Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.2% → 87.0%

Time: 15.2s

Alternatives: 12

Speedup: 0.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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: 81.2% 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: 87.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{-16}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{M \cdot D}{d \cdot -2}, \left(M \cdot D\right) \cdot \left(\frac{h}{\ell} \cdot \frac{0.5}{d}\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -2e-16)
   (*
    w0
    (sqrt (fma (/ (* M D) (* d -2.0)) (* (* M D) (* (/ h l) (/ 0.5 d))) 1.0)))
   w0))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -2e-16) {
		tmp = w0 * sqrt(fma(((M * D) / (d * -2.0)), ((M * D) * ((h / l) * (0.5 / d))), 1.0));
	} else {
		tmp = w0;
	}
	return tmp;
}

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -2e-16)
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(M * D) / Float64(d * -2.0)), Float64(Float64(M * D) * Float64(Float64(h / l) * Float64(0.5 / d))), 1.0)));
	else
		tmp = w0;
	end
	return tmp
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -2e-16], N[(w0 * N[Sqrt[N[(N[(N[(M * D), $MachinePrecision] / N[(d * -2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(M * D), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -2e-16

   1. Initial program 66.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

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

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

      14. *-commutative N/A

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

   4. Applied rewrites 67.6%

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

   if -2e-16 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 89.0%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 99.4%

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

      1. *-rgt-identity 99.4

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

   7. Applied rewrites 99.4%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{-16}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{M \cdot D}{d \cdot -2}, \left(M \cdot D\right) \cdot \left(\frac{h}{\ell} \cdot \frac{0.5}{d}\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{-16}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{M}{d}, \left(D \cdot -0.5\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{h \cdot 0.5}{d \cdot \ell}\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -2e-16)
   (*
    w0
    (sqrt (fma (/ M d) (* (* D -0.5) (* (* M D) (/ (* h 0.5) (* d l)))) 1.0)))
   w0))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -2e-16) {
		tmp = w0 * sqrt(fma((M / d), ((D * -0.5) * ((M * D) * ((h * 0.5) / (d * l)))), 1.0));
	} else {
		tmp = w0;
	}
	return tmp;
}

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -2e-16)
		tmp = Float64(w0 * sqrt(fma(Float64(M / d), Float64(Float64(D * -0.5) * Float64(Float64(M * D) * Float64(Float64(h * 0.5) / Float64(d * l)))), 1.0)));
	else
		tmp = w0;
	end
	return tmp
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -2e-16], N[(w0 * N[Sqrt[N[(N[(M / d), $MachinePrecision] * N[(N[(D * -0.5), $MachinePrecision] * N[(N[(M * D), $MachinePrecision] * N[(N[(h * 0.5), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -2e-16

   1. Initial program 66.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

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

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

      14. *-commutative N/A

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

   4. Applied rewrites 67.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. times-frac N/A

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

      13. associate-*l* N/A

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

      14. lower-fma.f64 N/A

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

   6. Applied rewrites 65.3%

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

   if -2e-16 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 89.0%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 99.4%

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

      1. *-rgt-identity 99.4

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

   7. Applied rewrites 99.4%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{-16}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{M}{d}, \left(D \cdot -0.5\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{h \cdot 0.5}{d \cdot \ell}\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -0.05:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{M}{d}, \frac{-0.25 \cdot \left(h \cdot \left(M \cdot \left(D \cdot D\right)\right)\right)}{d \cdot \ell}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -0.05)
   (* w0 (sqrt (fma (/ M d) (/ (* -0.25 (* h (* M (* D D)))) (* d l)) 1.0)))
   w0))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -0.05) {
		tmp = w0 * sqrt(fma((M / d), ((-0.25 * (h * (M * (D * D)))) / (d * l)), 1.0));
	} else {
		tmp = w0;
	}
	return tmp;
}

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -0.05)
		tmp = Float64(w0 * sqrt(fma(Float64(M / d), Float64(Float64(-0.25 * Float64(h * Float64(M * Float64(D * D)))) / Float64(d * l)), 1.0)));
	else
		tmp = w0;
	end
	return tmp
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -0.05], N[(w0 * N[Sqrt[N[(N[(M / d), $MachinePrecision] * N[(N[(-0.25 * N[(h * N[(M * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 65.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

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

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

      14. *-commutative N/A

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

   4. Applied rewrites 66.8%

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

      1. lift-*.f64 N/A

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

      2. frac-times N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 62.7

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

   6. Applied rewrites 62.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. times-frac N/A

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

      13. associate-*l* N/A

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

      14. lower-fma.f64 N/A

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

   8. Applied rewrites 59.3%

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

   9. Taylor expanded in D around 0

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. associate-*r* N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 N/A

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

       9. lower-*.f64 53.4

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

   11. Applied rewrites 53.4%

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

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

   1. Initial program 89.1%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 99.1%

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

      1. *-rgt-identity 99.1

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

   7. Applied rewrites 99.1%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -0.05:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{M}{d}, \frac{-0.25 \cdot \left(h \cdot \left(M \cdot \left(D \cdot D\right)\right)\right)}{d \cdot \ell}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -0.0005:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(-M \cdot D, \frac{\left(M \cdot D\right) \cdot h}{\left(d \cdot \ell\right) \cdot \left(d \cdot 4\right)}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -0.0005)
   (* w0 (sqrt (fma (- (* M D)) (/ (* (* M D) h) (* (* d l) (* d 4.0))) 1.0)))
   w0))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -0.0005) {
		tmp = w0 * sqrt(fma(-(M * D), (((M * D) * h) / ((d * l) * (d * 4.0))), 1.0));
	} else {
		tmp = w0;
	}
	return tmp;
}

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -0.0005)
		tmp = Float64(w0 * sqrt(fma(Float64(-Float64(M * D)), Float64(Float64(Float64(M * D) * h) / Float64(Float64(d * l) * Float64(d * 4.0))), 1.0)));
	else
		tmp = w0;
	end
	return tmp
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -0.0005], N[(w0 * N[Sqrt[N[((-N[(M * D), $MachinePrecision]) * N[(N[(N[(M * D), $MachinePrecision] * h), $MachinePrecision] / N[(N[(d * l), $MachinePrecision] * N[(d * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -5.0000000000000001e-4

   1. Initial program 65.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. swap-sqr N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. metadata-eval 54.0

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

   4. Applied rewrites 54.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

   6. Applied rewrites 52.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 55.6

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

   8. Applied rewrites 55.6%

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

   if -5.0000000000000001e-4 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 89.0%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 99.4%

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

      1. *-rgt-identity 99.4

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

   7. Applied rewrites 99.4%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -0.0005:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(-M \cdot D, \frac{\left(M \cdot D\right) \cdot h}{\left(d \cdot \ell\right) \cdot \left(d \cdot 4\right)}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -0.05:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(-M \cdot D, \frac{M \cdot \left(D \cdot h\right)}{4 \cdot \left(d \cdot \left(d \cdot \ell\right)\right)}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -0.05)
   (* w0 (sqrt (fma (- (* M D)) (/ (* M (* D h)) (* 4.0 (* d (* d l)))) 1.0)))
   w0))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -0.05) {
		tmp = w0 * sqrt(fma(-(M * D), ((M * (D * h)) / (4.0 * (d * (d * l)))), 1.0));
	} else {
		tmp = w0;
	}
	return tmp;
}

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -0.05)
		tmp = Float64(w0 * sqrt(fma(Float64(-Float64(M * D)), Float64(Float64(M * Float64(D * h)) / Float64(4.0 * Float64(d * Float64(d * l)))), 1.0)));
	else
		tmp = w0;
	end
	return tmp
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -0.05], N[(w0 * N[Sqrt[N[((-N[(M * D), $MachinePrecision]) * N[(N[(M * N[(D * h), $MachinePrecision]), $MachinePrecision] / N[(4.0 * N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 65.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. swap-sqr N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. metadata-eval 53.5

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

   4. Applied rewrites 53.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

   6. Applied rewrites 52.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 52.0

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 48.1

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lower-*.f64 48.1

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

      19. lift-*.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. associate-*l* N/A

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

      22. lift-*.f64 N/A

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

      23. lower-*.f64 51.2

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

   8. Applied rewrites 51.2%

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

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

   1. Initial program 89.1%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 99.1%

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

      1. *-rgt-identity 99.1

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

   7. Applied rewrites 99.1%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -0.05:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(-M \cdot D, \frac{M \cdot \left(D \cdot h\right)}{4 \cdot \left(d \cdot \left(d \cdot \ell\right)\right)}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -0.0005:\\ \;\;\;\;w0 \cdot \sqrt{1 - M \cdot \left(D \cdot \frac{\left(M \cdot D\right) \cdot h}{\ell \cdot \left(4 \cdot \left(d \cdot d\right)\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -0.0005)
   (* w0 (sqrt (- 1.0 (* M (* D (/ (* (* M D) h) (* l (* 4.0 (* d d)))))))))
   w0))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -0.0005) {
		tmp = w0 * sqrt((1.0 - (M * (D * (((M * D) * h) / (l * (4.0 * (d * d))))))));
	} else {
		tmp = w0;
	}
	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) / (2.0d0 * d_1)) ** 2.0d0) * (h / l)) <= (-0.0005d0)) then
        tmp = w0 * sqrt((1.0d0 - (m * (d * (((m * d) * h) / (l * (4.0d0 * (d_1 * d_1))))))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -0.0005) {
		tmp = w0 * Math.sqrt((1.0 - (M * (D * (((M * D) * h) / (l * (4.0 * (d * d))))))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -0.0005:
		tmp = w0 * math.sqrt((1.0 - (M * (D * (((M * D) * h) / (l * (4.0 * (d * d))))))))
	else:
		tmp = w0
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -0.0005)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(M * Float64(D * Float64(Float64(Float64(M * D) * h) / Float64(l * Float64(4.0 * Float64(d * d)))))))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (((((M * D) / (2.0 * d)) ^ 2.0) * (h / l)) <= -0.0005)
		tmp = w0 * sqrt((1.0 - (M * (D * (((M * D) * h) / (l * (4.0 * (d * d))))))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -0.0005], N[(w0 * N[Sqrt[N[(1.0 - N[(M * N[(D * N[(N[(N[(M * D), $MachinePrecision] * h), $MachinePrecision] / N[(l * N[(4.0 * N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -5.0000000000000001e-4

   1. Initial program 65.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. swap-sqr N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. metadata-eval 54.0

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

   4. Applied rewrites 54.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. frac-times N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 52.6

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

   6. Applied rewrites 52.6%

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

   if -5.0000000000000001e-4 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 89.0%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 99.4%

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

      1. *-rgt-identity 99.4

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

   7. Applied rewrites 99.4%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -0.0005:\\ \;\;\;\;w0 \cdot \sqrt{1 - M \cdot \left(D \cdot \frac{\left(M \cdot D\right) \cdot h}{\ell \cdot \left(4 \cdot \left(d \cdot d\right)\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.1% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -5e+114)
   (fma (* w0 -0.125) (* (/ (* (* M D) (* M D)) d) (/ h (* d l))) w0)
   w0))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -5e+114) {
		tmp = fma((w0 * -0.125), ((((M * D) * (M * D)) / d) * (h / (d * l))), w0);
	} else {
		tmp = w0;
	}
	return tmp;
}

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -5e+114)
		tmp = fma(Float64(w0 * -0.125), Float64(Float64(Float64(Float64(M * D) * Float64(M * D)) / d) * Float64(h / Float64(d * l))), w0);
	else
		tmp = w0;
	end
	return tmp
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -5e+114], N[(N[(w0 * -0.125), $MachinePrecision] * N[(N[(N[(N[(M * D), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * N[(h / N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + w0), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -5.0000000000000001e114

   1. Initial program 58.8%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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 41.8%

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

   6. Taylor expanded in w0 around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 37.5

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

   8. Applied rewrites 37.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. unswap-sqr N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 48.9

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

   10. Applied rewrites 48.9%

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

   if -5.0000000000000001e114 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 89.8%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 92.9%

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

      1. *-rgt-identity 92.9

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

   7. Applied rewrites 92.9%

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

Alternative 8: 79.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+178}:\\ \;\;\;\;\mathsf{fma}\left(w0 \cdot -0.125, \left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \frac{h}{d \cdot \left(d \cdot \ell\right)}, w0\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -2e+178)
   (fma (* w0 -0.125) (* (* (* M D) (* M D)) (/ h (* d (* d l)))) w0)
   w0))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -2e+178) {
		tmp = fma((w0 * -0.125), (((M * D) * (M * D)) * (h / (d * (d * l)))), w0);
	} else {
		tmp = w0;
	}
	return tmp;
}

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -2e+178)
		tmp = fma(Float64(w0 * -0.125), Float64(Float64(Float64(M * D) * Float64(M * D)) * Float64(h / Float64(d * Float64(d * l)))), w0);
	else
		tmp = w0;
	end
	return tmp
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -2e+178], N[(N[(w0 * -0.125), $MachinePrecision] * N[(N[(N[(M * D), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] * N[(h / N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + w0), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -2.0000000000000001e178

   1. Initial program 58.2%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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 42.3%

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

   6. Taylor expanded in w0 around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 37.9

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

   8. Applied rewrites 37.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. unswap-sqr N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 46.1

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

      19. lift-*.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. associate-*l* N/A

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

      22. lift-*.f64 N/A

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

      23. lower-*.f64 46.2

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

   10. Applied rewrites 46.2%

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

   if -2.0000000000000001e178 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 89.8%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 92.5%

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

      1. *-rgt-identity 92.5

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

   7. Applied rewrites 92.5%

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

Alternative 9: 79.8% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -5e+114)
   (fma D (* (* w0 -0.125) (/ (* M (* D (* M h))) (* d (* d l)))) w0)
   w0))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -5e+114) {
		tmp = fma(D, ((w0 * -0.125) * ((M * (D * (M * h))) / (d * (d * l)))), w0);
	} else {
		tmp = w0;
	}
	return tmp;
}

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -5e+114)
		tmp = fma(D, Float64(Float64(w0 * -0.125) * Float64(Float64(M * Float64(D * Float64(M * h))) / Float64(d * Float64(d * l)))), w0);
	else
		tmp = w0;
	end
	return tmp
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -5e+114], N[(D * N[(N[(w0 * -0.125), $MachinePrecision] * N[(N[(M * N[(D * N[(M * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + w0), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -5.0000000000000001e114

   1. Initial program 58.8%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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 41.8%

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

   6. Taylor expanded in w0 around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 37.5

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

   8. Applied rewrites 37.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lower-+.f64 N/A

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

   10. Applied rewrites 43.8%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. associate-/l* N/A

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

       9. lift-*.f64 N/A

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

       10. *-commutative N/A

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

       11. associate-*l* N/A

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

       12. lower-fma.f64 N/A

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

   12. Applied rewrites 47.1%

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

   if -5.0000000000000001e114 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 89.8%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 92.9%

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

      1. *-rgt-identity 92.9

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

   7. Applied rewrites 92.9%

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

4. Final simplification 80.6%

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

Alternative 10: 78.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+219}:\\ \;\;\;\;\left(\left(D \cdot D\right) \cdot \left(M \cdot \left(-0.125 \cdot \left(M \cdot h\right)\right)\right)\right) \cdot \frac{w0}{d \cdot \left(d \cdot \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -5e+219)
   (* (* (* D D) (* M (* -0.125 (* M h)))) (/ w0 (* d (* d l))))
   w0))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -5e+219) {
		tmp = ((D * D) * (M * (-0.125 * (M * h)))) * (w0 / (d * (d * l)));
	} else {
		tmp = w0;
	}
	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) / (2.0d0 * d_1)) ** 2.0d0) * (h / l)) <= (-5d+219)) then
        tmp = ((d * d) * (m * ((-0.125d0) * (m * h)))) * (w0 / (d_1 * (d_1 * l)))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -5e+219) {
		tmp = ((D * D) * (M * (-0.125 * (M * h)))) * (w0 / (d * (d * l)));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -5e+219:
		tmp = ((D * D) * (M * (-0.125 * (M * h)))) * (w0 / (d * (d * l)))
	else:
		tmp = w0
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -5e+219)
		tmp = Float64(Float64(Float64(D * D) * Float64(M * Float64(-0.125 * Float64(M * h)))) * Float64(w0 / Float64(d * Float64(d * l))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (((((M * D) / (2.0 * d)) ^ 2.0) * (h / l)) <= -5e+219)
		tmp = ((D * D) * (M * (-0.125 * (M * h)))) * (w0 / (d * (d * l)));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -5e+219], N[(N[(N[(D * D), $MachinePrecision] * N[(M * N[(-0.125 * N[(M * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(w0 / N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -5e219

   1. Initial program 57.6%

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

   3. Taylor expanded in M around 0

      \[\leadsto \color{blue}{w0 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}} \]
   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 42.9%

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

   6. Taylor expanded in w0 around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 38.5

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

   8. Applied rewrites 38.5%

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

   9. Taylor expanded in D around inf

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

       1. *-commutative N/A

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

       2. associate-/l* N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. associate-*r/ N/A

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

       9. lower-/.f64 N/A

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

       10. associate-*r* N/A

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

       11. associate-*r* N/A

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

       12. lower-*.f64 N/A

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

       13. lower-*.f64 N/A

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

       14. lower-*.f64 N/A

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

       15. unpow2 N/A

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

       16. lower-*.f64 N/A

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

       17. lower-*.f64 N/A

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

       18. unpow2 N/A

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

       19. lower-*.f64 38.5

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

   11. Applied rewrites 38.5%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. associate-*r* N/A

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. associate-/l* N/A

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

       11. associate-*r* N/A

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

       12. lower-*.f64 N/A

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

   13. Applied rewrites 40.2%

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

   if -5e219 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 89.9%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 92.0%

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

      1. *-rgt-identity 92.0

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

   7. Applied rewrites 92.0%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+219}:\\ \;\;\;\;\left(\left(D \cdot D\right) \cdot \left(M \cdot \left(-0.125 \cdot \left(M \cdot h\right)\right)\right)\right) \cdot \frac{w0}{d \cdot \left(d \cdot \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 78.3% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) (- INFINITY))
   (* (* D D) (* w0 (/ (* M (* -0.125 (* M h))) (* d (* d l)))))
   w0))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -inf.0

   1. Initial program 55.6%

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

   3. Taylor expanded in M around 0

      \[\leadsto \color{blue}{w0 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}} \]
   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 44.8%

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

   6. Taylor expanded in w0 around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 40.2

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

   8. Applied rewrites 40.2%

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

   9. Taylor expanded in D around inf

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

       1. *-commutative N/A

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

       2. associate-/l* N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. associate-*r/ N/A

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

       9. lower-/.f64 N/A

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

       10. associate-*r* N/A

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

       11. associate-*r* N/A

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

       12. lower-*.f64 N/A

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

       13. lower-*.f64 N/A

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

       14. lower-*.f64 N/A

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

       15. unpow2 N/A

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

       16. lower-*.f64 N/A

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

       17. lower-*.f64 N/A

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

       18. unpow2 N/A

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

       19. lower-*.f64 40.1

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

   11. Applied rewrites 40.1%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*r* N/A

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lift-*.f64 N/A

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

       8. associate-/l* N/A

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

       9. lower-*.f64 N/A

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

       10. lower-/.f64 40.3

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

       11. lift-*.f64 N/A

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

       12. *-commutative N/A

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

       13. lift-*.f64 N/A

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

       14. lift-*.f64 N/A

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

       15. associate-*r* N/A

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

       16. lift-*.f64 N/A

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

       17. associate-*l* N/A

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

       18. lower-*.f64 N/A

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

       19. lower-*.f64 42.0

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

   13. Applied rewrites 42.0%

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

   if -inf.0 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 90.1%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 90.7%

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

      1. *-rgt-identity 90.7

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

   7. Applied rewrites 90.7%

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

4. Final simplification 78.5%

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

Alternative 12: 67.9% accurate, 157.0× speedup

Math

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

FPCore

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

C

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

Fortran

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.4%

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

3. Taylor expanded in M around 0

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

5. Applied rewrites 69.1%

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

   1. *-rgt-identity 69.1

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

7. Applied rewrites 69.1%

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

Reproduce

herbie shell --seed 2024221 
(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))))))