Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.6% → 88.7%

Time: 17.4s

Alternatives: 21

Speedup: 2.1×

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: 81.6% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{M \cdot D}{d \cdot 2}\\ \mathbf{if}\;{t\_0}^{2} \cdot \frac{h}{\ell} \leq 4 \cdot 10^{-147}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(t\_0, \frac{M \cdot D}{d \cdot -2} \cdot \frac{h}{\ell}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(t\_0, \frac{\frac{\left(M \cdot D\right) \cdot h}{d \cdot 2}}{-\ell}, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (/ (* M D) (* d 2.0))))
   (if (<= (* (pow t_0 2.0) (/ h l)) 4e-147)
     (* w0 (sqrt (fma t_0 (* (/ (* M D) (* d -2.0)) (/ h l)) 1.0)))
     (* w0 (sqrt (fma t_0 (/ (/ (* (* M D) h) (* d 2.0)) (- l)) 1.0))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = (M * D) / (d * 2.0);
	double tmp;
	if ((pow(t_0, 2.0) * (h / l)) <= 4e-147) {
		tmp = w0 * sqrt(fma(t_0, (((M * D) / (d * -2.0)) * (h / l)), 1.0));
	} else {
		tmp = w0 * sqrt(fma(t_0, ((((M * D) * h) / (d * 2.0)) / -l), 1.0));
	}
	return tmp;
}

Julia

function code(w0, M, D, h, l, d)
	t_0 = Float64(Float64(M * D) / Float64(d * 2.0))
	tmp = 0.0
	if (Float64((t_0 ^ 2.0) * Float64(h / l)) <= 4e-147)
		tmp = Float64(w0 * sqrt(fma(t_0, Float64(Float64(Float64(M * D) / Float64(d * -2.0)) * Float64(h / l)), 1.0)));
	else
		tmp = Float64(w0 * sqrt(fma(t_0, Float64(Float64(Float64(Float64(M * D) * h) / Float64(d * 2.0)) / Float64(-l)), 1.0)));
	end
	return tmp
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := Block[{t$95$0 = N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Power[t$95$0, 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], 4e-147], N[(w0 * N[Sqrt[N[(t$95$0 * N[(N[(N[(M * D), $MachinePrecision] / N[(d * -2.0), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(t$95$0 * N[(N[(N[(N[(M * D), $MachinePrecision] * h), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision] / (-l)), $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)) < 3.9999999999999999e-147

   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. 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 91.1%

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

   if 3.9999999999999999e-147 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 27.6%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. distribute-neg-frac2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

   4. Applied rewrites 86.6%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq 4 \cdot 10^{-147}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot -2} \cdot \frac{h}{\ell}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{\frac{\left(M \cdot D\right) \cdot h}{d \cdot 2}}{-\ell}, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 88.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{M \cdot D}{d \cdot 2}\\ t_1 := \frac{M \cdot D}{d \cdot -2}\\ \mathbf{if}\;{t\_0}^{2} \cdot \frac{h}{\ell} \leq 2 \cdot 10^{-48}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(t\_0, t\_1 \cdot \frac{h}{\ell}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{t\_1}{\ell}, \left(M \cdot D\right) \cdot \left(h \cdot \frac{0.5}{d}\right), 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (/ (* M D) (* d 2.0))) (t_1 (/ (* M D) (* d -2.0))))
   (if (<= (* (pow t_0 2.0) (/ h l)) 2e-48)
     (* w0 (sqrt (fma t_0 (* t_1 (/ h l)) 1.0)))
     (* w0 (sqrt (fma (/ t_1 l) (* (* M D) (* h (/ 0.5 d))) 1.0))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = (M * D) / (d * 2.0);
	double t_1 = (M * D) / (d * -2.0);
	double tmp;
	if ((pow(t_0, 2.0) * (h / l)) <= 2e-48) {
		tmp = w0 * sqrt(fma(t_0, (t_1 * (h / l)), 1.0));
	} else {
		tmp = w0 * sqrt(fma((t_1 / l), ((M * D) * (h * (0.5 / d))), 1.0));
	}
	return tmp;
}

Julia

function code(w0, M, D, h, l, d)
	t_0 = Float64(Float64(M * D) / Float64(d * 2.0))
	t_1 = Float64(Float64(M * D) / Float64(d * -2.0))
	tmp = 0.0
	if (Float64((t_0 ^ 2.0) * Float64(h / l)) <= 2e-48)
		tmp = Float64(w0 * sqrt(fma(t_0, Float64(t_1 * Float64(h / l)), 1.0)));
	else
		tmp = Float64(w0 * sqrt(fma(Float64(t_1 / l), Float64(Float64(M * D) * Float64(h * Float64(0.5 / 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[(d * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(M * D), $MachinePrecision] / N[(d * -2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Power[t$95$0, 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], 2e-48], N[(w0 * N[Sqrt[N[(t$95$0 * N[(t$95$1 * N[(h / l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(N[(t$95$1 / l), $MachinePrecision] * N[(N[(M * D), $MachinePrecision] * N[(h * N[(0.5 / 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)) < 1.9999999999999999e-48

   1. Initial program 88.3%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-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 89.6%

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

   if 1.9999999999999999e-48 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 13.5%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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-/.f64 N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

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

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

      12. div-inv N/A

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

      13. times-frac N/A

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

      14. lower-fma.f64 N/A

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

   4. Applied rewrites 82.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. /-rgt-identity N/A

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

      5. lift-/.f64 N/A

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

      6. div-inv N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/r* N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 76.2

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

   6. Applied rewrites 76.2%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq 2 \cdot 10^{-48}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot -2} \cdot \frac{h}{\ell}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{\frac{M \cdot D}{d \cdot -2}}{\ell}, \left(M \cdot D\right) \cdot \left(h \cdot \frac{0.5}{d}\right), 1\right)}\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024229 
(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))))))