Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 82.2% → 88.7%

Time: 17.4s

Alternatives: 9

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: 82.2% 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: 88.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{M \cdot D}{2 \cdot d}\\ \mathbf{if}\;{t\_0}^{2} \cdot \frac{h}{\ell} \leq 5 \cdot 10^{-10}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(t\_0, \frac{h}{\ell} \cdot \frac{M \cdot D}{d \cdot -2}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\left(M \cdot h\right) \cdot \frac{D}{-2 \cdot \left(d \cdot \ell\right)}, 0.5 \cdot \left(D \cdot \frac{M}{d}\right), 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (/ (* M D) (* 2.0 d))))
   (if (<= (* (pow t_0 2.0) (/ h l)) 5e-10)
     (* w0 (sqrt (fma t_0 (* (/ h l) (/ (* M D) (* d -2.0))) 1.0)))
     (*
      w0
      (sqrt
       (fma (* (* M h) (/ D (* -2.0 (* d l)))) (* 0.5 (* D (/ M d))) 1.0))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = (M * D) / (2.0 * d);
	double tmp;
	if ((pow(t_0, 2.0) * (h / l)) <= 5e-10) {
		tmp = w0 * sqrt(fma(t_0, ((h / l) * ((M * D) / (d * -2.0))), 1.0));
	} else {
		tmp = w0 * sqrt(fma(((M * h) * (D / (-2.0 * (d * l)))), (0.5 * (D * (M / d))), 1.0));
	}
	return tmp;
}

Julia

function code(w0, M, D, h, l, d)
	t_0 = Float64(Float64(M * D) / Float64(2.0 * d))
	tmp = 0.0
	if (Float64((t_0 ^ 2.0) * Float64(h / l)) <= 5e-10)
		tmp = Float64(w0 * sqrt(fma(t_0, Float64(Float64(h / l) * Float64(Float64(M * D) / Float64(d * -2.0))), 1.0)));
	else
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(M * h) * Float64(D / Float64(-2.0 * Float64(d * l)))), Float64(0.5 * Float64(D * Float64(M / d))), 1.0)));
	end
	return tmp
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := Block[{t$95$0 = N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Power[t$95$0, 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], 5e-10], N[(w0 * N[Sqrt[N[(t$95$0 * N[(N[(h / l), $MachinePrecision] * N[(N[(M * D), $MachinePrecision] / N[(d * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(N[(N[(M * h), $MachinePrecision] * N[(D / N[(-2.0 * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[(D * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < 5.00000000000000031e-10

   1. Initial program 87.2%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. distribute-rgt-neg-in N/A

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

      9. associate-*l* N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 88.0%

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

   if 5.00000000000000031e-10 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 0.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. distribute-rgt-neg-in N/A

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

      9. associate-*l* N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 11.7%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 11.7

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

   6. Applied rewrites 84.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 89.7

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 89.7

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

   8. Applied rewrites 89.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 89.7

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

   10. Applied rewrites 89.7%

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

4. Final simplification 88.1%

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

Alternative 2: 88.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := M \cdot \frac{D}{d}\\ \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 5 \cdot 10^{-10}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(t\_0 \cdot -0.5, \frac{h}{\ell} \cdot \left(0.5 \cdot t\_0\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\left(M \cdot h\right) \cdot \frac{D}{-2 \cdot \left(d \cdot \ell\right)}, 0.5 \cdot \left(D \cdot \frac{M}{d}\right), 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (* M (/ D d))))
   (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) 5e-10)
     (* w0 (sqrt (fma (* t_0 -0.5) (* (/ h l) (* 0.5 t_0)) 1.0)))
     (*
      w0
      (sqrt
       (fma (* (* M h) (/ D (* -2.0 (* d l)))) (* 0.5 (* D (/ M d))) 1.0))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = M * (D / d);
	double tmp;
	if ((pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= 5e-10) {
		tmp = w0 * sqrt(fma((t_0 * -0.5), ((h / l) * (0.5 * t_0)), 1.0));
	} else {
		tmp = w0 * sqrt(fma(((M * h) * (D / (-2.0 * (d * l)))), (0.5 * (D * (M / d))), 1.0));
	}
	return tmp;
}

Julia

function code(w0, M, D, h, l, d)
	t_0 = Float64(M * Float64(D / d))
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= 5e-10)
		tmp = Float64(w0 * sqrt(fma(Float64(t_0 * -0.5), Float64(Float64(h / l) * Float64(0.5 * t_0)), 1.0)));
	else
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(M * h) * Float64(D / Float64(-2.0 * Float64(d * l)))), Float64(0.5 * Float64(D * Float64(M / d))), 1.0)));
	end
	return tmp
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := Block[{t$95$0 = N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], 5e-10], N[(w0 * N[Sqrt[N[(N[(t$95$0 * -0.5), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * N[(0.5 * t$95$0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(N[(N[(M * h), $MachinePrecision] * N[(D / N[(-2.0 * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[(D * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < 5.00000000000000031e-10

   1. Initial program 87.2%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. distribute-rgt-neg-in N/A

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

      9. associate-*l* N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 88.0%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

   6. Applied rewrites 88.3%

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

   if 5.00000000000000031e-10 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 0.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. distribute-rgt-neg-in N/A

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

      9. associate-*l* N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 11.7%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 11.7

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

   6. Applied rewrites 84.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 89.7

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 89.7

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

   8. Applied rewrites 89.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 89.7

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

   10. Applied rewrites 89.7%

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

4. Final simplification 88.4%

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

Alternative 3: 83.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{-8}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{\left(M \cdot D\right) \cdot \left(D \cdot \left(M \cdot h\right)\right)}{\left(-2 \cdot \left(d \cdot \ell\right)\right) \cdot \left(2 \cdot d\right)}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -5e-8)
   (*
    w0
    (sqrt
     (+ 1.0 (/ (* (* M D) (* D (* M h))) (* (* -2.0 (* d l)) (* 2.0 d))))))
   (* w0 1.0)))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -5e-8) {
		tmp = w0 * sqrt((1.0 + (((M * D) * (D * (M * h))) / ((-2.0 * (d * l)) * (2.0 * d)))));
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Fortran

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

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -5e-8) {
		tmp = w0 * Math.sqrt((1.0 + (((M * D) * (D * (M * h))) / ((-2.0 * (d * l)) * (2.0 * d)))));
	} else {
		tmp = w0 * 1.0;
	}
	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-8:
		tmp = w0 * math.sqrt((1.0 + (((M * D) * (D * (M * h))) / ((-2.0 * (d * l)) * (2.0 * d)))))
	else:
		tmp = w0 * 1.0
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -5e-8)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(Float64(M * D) * Float64(D * Float64(M * h))) / Float64(Float64(-2.0 * Float64(d * l)) * Float64(2.0 * d))))));
	else
		tmp = Float64(w0 * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (((((M * D) / (2.0 * d)) ^ 2.0) * (h / l)) <= -5e-8)
		tmp = w0 * sqrt((1.0 + (((M * D) * (D * (M * h))) / ((-2.0 * (d * l)) * (2.0 * d)))));
	else
		tmp = w0 * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -5e-8], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(N[(M * D), $MachinePrecision] * N[(D * N[(M * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(-2.0 * N[(d * l), $MachinePrecision]), $MachinePrecision] * N[(2.0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -4.9999999999999998e-8

   1. Initial program 67.2%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. distribute-rgt-neg-in N/A

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

      9. associate-*l* N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 69.2%

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

   6. Applied rewrites 53.3%

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

   if -4.9999999999999998e-8 < (*.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 h around 0

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

   5. Applied rewrites 97.9%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{-8}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{\left(M \cdot D\right) \cdot \left(D \cdot \left(M \cdot h\right)\right)}{\left(-2 \cdot \left(d \cdot \ell\right)\right) \cdot \left(2 \cdot d\right)}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -50000:\\ \;\;\;\;w0 \cdot \sqrt{D \cdot \left(D \cdot \left(\left(M \cdot -0.25\right) \cdot \frac{M \cdot h}{d \cdot \left(d \cdot \ell\right)}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -50000.0)
   (* w0 (sqrt (* D (* D (* (* M -0.25) (/ (* M h) (* d (* d l))))))))
   (* w0 1.0)))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -50000.0) {
		tmp = w0 * sqrt((D * (D * ((M * -0.25) * ((M * h) / (d * (d * l)))))));
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l)) <= (-50000.0d0)) then
        tmp = w0 * sqrt((d * (d * ((m * (-0.25d0)) * ((m * h) / (d_1 * (d_1 * l)))))))
    else
        tmp = w0 * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -50000.0) {
		tmp = w0 * Math.sqrt((D * (D * ((M * -0.25) * ((M * h) / (d * (d * l)))))));
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -50000.0:
		tmp = w0 * math.sqrt((D * (D * ((M * -0.25) * ((M * h) / (d * (d * l)))))))
	else:
		tmp = w0 * 1.0
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -50000.0)
		tmp = Float64(w0 * sqrt(Float64(D * Float64(D * Float64(Float64(M * -0.25) * Float64(Float64(M * h) / Float64(d * Float64(d * l))))))));
	else
		tmp = Float64(w0 * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (((((M * D) / (2.0 * d)) ^ 2.0) * (h / l)) <= -50000.0)
		tmp = w0 * sqrt((D * (D * ((M * -0.25) * ((M * h) / (d * (d * l)))))));
	else
		tmp = w0 * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -50000.0], N[(w0 * N[Sqrt[N[(D * N[(D * N[(N[(M * -0.25), $MachinePrecision] * N[(N[(M * h), $MachinePrecision] / N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -5e4

   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. Taylor expanded in h around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 39.0

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

   5. Applied rewrites 39.0%

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

   7. Applied rewrites 43.6%

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

   9. Applied rewrites 46.5%

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

   11. Applied rewrites 51.1%

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

   if -5e4 < (*.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 h around 0

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

   5. Applied rewrites 96.9%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -50000:\\ \;\;\;\;w0 \cdot \sqrt{D \cdot \left(D \cdot \left(\left(M \cdot -0.25\right) \cdot \frac{M \cdot h}{d \cdot \left(d \cdot \ell\right)}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -4 \cdot 10^{+188}:\\ \;\;\;\;\mathsf{fma}\left(D \cdot D, \left(M \cdot -0.125\right) \cdot \left(M \cdot \frac{h \cdot w0}{d \cdot \left(d \cdot \ell\right)}\right), w0\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -4e+188)
   (fma (* D D) (* (* M -0.125) (* M (/ (* h w0) (* d (* d l))))) w0)
   (* w0 1.0)))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -4e+188) {
		tmp = fma((D * D), ((M * -0.125) * (M * ((h * w0) / (d * (d * l))))), w0);
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -4e+188)
		tmp = fma(Float64(D * D), Float64(Float64(M * -0.125) * Float64(M * Float64(Float64(h * w0) / Float64(d * Float64(d * l))))), w0);
	else
		tmp = Float64(w0 * 1.0);
	end
	return tmp
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -4e+188], N[(N[(D * D), $MachinePrecision] * N[(N[(M * -0.125), $MachinePrecision] * N[(M * N[(N[(h * w0), $MachinePrecision] / N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + w0), $MachinePrecision], N[(w0 * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -4.0000000000000001e188

   1. Initial program 61.0%

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

   3. Taylor expanded in h 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 42.4%

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

   7. Applied rewrites 49.0%

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

   if -4.0000000000000001e188 < (*.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 h around 0

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

   5. Applied rewrites 91.0%

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

Alternative 6: 80.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -4 \cdot 10^{+188}:\\ \;\;\;\;D \cdot \left(\left(D \cdot -0.125\right) \cdot \frac{M \cdot \left(M \cdot \left(h \cdot w0\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -4e+188)
   (* D (* (* D -0.125) (/ (* M (* M (* h w0))) (* d (* d l)))))
   (* w0 1.0)))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -4e+188) {
		tmp = D * ((D * -0.125) * ((M * (M * (h * w0))) / (d * (d * l))));
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l)) <= (-4d+188)) then
        tmp = d * ((d * (-0.125d0)) * ((m * (m * (h * w0))) / (d_1 * (d_1 * l))))
    else
        tmp = w0 * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -4e+188) {
		tmp = D * ((D * -0.125) * ((M * (M * (h * w0))) / (d * (d * l))));
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -4e+188:
		tmp = D * ((D * -0.125) * ((M * (M * (h * w0))) / (d * (d * l))))
	else:
		tmp = w0 * 1.0
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -4e+188)
		tmp = Float64(D * Float64(Float64(D * -0.125) * Float64(Float64(M * Float64(M * Float64(h * w0))) / Float64(d * Float64(d * l)))));
	else
		tmp = Float64(w0 * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (((((M * D) / (2.0 * d)) ^ 2.0) * (h / l)) <= -4e+188)
		tmp = D * ((D * -0.125) * ((M * (M * (h * w0))) / (d * (d * l))));
	else
		tmp = w0 * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -4e+188], N[(D * N[(N[(D * -0.125), $MachinePrecision] * N[(N[(M * N[(M * N[(h * w0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(w0 * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -4.0000000000000001e188

   1. Initial program 61.0%

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

   3. Taylor expanded in h 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 42.4%

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

   6. Taylor expanded in h around inf

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

   8. Applied rewrites 42.5%

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

   10. Applied rewrites 50.6%

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

   if -4.0000000000000001e188 < (*.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 h around 0

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

   5. Applied rewrites 91.0%

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

Alternative 7: 78.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq 5 \cdot 10^{-141}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{D \cdot \left(M \cdot h\right)}{-2 \cdot \left(d \cdot \ell\right)}, 0.5 \cdot \left(D \cdot \frac{M}{d}\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{h}{\ell} \cdot \left(\left(\left(M \cdot M\right) \cdot 0.25\right) \cdot \frac{D}{-d}\right), \frac{D}{d}, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M 5e-141)
   (*
    w0
    (sqrt (fma (/ (* D (* M h)) (* -2.0 (* d l))) (* 0.5 (* D (/ M d))) 1.0)))
   (*
    w0
    (sqrt (fma (* (/ h l) (* (* (* M M) 0.25) (/ D (- d)))) (/ D d) 1.0)))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 5e-141) {
		tmp = w0 * sqrt(fma(((D * (M * h)) / (-2.0 * (d * l))), (0.5 * (D * (M / d))), 1.0));
	} else {
		tmp = w0 * sqrt(fma(((h / l) * (((M * M) * 0.25) * (D / -d))), (D / d), 1.0));
	}
	return tmp;
}

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 5e-141)
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(D * Float64(M * h)) / Float64(-2.0 * Float64(d * l))), Float64(0.5 * Float64(D * Float64(M / d))), 1.0)));
	else
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(h / l) * Float64(Float64(Float64(M * M) * 0.25) * Float64(D / Float64(-d)))), Float64(D / d), 1.0)));
	end
	return tmp
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 5e-141], N[(w0 * N[Sqrt[N[(N[(N[(D * N[(M * h), $MachinePrecision]), $MachinePrecision] / N[(-2.0 * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[(D * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(N[(N[(h / l), $MachinePrecision] * N[(N[(N[(M * M), $MachinePrecision] * 0.25), $MachinePrecision] * N[(D / (-d)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(D / d), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 4.9999999999999999e-141

   1. Initial program 82.5%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. distribute-rgt-neg-in N/A

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

      9. associate-*l* N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 84.3%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 84.3

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

   6. Applied rewrites 83.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 83.8

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

   8. Applied rewrites 83.8%

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

   if 4.9999999999999999e-141 < M

   1. Initial program 78.3%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

   4. Applied rewrites 71.7%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 5 \cdot 10^{-141}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{D \cdot \left(M \cdot h\right)}{-2 \cdot \left(d \cdot \ell\right)}, 0.5 \cdot \left(D \cdot \frac{M}{d}\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{h}{\ell} \cdot \left(\left(\left(M \cdot M\right) \cdot 0.25\right) \cdot \frac{D}{-d}\right), \frac{D}{d}, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 81.6% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 81.0%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

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

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

   6. lift-pow.f64 N/A

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

   7. unpow2 N/A

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

   8. distribute-rgt-neg-in N/A

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

   9. associate-*l* N/A

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

   10. lower-fma.f64 N/A

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

4. Applied rewrites 82.6%

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

   1. lift-fma.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 82.6

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

6. Applied rewrites 79.6%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 80.3

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 80.3

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

8. Applied rewrites 80.3%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. *-commutative N/A

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

   5. associate-/l* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 80.7

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

10. Applied rewrites 80.7%

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

11. Final simplification 80.7%

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

Alternative 9: 69.1% accurate, 26.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.0%

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

3. Taylor expanded in h around 0

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

5. Applied rewrites 64.6%

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

Reproduce

herbie shell --seed 2024235 
(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))))))