Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.3% → 89.1%

Time: 14.7s

Alternatives: 15

Speedup: 2.0×

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: 89.1% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (*
  w0
  (sqrt (fma (/ (/ (* M D) (* d -2.0)) l) (* (/ (* M D) (* d 2.0)) 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)) * 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)) * 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] * h), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.8%

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

   1. lift--.f64 N/A

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

   2. 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 89.2%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/r/ N/A

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

   4. /-rgt-identity N/A

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

   5. lower-*.f64 89.2

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

6. Applied rewrites 89.2%

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

7. Final simplification 89.2%

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

Alternative 2: 85.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq -20000000000:\\ \;\;\;\;w0 \cdot \sqrt{D \cdot \frac{\frac{M \cdot \left(D \cdot -0.25\right)}{d \cdot \ell} \cdot \left(M \cdot h\right)}{d}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* d 2.0)) 2.0) (/ h l)) -20000000000.0)
   (* w0 (sqrt (* D (/ (* (/ (* M (* D -0.25)) (* d l)) (* M h)) 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)) <= -20000000000.0) {
		tmp = w0 * sqrt((D * ((((M * (D * -0.25)) / (d * l)) * (M * h)) / 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)) <= (-20000000000.0d0)) then
        tmp = w0 * sqrt((d * ((((m * (d * (-0.25d0))) / (d_1 * l)) * (m * h)) / 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)) <= -20000000000.0) {
		tmp = w0 * Math.sqrt((D * ((((M * (D * -0.25)) / (d * l)) * (M * h)) / 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)) <= -20000000000.0:
		tmp = w0 * math.sqrt((D * ((((M * (D * -0.25)) / (d * l)) * (M * h)) / 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)) <= -20000000000.0)
		tmp = Float64(w0 * sqrt(Float64(D * Float64(Float64(Float64(Float64(M * Float64(D * -0.25)) / Float64(d * l)) * Float64(M * h)) / 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)) <= -20000000000.0)
		tmp = w0 * sqrt((D * ((((M * (D * -0.25)) / (d * l)) * (M * h)) / 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], -20000000000.0], N[(w0 * N[Sqrt[N[(D * N[(N[(N[(N[(M * N[(D * -0.25), $MachinePrecision]), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision] * N[(M * h), $MachinePrecision]), $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)) < -2e10

   1. Initial program 64.2%

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

   3. Taylor expanded in M around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 44.0

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

   5. Applied rewrites 44.0%

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

   7. Applied rewrites 48.3%

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

   9. Applied rewrites 58.9%

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

   11. Applied rewrites 62.7%

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

   if -2e10 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 85.9%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 96.4%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq -20000000000:\\ \;\;\;\;w0 \cdot \sqrt{D \cdot \frac{\frac{M \cdot \left(D \cdot -0.25\right)}{d \cdot \ell} \cdot \left(M \cdot h\right)}{d}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq -20000000000:\\ \;\;\;\;w0 \cdot \sqrt{D \cdot \left(\frac{D \cdot -0.25}{d \cdot \ell} \cdot \left(M \cdot \frac{M \cdot h}{d}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* d 2.0)) 2.0) (/ h l)) -20000000000.0)
   (* w0 (sqrt (* D (* (/ (* D -0.25) (* d l)) (* M (/ (* M h) 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)) <= -20000000000.0) {
		tmp = w0 * sqrt((D * (((D * -0.25) / (d * l)) * (M * ((M * h) / 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)) <= (-20000000000.0d0)) then
        tmp = w0 * sqrt((d * (((d * (-0.25d0)) / (d_1 * l)) * (m * ((m * h) / 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)) <= -20000000000.0) {
		tmp = w0 * Math.sqrt((D * (((D * -0.25) / (d * l)) * (M * ((M * h) / 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)) <= -20000000000.0:
		tmp = w0 * math.sqrt((D * (((D * -0.25) / (d * l)) * (M * ((M * h) / 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)) <= -20000000000.0)
		tmp = Float64(w0 * sqrt(Float64(D * Float64(Float64(Float64(D * -0.25) / Float64(d * l)) * Float64(M * Float64(Float64(M * h) / 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)) <= -20000000000.0)
		tmp = w0 * sqrt((D * (((D * -0.25) / (d * l)) * (M * ((M * h) / 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], -20000000000.0], N[(w0 * N[Sqrt[N[(D * N[(N[(N[(D * -0.25), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision] * N[(M * N[(N[(M * h), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -2e10

   1. Initial program 64.2%

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

   3. Taylor expanded in M around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 44.0

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

   5. Applied rewrites 44.0%

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

   7. Applied rewrites 48.3%

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

   9. Applied rewrites 58.9%

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

   if -2e10 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 85.9%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 96.4%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq -20000000000:\\ \;\;\;\;w0 \cdot \sqrt{D \cdot \left(\frac{D \cdot -0.25}{d \cdot \ell} \cdot \left(M \cdot \frac{M \cdot h}{d}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+57}:\\ \;\;\;\;w0 \cdot \sqrt{\frac{\left(D \cdot -0.25\right) \cdot \left(M \cdot \left(D \cdot \left(M \cdot h\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* d 2.0)) 2.0) (/ h l)) -5e+57)
   (* w0 (sqrt (/ (* (* D -0.25) (* M (* D (* M h)))) (* d (* 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+57) {
		tmp = w0 * sqrt((((D * -0.25) * (M * (D * (M * h)))) / (d * (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+57)) then
        tmp = w0 * sqrt((((d * (-0.25d0)) * (m * (d * (m * h)))) / (d_1 * (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+57) {
		tmp = w0 * Math.sqrt((((D * -0.25) * (M * (D * (M * h)))) / (d * (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+57:
		tmp = w0 * math.sqrt((((D * -0.25) * (M * (D * (M * h)))) / (d * (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+57)
		tmp = Float64(w0 * sqrt(Float64(Float64(Float64(D * -0.25) * Float64(M * Float64(D * Float64(M * h)))) / Float64(d * 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+57)
		tmp = w0 * sqrt((((D * -0.25) * (M * (D * (M * h)))) / (d * (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+57], N[(w0 * N[Sqrt[N[(N[(N[(D * -0.25), $MachinePrecision] * N[(M * N[(D * N[(M * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * N[(d * l), $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)) < -4.99999999999999972e57

   1. Initial program 62.7%

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

   3. Taylor expanded in M around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 45.9

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

   5. Applied rewrites 45.9%

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

   7. Applied rewrites 50.4%

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

   9. Applied rewrites 59.4%

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

   if -4.99999999999999972e57 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 86.1%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 95.1%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+57}:\\ \;\;\;\;w0 \cdot \sqrt{\frac{\left(D \cdot -0.25\right) \cdot \left(M \cdot \left(D \cdot \left(M \cdot h\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+57}:\\ \;\;\;\;w0 \cdot \sqrt{\frac{M \cdot \left(D \cdot \left(M \cdot \left(h \cdot \left(D \cdot -0.25\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* d 2.0)) 2.0) (/ h l)) -5e+57)
   (* w0 (sqrt (/ (* M (* D (* M (* h (* D -0.25))))) (* d (* 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+57) {
		tmp = w0 * sqrt(((M * (D * (M * (h * (D * -0.25))))) / (d * (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+57)) then
        tmp = w0 * sqrt(((m * (d * (m * (h * (d * (-0.25d0)))))) / (d_1 * (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+57) {
		tmp = w0 * Math.sqrt(((M * (D * (M * (h * (D * -0.25))))) / (d * (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+57:
		tmp = w0 * math.sqrt(((M * (D * (M * (h * (D * -0.25))))) / (d * (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+57)
		tmp = Float64(w0 * sqrt(Float64(Float64(M * Float64(D * Float64(M * Float64(h * Float64(D * -0.25))))) / Float64(d * 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+57)
		tmp = w0 * sqrt(((M * (D * (M * (h * (D * -0.25))))) / (d * (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+57], N[(w0 * N[Sqrt[N[(N[(M * N[(D * N[(M * N[(h * N[(D * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * N[(d * l), $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)) < -4.99999999999999972e57

   1. Initial program 62.7%

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

   3. Taylor expanded in M around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 45.9

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

   5. Applied rewrites 45.9%

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

   7. Applied rewrites 50.4%

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

   9. Applied rewrites 58.2%

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

   if -4.99999999999999972e57 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 86.1%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 95.1%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+57}:\\ \;\;\;\;w0 \cdot \sqrt{\frac{M \cdot \left(D \cdot \left(M \cdot \left(h \cdot \left(D \cdot -0.25\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+57}:\\ \;\;\;\;w0 \cdot \sqrt{\frac{D \cdot \left(D \cdot -0.25\right)}{d \cdot \left(d \cdot \ell\right)} \cdot \left(M \cdot \left(M \cdot h\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* d 2.0)) 2.0) (/ h l)) -5e+57)
   (* w0 (sqrt (* (/ (* D (* D -0.25)) (* d (* d l))) (* M (* M h)))))
   (* 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+57) {
		tmp = w0 * sqrt((((D * (D * -0.25)) / (d * (d * l))) * (M * (M * h))));
	} 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+57)) then
        tmp = w0 * sqrt((((d * (d * (-0.25d0))) / (d_1 * (d_1 * l))) * (m * (m * h))))
    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+57) {
		tmp = w0 * Math.sqrt((((D * (D * -0.25)) / (d * (d * l))) * (M * (M * h))));
	} 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+57:
		tmp = w0 * math.sqrt((((D * (D * -0.25)) / (d * (d * l))) * (M * (M * h))))
	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+57)
		tmp = Float64(w0 * sqrt(Float64(Float64(Float64(D * Float64(D * -0.25)) / Float64(d * Float64(d * l))) * Float64(M * Float64(M * h)))));
	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+57)
		tmp = w0 * sqrt((((D * (D * -0.25)) / (d * (d * l))) * (M * (M * h))));
	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+57], N[(w0 * N[Sqrt[N[(N[(N[(D * N[(D * -0.25), $MachinePrecision]), $MachinePrecision] / N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(M * N[(M * h), $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)) < -4.99999999999999972e57

   1. Initial program 62.7%

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

   3. Taylor expanded in M around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 45.9

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

   5. Applied rewrites 45.9%

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

   7. Applied rewrites 50.4%

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

   9. Applied rewrites 53.5%

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

   if -4.99999999999999972e57 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 86.1%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 95.1%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+57}:\\ \;\;\;\;w0 \cdot \sqrt{\frac{D \cdot \left(D \cdot -0.25\right)}{d \cdot \left(d \cdot \ell\right)} \cdot \left(M \cdot \left(M \cdot h\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq -20000000000:\\ \;\;\;\;\mathsf{fma}\left(D \cdot D, \frac{h \cdot -0.125}{d} \cdot \frac{M \cdot \left(w0 \cdot M\right)}{d \cdot \ell}, w0\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* d 2.0)) 2.0) (/ h l)) -20000000000.0)
   (fma (* D D) (* (/ (* h -0.125) d) (/ (* M (* w0 M)) (* 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)) <= -20000000000.0) {
		tmp = fma((D * D), (((h * -0.125) / d) * ((M * (w0 * M)) / (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)) <= -20000000000.0)
		tmp = fma(Float64(D * D), Float64(Float64(Float64(h * -0.125) / d) * Float64(Float64(M * Float64(w0 * M)) / 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], -20000000000.0], N[(N[(D * D), $MachinePrecision] * N[(N[(N[(h * -0.125), $MachinePrecision] / d), $MachinePrecision] * N[(N[(M * N[(w0 * M), $MachinePrecision]), $MachinePrecision] / 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)) < -2e10

   1. Initial program 64.2%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      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

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

   5. Applied rewrites 41.6%

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

   7. Applied rewrites 48.0%

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

   if -2e10 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 85.9%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 96.4%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq -20000000000:\\ \;\;\;\;\mathsf{fma}\left(D \cdot D, \frac{h \cdot -0.125}{d} \cdot \frac{M \cdot \left(w0 \cdot M\right)}{d \cdot \ell}, w0\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+110}:\\ \;\;\;\;\mathsf{fma}\left(D \cdot D, \left(h \cdot -0.125\right) \cdot \frac{M \cdot \left(w0 \cdot M\right)}{d \cdot \left(d \cdot \ell\right)}, w0\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* d 2.0)) 2.0) (/ h l)) -2e+110)
   (fma (* D D) (* (* h -0.125) (/ (* 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+110) {
		tmp = fma((D * D), ((h * -0.125) * ((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+110)
		tmp = fma(Float64(D * D), Float64(Float64(h * -0.125) * Float64(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+110], N[(N[(D * D), $MachinePrecision] * N[(N[(h * -0.125), $MachinePrecision] * N[(N[(M * N[(w0 * M), $MachinePrecision]), $MachinePrecision] / N[(d * N[(d * l), $MachinePrecision]), $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)) < -2e110

   1. Initial program 61.0%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      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

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

   5. Applied rewrites 45.0%

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

   7. Applied rewrites 48.6%

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

   if -2e110 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 86.3%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 93.7%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+110}:\\ \;\;\;\;\mathsf{fma}\left(D \cdot D, \left(h \cdot -0.125\right) \cdot \frac{M \cdot \left(w0 \cdot M\right)}{d \cdot \left(d \cdot \ell\right)}, w0\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 78.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq -4 \cdot 10^{+208}:\\ \;\;\;\;\frac{\left(D \cdot D\right) \cdot \left(-0.125 \cdot \left(w0 \cdot \left(M \cdot \left(M \cdot h\right)\right)\right)\right)}{\ell \cdot \left(d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* d 2.0)) 2.0) (/ h l)) -4e+208)
   (/ (* (* D D) (* -0.125 (* w0 (* M (* M h))))) (* 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)) <= -4e+208) {
		tmp = ((D * D) * (-0.125 * (w0 * (M * (M * h))))) / (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)) <= (-4d+208)) then
        tmp = ((d * d) * ((-0.125d0) * (w0 * (m * (m * h))))) / (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)) <= -4e+208) {
		tmp = ((D * D) * (-0.125 * (w0 * (M * (M * h))))) / (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)) <= -4e+208:
		tmp = ((D * D) * (-0.125 * (w0 * (M * (M * h))))) / (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)) <= -4e+208)
		tmp = Float64(Float64(Float64(D * D) * Float64(-0.125 * Float64(w0 * Float64(M * Float64(M * h))))) / 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)) <= -4e+208)
		tmp = ((D * D) * (-0.125 * (w0 * (M * (M * h))))) / (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], -4e+208], N[(N[(N[(D * D), $MachinePrecision] * N[(-0.125 * N[(w0 * N[(M * N[(M * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[(d * d), $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)) < -3.9999999999999999e208

   1. Initial program 59.8%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      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

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

   5. Applied rewrites 44.9%

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

   6. Taylor expanded in D around inf

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

   8. Applied rewrites 44.9%

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

   10. Applied rewrites 48.2%

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

   if -3.9999999999999999e208 < (*.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 92.8%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq -4 \cdot 10^{+208}:\\ \;\;\;\;\frac{\left(D \cdot D\right) \cdot \left(-0.125 \cdot \left(w0 \cdot \left(M \cdot \left(M \cdot h\right)\right)\right)\right)}{\ell \cdot \left(d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 78.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+228}:\\ \;\;\;\;\frac{\left(D \cdot D\right) \cdot \left(-0.125 \cdot \left(w0 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* d 2.0)) 2.0) (/ h l)) -2e+228)
   (/ (* (* D D) (* -0.125 (* w0 (* h (* M M))))) (* d (* 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)) <= -2e+228) {
		tmp = ((D * D) * (-0.125 * (w0 * (h * (M * M))))) / (d * (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)) <= (-2d+228)) then
        tmp = ((d * d) * ((-0.125d0) * (w0 * (h * (m * m))))) / (d_1 * (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)) <= -2e+228) {
		tmp = ((D * D) * (-0.125 * (w0 * (h * (M * M))))) / (d * (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)) <= -2e+228:
		tmp = ((D * D) * (-0.125 * (w0 * (h * (M * M))))) / (d * (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)) <= -2e+228)
		tmp = Float64(Float64(Float64(D * D) * Float64(-0.125 * Float64(w0 * Float64(h * Float64(M * M))))) / Float64(d * 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)) <= -2e+228)
		tmp = ((D * D) * (-0.125 * (w0 * (h * (M * M))))) / (d * (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], -2e+228], N[(N[(N[(D * D), $MachinePrecision] * N[(-0.125 * N[(w0 * N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * N[(d * l), $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)) < -1.9999999999999998e228

   1. Initial program 57.9%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      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

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

   5. Applied rewrites 45.4%

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

   6. Taylor expanded in D around inf

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

   8. Applied rewrites 47.0%

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

   10. Applied rewrites 50.6%

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

   if -1.9999999999999998e228 < (*.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 91.4%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+228}:\\ \;\;\;\;\frac{\left(D \cdot D\right) \cdot \left(-0.125 \cdot \left(w0 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 79.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+225}:\\ \;\;\;\;D \cdot \frac{-0.125 \cdot \left(D \cdot \left(w0 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\ell \cdot \left(d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* d 2.0)) 2.0) (/ h l)) -5e+225)
   (* D (/ (* -0.125 (* D (* w0 (* h (* M M))))) (* 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)) <= -5e+225) {
		tmp = D * ((-0.125 * (D * (w0 * (h * (M * M))))) / (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)) <= (-5d+225)) then
        tmp = d * (((-0.125d0) * (d * (w0 * (h * (m * m))))) / (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)) <= -5e+225) {
		tmp = D * ((-0.125 * (D * (w0 * (h * (M * M))))) / (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)) <= -5e+225:
		tmp = D * ((-0.125 * (D * (w0 * (h * (M * M))))) / (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)) <= -5e+225)
		tmp = Float64(D * Float64(Float64(-0.125 * Float64(D * Float64(w0 * Float64(h * Float64(M * M))))) / 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)) <= -5e+225)
		tmp = D * ((-0.125 * (D * (w0 * (h * (M * M))))) / (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], -5e+225], N[(D * N[(N[(-0.125 * N[(D * N[(w0 * N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[(d * 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)) < -4.99999999999999981e225

   1. Initial program 58.5%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      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

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

   5. Applied rewrites 44.7%

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

   6. Taylor expanded in D around inf

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

   8. Applied rewrites 46.3%

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

   10. Applied rewrites 48.1%

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

   if -4.99999999999999981e225 < (*.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 91.9%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+225}:\\ \;\;\;\;D \cdot \frac{-0.125 \cdot \left(D \cdot \left(w0 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\ell \cdot \left(d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 87.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \cdot 2 \leq 5 \cdot 10^{+103}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{M \cdot D}{d \cdot -2}, \frac{\frac{\left(M \cdot D\right) \cdot h}{d \cdot 2}}{\ell}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, h \cdot \frac{M \cdot D}{-2 \cdot \left(d \cdot \ell\right)}, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* d 2.0) 5e+103)
   (*
    w0
    (sqrt (fma (/ (* M D) (* d -2.0)) (/ (/ (* (* M D) h) (* d 2.0)) l) 1.0)))
   (*
    w0
    (sqrt
     (fma (/ (* M D) (* d 2.0)) (* h (/ (* M D) (* -2.0 (* d l)))) 1.0)))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((d * 2.0) <= 5e+103) {
		tmp = w0 * sqrt(fma(((M * D) / (d * -2.0)), ((((M * D) * h) / (d * 2.0)) / l), 1.0));
	} else {
		tmp = w0 * sqrt(fma(((M * D) / (d * 2.0)), (h * ((M * D) / (-2.0 * (d * l)))), 1.0));
	}
	return tmp;
}

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64(d * 2.0) <= 5e+103)
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(M * D) / Float64(d * -2.0)), Float64(Float64(Float64(Float64(M * D) * h) / Float64(d * 2.0)) / l), 1.0)));
	else
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(M * D) / Float64(d * 2.0)), Float64(h * Float64(Float64(M * D) / Float64(-2.0 * Float64(d * l)))), 1.0)));
	end
	return tmp
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(d * 2.0), $MachinePrecision], 5e+103], N[(w0 * N[Sqrt[N[(N[(N[(M * D), $MachinePrecision] / N[(d * -2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(M * D), $MachinePrecision] * h), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision] * N[(h * N[(N[(M * D), $MachinePrecision] / N[(-2.0 * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) d) < 5e103

   1. Initial program 80.4%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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. Applied rewrites 73.7%

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

   6. Applied rewrites 88.6%

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

   if 5e103 < (*.f64 #s(literal 2 binary64) d)

   1. Initial program 77.2%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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. Applied rewrites 77.2%

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

   6. Applied rewrites 85.6%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \cdot 2 \leq 5 \cdot 10^{+103}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{M \cdot D}{d \cdot -2}, \frac{\frac{\left(M \cdot D\right) \cdot h}{d \cdot 2}}{\ell}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, h \cdot \frac{M \cdot D}{-2 \cdot \left(d \cdot \ell\right)}, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 78.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \cdot D \leq 2 \cdot 10^{-284}:\\ \;\;\;\;w0 \cdot 1\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, h \cdot \frac{M \cdot D}{-2 \cdot \left(d \cdot \ell\right)}, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* M D) 2e-284)
   (* w0 1.0)
   (*
    w0
    (sqrt
     (fma (/ (* M D) (* d 2.0)) (* h (/ (* M D) (* -2.0 (* 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-284) {
		tmp = w0 * 1.0;
	} else {
		tmp = w0 * sqrt(fma(((M * D) / (d * 2.0)), (h * ((M * D) / (-2.0 * (d * l)))), 1.0));
	}
	return tmp;
}

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64(M * D) <= 2e-284)
		tmp = Float64(w0 * 1.0);
	else
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(M * D) / Float64(d * 2.0)), Float64(h * Float64(Float64(M * D) / Float64(-2.0 * Float64(d * l)))), 1.0)));
	end
	return tmp
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(M * D), $MachinePrecision], 2e-284], N[(w0 * 1.0), $MachinePrecision], N[(w0 * N[Sqrt[N[(N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision] * N[(h * N[(N[(M * D), $MachinePrecision] / N[(-2.0 * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M D) < 2.00000000000000007e-284

   1. Initial program 82.7%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 78.7%

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

   if 2.00000000000000007e-284 < (*.f64 M D)

   1. Initial program 74.9%

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

      1. lift--.f64 N/A

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

      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. Applied rewrites 62.4%

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

   6. Applied rewrites 82.1%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot D \leq 2 \cdot 10^{-284}:\\ \;\;\;\;w0 \cdot 1\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, h \cdot \frac{M \cdot D}{-2 \cdot \left(d \cdot \ell\right)}, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 73.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{d \cdot 2} \leq 2 \cdot 10^{+34}:\\ \;\;\;\;w0 \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{D \cdot \left(-0.125 \cdot \left(h \cdot \left(M \cdot \left(w0 \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}, D, w0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (/ (* M D) (* d 2.0)) 2e+34)
   (* w0 1.0)
   (fma (/ (* D (* -0.125 (* h (* M (* w0 M))))) (* d (* d l))) D w0)))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (((M * D) / (d * 2.0)) <= 2e+34) {
		tmp = w0 * 1.0;
	} else {
		tmp = fma(((D * (-0.125 * (h * (M * (w0 * M))))) / (d * (d * l))), D, w0);
	}
	return tmp;
}

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64(Float64(M * D) / Float64(d * 2.0)) <= 2e+34)
		tmp = Float64(w0 * 1.0);
	else
		tmp = fma(Float64(Float64(D * Float64(-0.125 * Float64(h * Float64(M * Float64(w0 * M))))) / Float64(d * Float64(d * l))), D, w0);
	end
	return tmp
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2e+34], N[(w0 * 1.0), $MachinePrecision], N[(N[(N[(D * N[(-0.125 * N[(h * N[(M * N[(w0 * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * D + w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) < 1.99999999999999989e34

   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. Taylor expanded in M around 0

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

   5. Applied rewrites 80.4%

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

   if 1.99999999999999989e34 < (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d))

   1. Initial program 61.0%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      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

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

   5. Applied rewrites 33.9%

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

   7. Applied rewrites 53.9%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{d \cdot 2} \leq 2 \cdot 10^{+34}:\\ \;\;\;\;w0 \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{D \cdot \left(-0.125 \cdot \left(h \cdot \left(M \cdot \left(w0 \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}, D, w0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 68.6% 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 79.8%

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

3. Taylor expanded in M around 0

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

5. Applied rewrites 70.7%

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

Reproduce

herbie shell --seed 2024237 
(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))))))