, N[(w0 * N[Sqrt[N[(N[(t$95$0 * N[(t$95$0 / N[(l * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * h + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 79.4%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

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

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

   6. lift-/.f64 N/A

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

   7. clear-num N/A

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

   8. un-div-inv N/A

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

   9. lift-pow.f64 N/A

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

   10. unpow2 N/A

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

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

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

   12. div-inv N/A

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

   13. times-frac N/A

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

   14. lower-fma.f64 N/A

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

4. Applied rewrites 91.8%

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

6. Applied rewrites 73.2%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*r* N/A

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

   9. times-frac N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. associate-*l/ N/A

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

   13. lift-/.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. lower-*.f64 N/A

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

8. Applied rewrites 88.8%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. associate-/r* N/A

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

   6. lift-/.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-*.f64 91.0

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

10. Applied rewrites 91.0%

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

11. Final simplification 91.0%

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

Alternative 9: 75.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \cdot D \leq 10^{-187}:\\ \;\;\;\;w0 \cdot 1\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\left(M \cdot D\right) \cdot \frac{M \cdot D}{d \cdot \left(-4 \cdot \left(d \cdot \ell\right)\right)}, h, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* M D) 1e-187)
   (* w0 1.0)
   (* w0 (sqrt (fma (* (* M D) (/ (* M D) (* d (* -4.0 (* d l))))) h 1.0)))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((M * D) <= 1e-187) {
		tmp = w0 * 1.0;
	} else {
		tmp = w0 * sqrt(fma(((M * D) * ((M * D) / (d * (-4.0 * (d * l))))), h, 1.0));
	}
	return tmp;
}

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64(M * D) <= 1e-187)
		tmp = Float64(w0 * 1.0);
	else
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(M * D) * Float64(Float64(M * D) / Float64(d * Float64(-4.0 * Float64(d * l))))), h, 1.0)));
	end
	return tmp
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(M * D), $MachinePrecision], 1e-187], N[(w0 * 1.0), $MachinePrecision], N[(w0 * N[Sqrt[N[(N[(N[(M * D), $MachinePrecision] * N[(N[(M * D), $MachinePrecision] / N[(d * N[(-4.0 * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * h + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M D) < 1e-187

   1. Initial program 80.0%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 76.5%

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

   if 1e-187 < (*.f64 M D)

   1. Initial program 78.3%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

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

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

      12. div-inv N/A

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

      13. times-frac N/A

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

      14. lower-fma.f64 N/A

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

   4. Applied rewrites 89.9%

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

   6. Applied rewrites 72.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 81.1

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. associate-*l* N/A

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

      20. metadata-eval N/A

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

      21. metadata-eval N/A

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

      22. lower-*.f64 N/A

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

      23. metadata-eval 81.1

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

   8. Applied rewrites 81.1%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot D \leq 10^{-187}:\\ \;\;\;\;w0 \cdot 1\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\left(M \cdot D\right) \cdot \frac{M \cdot D}{d \cdot \left(-4 \cdot \left(d \cdot \ell\right)\right)}, h, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 73.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \cdot D \leq 2 \cdot 10^{-120}:\\ \;\;\;\;w0 \cdot 1\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(D, \left(h \cdot -0.25\right) \cdot \frac{M \cdot \left(M \cdot D\right)}{d \cdot \left(d \cdot \ell\right)}, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* M D) 2e-120)
   (* w0 1.0)
   (* w0 (sqrt (fma D (* (* h -0.25) (/ (* M (* M D)) (* d (* d l)))) 1.0)))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((M * D) <= 2e-120) {
		tmp = w0 * 1.0;
	} else {
		tmp = w0 * sqrt(fma(D, ((h * -0.25) * ((M * (M * D)) / (d * (d * l)))), 1.0));
	}
	return tmp;
}

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64(M * D) <= 2e-120)
		tmp = Float64(w0 * 1.0);
	else
		tmp = Float64(w0 * sqrt(fma(D, Float64(Float64(h * -0.25) * Float64(Float64(M * Float64(M * D)) / Float64(d * Float64(d * l)))), 1.0)));
	end
	return tmp
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(M * D), $MachinePrecision], 2e-120], N[(w0 * 1.0), $MachinePrecision], N[(w0 * N[Sqrt[N[(D * N[(N[(h * -0.25), $MachinePrecision] * N[(N[(M * N[(M * D), $MachinePrecision]), $MachinePrecision] / N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M D) < 1.99999999999999996e-120

   1. Initial program 78.5%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 76.4%

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

   if 1.99999999999999996e-120 < (*.f64 M D)

   1. Initial program 81.8%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. swap-sqr N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. metadata-eval 76.6

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

   4. Applied rewrites 76.6%

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

   5. Taylor expanded in h around inf

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lft-mult-inverse N/A

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

      8. lower-fma.f64 N/A

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

   7. Applied rewrites 57.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 57.8

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

   9. Applied rewrites 70.8%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot D \leq 2 \cdot 10^{-120}:\\ \;\;\;\;w0 \cdot 1\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(D, \left(h \cdot -0.25\right) \cdot \frac{M \cdot \left(M \cdot D\right)}{d \cdot \left(d \cdot \ell\right)}, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 74.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \cdot D \leq 10^{-187}:\\ \;\;\;\;w0 \cdot 1\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(D, M \cdot \left(\left(h \cdot -0.25\right) \cdot \frac{M \cdot D}{\ell \cdot \left(d \cdot d\right)}\right), 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* M D) 1e-187)
   (* w0 1.0)
   (* w0 (sqrt (fma D (* M (* (* h -0.25) (/ (* M D) (* l (* d d))))) 1.0)))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((M * D) <= 1e-187) {
		tmp = w0 * 1.0;
	} else {
		tmp = w0 * sqrt(fma(D, (M * ((h * -0.25) * ((M * D) / (l * (d * d))))), 1.0));
	}
	return tmp;
}

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64(M * D) <= 1e-187)
		tmp = Float64(w0 * 1.0);
	else
		tmp = Float64(w0 * sqrt(fma(D, Float64(M * Float64(Float64(h * -0.25) * Float64(Float64(M * D) / Float64(l * Float64(d * d))))), 1.0)));
	end
	return tmp
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(M * D), $MachinePrecision], 1e-187], N[(w0 * 1.0), $MachinePrecision], N[(w0 * N[Sqrt[N[(D * N[(M * N[(N[(h * -0.25), $MachinePrecision] * N[(N[(M * D), $MachinePrecision] / N[(l * N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M D) < 1e-187

   1. Initial program 80.0%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 76.5%

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

   if 1e-187 < (*.f64 M D)

   1. Initial program 78.3%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. swap-sqr N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. metadata-eval 73.8

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

   4. Applied rewrites 73.8%

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

   5. Taylor expanded in h around inf

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lft-mult-inverse N/A

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

      8. lower-fma.f64 N/A

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

   7. Applied rewrites 59.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 59.8

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

   9. Applied rewrites 71.2%

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

   11. Applied rewrites 78.1%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot D \leq 10^{-187}:\\ \;\;\;\;w0 \cdot 1\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(D, M \cdot \left(\left(h \cdot -0.25\right) \cdot \frac{M \cdot D}{\ell \cdot \left(d \cdot d\right)}\right), 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 85.3% accurate, 2.1× speedup

Math

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

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (*
  w0
  (sqrt (fma (/ (* (* M D) 0.5) d) (* h (/ (* M D) (* (* d -2.0) l))) 1.0))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * sqrt(fma((((M * D) * 0.5) / d), (h * ((M * D) / ((d * -2.0) * l))), 1.0));
}

Julia

function code(w0, M, D, h, l, d)
	return Float64(w0 * sqrt(fma(Float64(Float64(Float64(M * D) * 0.5) / d), Float64(h * Float64(Float64(M * D) / Float64(Float64(d * -2.0) * l))), 1.0)))
end

Wolfram

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

Derivation

1. Initial program 79.4%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

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

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

   6. lift-/.f64 N/A

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

   7. clear-num N/A

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

   8. un-div-inv N/A

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

   9. lift-pow.f64 N/A

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

   10. unpow2 N/A

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

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

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

   12. div-inv N/A

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

   13. times-frac N/A

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

   14. lower-fma.f64 N/A

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

4. Applied rewrites 91.8%

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

   1. lift-fma.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. div-inv N/A

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

   5. lift-/.f64 N/A

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

   6. remove-double-div N/A

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

   7. associate-*l* N/A

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

   8. lower-fma.f64 N/A

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

6. Applied rewrites 88.8%

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

7. Final simplification 88.8%

   \[\leadsto w0 \cdot \sqrt{\mathsf{fma}\left(\frac{\left(M \cdot D\right) \cdot 0.5}{d}, h \cdot \frac{M \cdot D}{\left(d \cdot -2\right) \cdot \ell}, 1\right)} \]
8. Add Preprocessing

Alternative 13: 84.6% accurate, 2.3× speedup

Math

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

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (* w0 (sqrt (fma (* (/ (* M D) d) (/ (* M D) (* -4.0 (* d l)))) h 1.0))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * sqrt(fma((((M * D) / d) * ((M * D) / (-4.0 * (d * l)))), h, 1.0));
}

Julia

function code(w0, M, D, h, l, d)
	return Float64(w0 * sqrt(fma(Float64(Float64(Float64(M * D) / d) * Float64(Float64(M * D) / Float64(-4.0 * Float64(d * l)))), h, 1.0)))
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := N[(w0 * N[Sqrt[N[(N[(N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision] * N[(N[(M * D), $MachinePrecision] / N[(-4.0 * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * h + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.4%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

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

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

   6. lift-/.f64 N/A

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

   7. clear-num N/A

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

   8. un-div-inv N/A

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

   9. lift-pow.f64 N/A

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

   10. unpow2 N/A

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

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

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

   12. div-inv N/A

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

   13. times-frac N/A

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

   14. lower-fma.f64 N/A

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

4. Applied rewrites 91.8%

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

6. Applied rewrites 73.2%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*r* N/A

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

   9. times-frac N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. associate-*l/ N/A

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

   13. lift-/.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. lower-*.f64 N/A

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

8. Applied rewrites 88.8%

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

9. Final simplification 88.8%

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

Alternative 14: 83.5% accurate, 2.3× speedup

Math

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

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (* w0 (sqrt (fma (* (/ (* M D) (* -4.0 (* d l))) (* D (/ M d))) h 1.0))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * sqrt(fma((((M * D) / (-4.0 * (d * l))) * (D * (M / d))), h, 1.0));
}

Julia

function code(w0, M, D, h, l, d)
	return Float64(w0 * sqrt(fma(Float64(Float64(Float64(M * D) / Float64(-4.0 * Float64(d * l))) * Float64(D * Float64(M / d))), h, 1.0)))
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := N[(w0 * N[Sqrt[N[(N[(N[(N[(M * D), $MachinePrecision] / N[(-4.0 * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(D * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * h + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.4%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

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

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

   6. lift-/.f64 N/A

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

   7. clear-num N/A

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

   8. un-div-inv N/A

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

   9. lift-pow.f64 N/A

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

   10. unpow2 N/A

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

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

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

   12. div-inv N/A

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

   13. times-frac N/A

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

   14. lower-fma.f64 N/A

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

4. Applied rewrites 91.8%

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

6. Applied rewrites 73.2%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*r* N/A

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

   9. times-frac N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. associate-*l/ N/A

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

   13. lift-/.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. lower-*.f64 N/A

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

8. Applied rewrites 88.8%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 87.7

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

10. Applied rewrites 87.7%

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

11. Final simplification 87.7%

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

Alternative 15: 77.9% accurate, 2.3× speedup

Math

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

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (* w0 (sqrt (fma (/ D d) (* (/ (* M (* M D)) (* d l)) (* h -0.25)) 1.0))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * sqrt(fma((D / d), (((M * (M * D)) / (d * l)) * (h * -0.25)), 1.0));
}

Julia

function code(w0, M, D, h, l, d)
	return Float64(w0 * sqrt(fma(Float64(D / d), Float64(Float64(Float64(M * Float64(M * D)) / Float64(d * l)) * Float64(h * -0.25)), 1.0)))
end

Wolfram

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

Derivation

1. Initial program 79.4%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. frac-times N/A

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

   6. associate-/l* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. swap-sqr N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. lower-*.f64 N/A

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

   17. lower-*.f64 N/A

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

   18. metadata-eval 71.3

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

4. Applied rewrites 71.3%

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

5. Taylor expanded in h around inf

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

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

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

   2. metadata-eval N/A

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

   3. +-commutative N/A

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

   4. distribute-rgt-in N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. lft-mult-inverse N/A

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

   8. lower-fma.f64 N/A

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

7. Applied rewrites 64.4%

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

9. Applied rewrites 79.5%

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

Alternative 16: 67.6% accurate, 26.2× speedup

Math

\[\begin{array}{l} \\ w0 \cdot 1 \end{array} \]

FPCore

(FPCore (w0 M D h l d) :precision binary64 (* w0 1.0))

C

double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * 1.0;
}

Fortran

real(8) function code(w0, m, d, h, l, d_1)
    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 * 1.0d0
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * 1.0;
}

Python

def code(w0, M, D, h, l, d):
	return w0 * 1.0

Julia

function code(w0, M, D, h, l, d)
	return Float64(w0 * 1.0)
end

MATLAB

function tmp = code(w0, M, D, h, l, d)
	tmp = w0 * 1.0;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := N[(w0 * 1.0), $MachinePrecision]

Derivation

1. Initial program 79.4%

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

3. Taylor expanded in M around 0

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

5. Applied rewrites 71.2%

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

Reproduce

herbie shell --seed 2024222 
(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))))))