Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.1% → 88.6%

Time: 15.1s

Alternatives: 16

Speedup: 0.7×

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: 81.1% 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: 88.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{M \cdot D}{2 \cdot d}\\ \mathbf{if}\;{t\_0}^{2} \cdot \frac{h}{\ell} \leq 10^{-27}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(t\_0, \frac{h}{\ell} \cdot \frac{M \cdot D}{d \cdot -2}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(t\_0, \frac{\frac{\left(M \cdot D\right) \cdot h}{2 \cdot d}}{-\ell}, 1\right)}\\ \end{array} \end{array} \]

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)) 1e-27)
     (* w0 (sqrt (fma t_0 (* (/ h l) (/ (* M D) (* d -2.0))) 1.0)))
     (* w0 (sqrt (fma t_0 (/ (/ (* (* M D) h) (* 2.0 d)) (- l)) 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)) <= 1e-27) {
		tmp = w0 * sqrt(fma(t_0, ((h / l) * ((M * D) / (d * -2.0))), 1.0));
	} else {
		tmp = w0 * sqrt(fma(t_0, ((((M * D) * h) / (2.0 * d)) / -l), 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)) <= 1e-27)
		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(t_0, Float64(Float64(Float64(Float64(M * D) * h) / Float64(2.0 * d)) / 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[(2.0 * d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Power[t$95$0, 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], 1e-27], 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[(t$95$0 * N[(N[(N[(N[(M * D), $MachinePrecision] * h), $MachinePrecision] / N[(2.0 * d), $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)) < 1e-27

   1. Initial program 86.0%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

      4. unpow2 N/A

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

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

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

      6. associate-*l* N/A

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

      7. accelerator-lowering-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 egg-rr 86.3%

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

   if 1e-27 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 20.0%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. distribute-neg-frac2 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. accelerator-lowering-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 egg-rr 90.6%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 10^{-27}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{M \cdot D}{2 \cdot d}, \frac{h}{\ell} \cdot \frac{M \cdot D}{d \cdot -2}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{M \cdot D}{2 \cdot d}, \frac{\frac{\left(M \cdot D\right) \cdot h}{2 \cdot d}}{-\ell}, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 78.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\\ \mathbf{if}\;t\_0 \leq 10^{-86}:\\ \;\;\;\;w0\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+121}:\\ \;\;\;\;w0 \cdot \sqrt{1 - D \cdot \left(M \cdot \frac{\left(M \cdot D\right) \cdot h}{\ell \cdot \left(\left(d \cdot d\right) \cdot 4\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot \left(w0 \cdot \left(h \cdot -0.125\right)\right)\right)}{d \cdot \ell}, \frac{D}{d}, w0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (pow (/ (* M D) (* 2.0 d)) 2.0)))
   (if (<= t_0 1e-86)
     w0
     (if (<= t_0 5e+121)
       (*
        w0
        (sqrt (- 1.0 (* D (* M (/ (* (* M D) h) (* l (* (* d d) 4.0))))))))
       (fma (/ (* (* M D) (* M (* w0 (* h -0.125)))) (* d l)) (/ D d) w0)))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = pow(((M * D) / (2.0 * d)), 2.0);
	double tmp;
	if (t_0 <= 1e-86) {
		tmp = w0;
	} else if (t_0 <= 5e+121) {
		tmp = w0 * sqrt((1.0 - (D * (M * (((M * D) * h) / (l * ((d * d) * 4.0)))))));
	} else {
		tmp = fma((((M * D) * (M * (w0 * (h * -0.125)))) / (d * l)), (D / d), w0);
	}
	return tmp;
}

Julia

function code(w0, M, D, h, l, d)
	t_0 = Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0
	tmp = 0.0
	if (t_0 <= 1e-86)
		tmp = w0;
	elseif (t_0 <= 5e+121)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(D * Float64(M * Float64(Float64(Float64(M * D) * h) / Float64(l * Float64(Float64(d * d) * 4.0))))))));
	else
		tmp = fma(Float64(Float64(Float64(M * D) * Float64(M * Float64(w0 * Float64(h * -0.125)))) / Float64(d * l)), Float64(D / d), w0);
	end
	return tmp
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := Block[{t$95$0 = N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[t$95$0, 1e-86], w0, If[LessEqual[t$95$0, 5e+121], N[(w0 * N[Sqrt[N[(1.0 - N[(D * N[(M * N[(N[(N[(M * D), $MachinePrecision] * h), $MachinePrecision] / N[(l * N[(N[(d * d), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(M * D), $MachinePrecision] * N[(M * N[(w0 * N[(h * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision] * N[(D / d), $MachinePrecision] + w0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) < 1.00000000000000008e-86

   1. Initial program 93.5%

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

   3. Taylor expanded in M around 0

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

   5. Simplified 98.0%

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

   if 1.00000000000000008e-86 < (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) < 5.00000000000000007e121

   1. Initial program 81.5%

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

      1. unpow2 N/A

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

      2. frac-times N/A

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

      3. associate-/l* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. swap-sqr N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. *-lowering-*.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. metadata-eval 72.3

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

   4. Applied egg-rr 72.3%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. frac-times N/A

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

      7. /-lowering-/.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-lowering-*.f64 59.9

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

   6. Applied egg-rr 59.9%

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

   if 5.00000000000000007e121 < (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))

   1. Initial program 60.1%

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

   3. Taylor expanded in M around 0

      \[\leadsto \color{blue}{w0 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}} \]
   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. accelerator-lowering-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. Simplified 44.2%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. associate-*r/ N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. *-lowering-*.f64 N/A

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

      16. *-lowering-*.f64 49.5

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

   7. Applied egg-rr 49.5%

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

      1. associate-*l/ N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. associate-*r/ N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

   9. Applied egg-rr 56.7%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq 10^{-86}:\\ \;\;\;\;w0\\ \mathbf{elif}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq 5 \cdot 10^{+121}:\\ \;\;\;\;w0 \cdot \sqrt{1 - D \cdot \left(M \cdot \frac{\left(M \cdot D\right) \cdot h}{\ell \cdot \left(\left(d \cdot d\right) \cdot 4\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot \left(w0 \cdot \left(h \cdot -0.125\right)\right)\right)}{d \cdot \ell}, \frac{D}{d}, w0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{M \cdot D}{2 \cdot d}\\ \mathbf{if}\;{t\_0}^{2} \cdot \frac{h}{\ell} \leq 5 \cdot 10^{-7}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(t\_0, \frac{h}{\ell} \cdot \frac{M \cdot D}{d \cdot -2}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{D \cdot D}{d \cdot d}, \frac{\left(M \cdot \left(M \cdot w0\right)\right) \cdot \left(h \cdot -0.125\right)}{\ell}, w0\right)\\ \end{array} \end{array} \]

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-7)
     (* w0 (sqrt (fma t_0 (* (/ h l) (/ (* M D) (* d -2.0))) 1.0)))
     (fma (/ (* D D) (* d d)) (/ (* (* M (* M w0)) (* h -0.125)) l) w0))))

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-7) {
		tmp = w0 * sqrt(fma(t_0, ((h / l) * ((M * D) / (d * -2.0))), 1.0));
	} else {
		tmp = fma(((D * D) / (d * d)), (((M * (M * w0)) * (h * -0.125)) / l), w0);
	}
	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-7)
		tmp = Float64(w0 * sqrt(fma(t_0, Float64(Float64(h / l) * Float64(Float64(M * D) / Float64(d * -2.0))), 1.0)));
	else
		tmp = fma(Float64(Float64(D * D) / Float64(d * d)), Float64(Float64(Float64(M * Float64(M * w0)) * Float64(h * -0.125)) / l), w0);
	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-7], 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[(N[(N[(D * D), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M * N[(M * w0), $MachinePrecision]), $MachinePrecision] * N[(h * -0.125), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] + w0), $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)) < 4.99999999999999977e-7

   1. Initial program 86.2%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

      4. unpow2 N/A

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

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

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

      6. associate-*l* N/A

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

      7. accelerator-lowering-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 egg-rr 86.6%

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

   if 4.99999999999999977e-7 < (*.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. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. accelerator-lowering-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. Simplified 44.5%

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

      1. associate-*r/ N/A

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

      2. times-frac N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. *-commutative N/A

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

      14. *-lowering-*.f64 63.3

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

   7. Applied egg-rr 63.3%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 5 \cdot 10^{-7}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{M \cdot D}{2 \cdot d}, \frac{h}{\ell} \cdot \frac{M \cdot D}{d \cdot -2}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{D \cdot D}{d \cdot d}, \frac{\left(M \cdot \left(M \cdot w0\right)\right) \cdot \left(h \cdot -0.125\right)}{\ell}, w0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 10^{-27}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{M \cdot D}{d}}{d \cdot 4}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{\left(M \cdot D\right) \cdot h}{d \cdot \left(\ell \cdot 4\right)} \cdot \frac{M}{d}, -D, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) 1e-27)
   (* w0 (sqrt (- 1.0 (* (/ h l) (* (* M D) (/ (/ (* M D) d) (* d 4.0)))))))
   (*
    w0
    (sqrt (fma (* (/ (* (* M D) h) (* d (* l 4.0))) (/ M d)) (- D) 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)) <= 1e-27) {
		tmp = w0 * sqrt((1.0 - ((h / l) * ((M * D) * (((M * D) / d) / (d * 4.0))))));
	} else {
		tmp = w0 * sqrt(fma(((((M * D) * h) / (d * (l * 4.0))) * (M / d)), -D, 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)) <= 1e-27)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * Float64(Float64(M * D) * Float64(Float64(Float64(M * D) / d) / Float64(d * 4.0)))))));
	else
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(Float64(Float64(M * D) * h) / Float64(d * Float64(l * 4.0))) * Float64(M / d)), Float64(-D), 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], 1e-27], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[(M * D), $MachinePrecision] * N[(N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision] / N[(d * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(N[(N[(N[(N[(M * D), $MachinePrecision] * h), $MachinePrecision] / N[(d * N[(l * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(M / d), $MachinePrecision]), $MachinePrecision] * (-D) + 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)) < 1e-27

   1. Initial program 86.0%

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

      1. unpow2 N/A

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

      2. frac-times N/A

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

      3. associate-/l* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. swap-sqr N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. *-lowering-*.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. metadata-eval 76.3

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

   4. Applied egg-rr 76.3%

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

      1. associate-*l* N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-lowering-*.f64 86.0

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

   6. Applied egg-rr 86.0%

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

   if 1e-27 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 20.0%

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

      1. unpow2 N/A

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

      2. frac-times N/A

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

      3. associate-/l* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. swap-sqr N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. *-lowering-*.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. metadata-eval 14.9

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

   4. Applied egg-rr 14.9%

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

   6. Applied egg-rr 63.3%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. associate-*l* N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. *-commutative N/A

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

      14. *-lowering-*.f64 N/A

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

      15. neg-lowering-neg.f64 64.2

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

   8. Applied egg-rr 64.2%

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

      1. associate-*l/ N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 N/A

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

      12. /-lowering-/.f64 75.6

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

   10. Applied egg-rr 75.6%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 10^{-27}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{M \cdot D}{d}}{d \cdot 4}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{\left(M \cdot D\right) \cdot h}{d \cdot \left(\ell \cdot 4\right)} \cdot \frac{M}{d}, -D, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+37}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot \left(w0 \cdot \left(h \cdot -0.125\right)\right)\right)}{d \cdot \ell}, \frac{D}{d}, w0\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -5e+37)
   (fma (/ (* (* M D) (* M (* w0 (* h -0.125)))) (* d l)) (/ D d) w0)
   w0))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -5e+37) {
		tmp = fma((((M * D) * (M * (w0 * (h * -0.125)))) / (d * l)), (D / d), w0);
	} else {
		tmp = w0;
	}
	return tmp;
}

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -5e+37)
		tmp = fma(Float64(Float64(Float64(M * D) * Float64(M * Float64(w0 * Float64(h * -0.125)))) / Float64(d * l)), Float64(D / d), w0);
	else
		tmp = w0;
	end
	return tmp
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -5e+37], N[(N[(N[(N[(M * D), $MachinePrecision] * N[(M * N[(w0 * N[(h * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision] * N[(D / d), $MachinePrecision] + w0), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -4.99999999999999989e37

   1. Initial program 64.1%

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

   3. Taylor expanded in M around 0

      \[\leadsto \color{blue}{w0 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}} \]
   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. accelerator-lowering-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. Simplified 39.6%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. associate-*r/ N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. *-lowering-*.f64 N/A

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

      16. *-lowering-*.f64 42.6

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

   7. Applied egg-rr 42.6%

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

      1. associate-*l/ N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. associate-*r/ N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

   9. Applied egg-rr 47.3%

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

   if -4.99999999999999989e37 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 90.2%

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

   3. Taylor expanded in M around 0

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

   5. Simplified 96.1%

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

Alternative 6: 80.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+85}:\\ \;\;\;\;\mathsf{fma}\left(\frac{D \cdot D}{d}, \left(M \cdot w0\right) \cdot \left(M \cdot \frac{h \cdot -0.125}{d \cdot \ell}\right), w0\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -5e+85)
   (fma (/ (* D D) d) (* (* M w0) (* M (/ (* h -0.125) (* d l)))) w0)
   w0))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -5e+85) {
		tmp = fma(((D * D) / d), ((M * w0) * (M * ((h * -0.125) / (d * l)))), w0);
	} else {
		tmp = w0;
	}
	return tmp;
}

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -5e+85)
		tmp = fma(Float64(Float64(D * D) / d), Float64(Float64(M * w0) * Float64(M * Float64(Float64(h * -0.125) / Float64(d * l)))), w0);
	else
		tmp = w0;
	end
	return tmp
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -5e+85], N[(N[(N[(D * D), $MachinePrecision] / d), $MachinePrecision] * N[(N[(M * w0), $MachinePrecision] * N[(M * N[(N[(h * -0.125), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + w0), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -5.0000000000000001e85

   1. Initial program 61.6%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. accelerator-lowering-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. Simplified 42.1%

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

      1. associate-*r/ N/A

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

      2. associate-*l* N/A

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

      3. times-frac N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. *-commutative N/A

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

      14. *-lowering-*.f64 N/A

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

      15. *-lowering-*.f64 47.1

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

   7. Applied egg-rr 47.1%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 48.3

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

   9. Applied egg-rr 48.3%

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

   if -5.0000000000000001e85 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 90.6%

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

   3. Taylor expanded in M around 0

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

   5. Simplified 93.0%

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

Alternative 7: 80.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+85}:\\ \;\;\;\;\mathsf{fma}\left(D \cdot \frac{D}{d}, \frac{\left(M \cdot \left(M \cdot w0\right)\right) \cdot \left(h \cdot -0.125\right)}{d \cdot \ell}, w0\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -5e+85)
   (fma (* D (/ D d)) (/ (* (* M (* M w0)) (* h -0.125)) (* d l)) w0)
   w0))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -5e+85) {
		tmp = fma((D * (D / d)), (((M * (M * w0)) * (h * -0.125)) / (d * l)), w0);
	} else {
		tmp = w0;
	}
	return tmp;
}

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -5e+85)
		tmp = fma(Float64(D * Float64(D / d)), Float64(Float64(Float64(M * Float64(M * w0)) * Float64(h * -0.125)) / Float64(d * l)), w0);
	else
		tmp = w0;
	end
	return tmp
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -5e+85], N[(N[(D * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M * N[(M * w0), $MachinePrecision]), $MachinePrecision] * N[(h * -0.125), $MachinePrecision]), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision] + w0), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -5.0000000000000001e85

   1. Initial program 61.6%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. accelerator-lowering-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. Simplified 42.1%

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

      1. associate-*r/ N/A

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

      2. associate-*l* N/A

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

      3. times-frac N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. *-commutative N/A

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

      14. *-lowering-*.f64 N/A

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

      15. *-lowering-*.f64 47.1

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

   7. Applied egg-rr 47.1%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 48.7

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

   9. Applied egg-rr 48.7%

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

   if -5.0000000000000001e85 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 90.6%

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

   3. Taylor expanded in M around 0

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

   5. Simplified 93.0%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+85}:\\ \;\;\;\;\mathsf{fma}\left(D \cdot \frac{D}{d}, \frac{\left(M \cdot \left(M \cdot w0\right)\right) \cdot \left(h \cdot -0.125\right)}{d \cdot \ell}, w0\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+85}:\\ \;\;\;\;\left(D \cdot D\right) \cdot \left(\frac{w0 \cdot \left(M \cdot \left(M \cdot h\right)\right)}{d} \cdot \frac{-0.125}{d \cdot \ell}\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -5e+85)
   (* (* D D) (* (/ (* w0 (* M (* M h))) d) (/ -0.125 (* d l))))
   w0))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -5e+85) {
		tmp = (D * D) * (((w0 * (M * (M * h))) / d) * (-0.125 / (d * l)));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l)) <= (-5d+85)) then
        tmp = (d * d) * (((w0 * (m * (m * h))) / d_1) * ((-0.125d0) / (d_1 * l)))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -5e+85) {
		tmp = (D * D) * (((w0 * (M * (M * h))) / d) * (-0.125 / (d * l)));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -5e+85:
		tmp = (D * D) * (((w0 * (M * (M * h))) / d) * (-0.125 / (d * l)))
	else:
		tmp = w0
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -5e+85)
		tmp = Float64(Float64(D * D) * Float64(Float64(Float64(w0 * Float64(M * Float64(M * h))) / d) * Float64(-0.125 / Float64(d * l))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (((((M * D) / (2.0 * d)) ^ 2.0) * (h / l)) <= -5e+85)
		tmp = (D * D) * (((w0 * (M * (M * h))) / d) * (-0.125 / (d * l)));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -5e+85], N[(N[(D * D), $MachinePrecision] * N[(N[(N[(w0 * N[(M * N[(M * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * N[(-0.125 / N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -5.0000000000000001e85

   1. Initial program 61.6%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. accelerator-lowering-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. Simplified 42.1%

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

   6. Taylor expanded in D around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. associate-*r/ N/A

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

      9. /-lowering-/.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. *-lowering-*.f64 N/A

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

      13. *-lowering-*.f64 N/A

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 N/A

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

      16. *-lowering-*.f64 N/A

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

      17. unpow2 N/A

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

      18. *-lowering-*.f64 39.8

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

   8. Simplified 39.8%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. times-frac N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. *-lowering-*.f64 46.0

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

   10. Applied egg-rr 46.0%

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

   if -5.0000000000000001e85 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 90.6%

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

   3. Taylor expanded in M around 0

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

   5. Simplified 93.0%

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

Alternative 9: 79.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+85}:\\ \;\;\;\;\left(D \cdot D\right) \cdot \left(\left(M \cdot h\right) \cdot \left(\frac{w0}{d \cdot \left(d \cdot \ell\right)} \cdot \left(M \cdot -0.125\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -5e+85)
   (* (* D D) (* (* M h) (* (/ w0 (* d (* d l))) (* M -0.125))))
   w0))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -5e+85) {
		tmp = (D * D) * ((M * h) * ((w0 / (d * (d * l))) * (M * -0.125)));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l)) <= (-5d+85)) then
        tmp = (d * d) * ((m * h) * ((w0 / (d_1 * (d_1 * l))) * (m * (-0.125d0))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -5e+85) {
		tmp = (D * D) * ((M * h) * ((w0 / (d * (d * l))) * (M * -0.125)));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -5e+85:
		tmp = (D * D) * ((M * h) * ((w0 / (d * (d * l))) * (M * -0.125)))
	else:
		tmp = w0
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -5e+85)
		tmp = Float64(Float64(D * D) * Float64(Float64(M * h) * Float64(Float64(w0 / Float64(d * Float64(d * l))) * Float64(M * -0.125))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (((((M * D) / (2.0 * d)) ^ 2.0) * (h / l)) <= -5e+85)
		tmp = (D * D) * ((M * h) * ((w0 / (d * (d * l))) * (M * -0.125)));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -5e+85], N[(N[(D * D), $MachinePrecision] * N[(N[(M * h), $MachinePrecision] * N[(N[(w0 / N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(M * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -5.0000000000000001e85

   1. Initial program 61.6%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. accelerator-lowering-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. Simplified 42.1%

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

   6. Taylor expanded in D around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. associate-*r/ N/A

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

      9. /-lowering-/.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. *-lowering-*.f64 N/A

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

      13. *-lowering-*.f64 N/A

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 N/A

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

      16. *-lowering-*.f64 N/A

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

      17. unpow2 N/A

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

      18. *-lowering-*.f64 39.8

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

   8. Simplified 39.8%

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

      1. associate-*r* N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 41.1

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

   10. Applied egg-rr 41.1%

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. associate-*r* N/A

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

       4. *-lowering-*.f64 N/A

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

       5. *-lowering-*.f64 N/A

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

       6. /-lowering-/.f64 N/A

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

       7. *-lowering-*.f64 N/A

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

       8. *-lowering-*.f64 N/A

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

       9. *-commutative N/A

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

       10. *-lowering-*.f64 N/A

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

       11. *-lowering-*.f64 44.7

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

   12. Applied egg-rr 44.7%

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

   if -5.0000000000000001e85 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 90.6%

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

   3. Taylor expanded in M around 0

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

   5. Simplified 93.0%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+85}:\\ \;\;\;\;\left(D \cdot D\right) \cdot \left(\left(M \cdot h\right) \cdot \left(\frac{w0}{d \cdot \left(d \cdot \ell\right)} \cdot \left(M \cdot -0.125\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 79.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+85}:\\ \;\;\;\;\left(D \cdot D\right) \cdot \left(\frac{w0}{d \cdot \left(d \cdot \ell\right)} \cdot \left(-0.125 \cdot \left(M \cdot \left(M \cdot h\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -5e+85)
   (* (* D D) (* (/ w0 (* d (* d l))) (* -0.125 (* M (* M h)))))
   w0))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -5e+85) {
		tmp = (D * D) * ((w0 / (d * (d * l))) * (-0.125 * (M * (M * h))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l)) <= (-5d+85)) then
        tmp = (d * d) * ((w0 / (d_1 * (d_1 * l))) * ((-0.125d0) * (m * (m * h))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -5e+85) {
		tmp = (D * D) * ((w0 / (d * (d * l))) * (-0.125 * (M * (M * h))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -5e+85:
		tmp = (D * D) * ((w0 / (d * (d * l))) * (-0.125 * (M * (M * h))))
	else:
		tmp = w0
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -5e+85)
		tmp = Float64(Float64(D * D) * Float64(Float64(w0 / Float64(d * Float64(d * l))) * Float64(-0.125 * Float64(M * Float64(M * h)))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (((((M * D) / (2.0 * d)) ^ 2.0) * (h / l)) <= -5e+85)
		tmp = (D * D) * ((w0 / (d * (d * l))) * (-0.125 * (M * (M * h))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -5e+85], N[(N[(D * D), $MachinePrecision] * N[(N[(w0 / N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.125 * N[(M * N[(M * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -5.0000000000000001e85

   1. Initial program 61.6%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. accelerator-lowering-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. Simplified 42.1%

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

   6. Taylor expanded in D around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. associate-*r/ N/A

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

      9. /-lowering-/.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. *-lowering-*.f64 N/A

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

      13. *-lowering-*.f64 N/A

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 N/A

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

      16. *-lowering-*.f64 N/A

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

      17. unpow2 N/A

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

      18. *-lowering-*.f64 39.8

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

   8. Simplified 39.8%

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

      1. associate-*r* N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 41.1

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

   10. Applied egg-rr 41.1%

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

   if -5.0000000000000001e85 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 90.6%

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

   3. Taylor expanded in M around 0

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

   5. Simplified 93.0%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+85}:\\ \;\;\;\;\left(D \cdot D\right) \cdot \left(\frac{w0}{d \cdot \left(d \cdot \ell\right)} \cdot \left(-0.125 \cdot \left(M \cdot \left(M \cdot h\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 76.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{M \cdot D}{2 \cdot d}\\ \mathbf{if}\;t\_0 \leq 10^{-43}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(t\_0, \frac{\left(\left(M \cdot D\right) \cdot h\right) \cdot -0.5}{d \cdot \ell}, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (/ (* M D) (* 2.0 d))))
   (if (<= t_0 1e-43)
     w0
     (* w0 (sqrt (fma t_0 (/ (* (* (* M D) h) -0.5) (* d l)) 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 (t_0 <= 1e-43) {
		tmp = w0;
	} else {
		tmp = w0 * sqrt(fma(t_0, ((((M * D) * h) * -0.5) / (d * l)), 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 (t_0 <= 1e-43)
		tmp = w0;
	else
		tmp = Float64(w0 * sqrt(fma(t_0, Float64(Float64(Float64(Float64(M * D) * h) * -0.5) / Float64(d * 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[(2.0 * d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e-43], w0, N[(w0 * N[Sqrt[N[(t$95$0 * N[(N[(N[(N[(M * D), $MachinePrecision] * h), $MachinePrecision] * -0.5), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) < 1.00000000000000008e-43

   1. Initial program 84.4%

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

   3. Taylor expanded in M around 0

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

   5. Simplified 75.2%

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

   if 1.00000000000000008e-43 < (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d))

   1. Initial program 68.3%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. distribute-neg-frac2 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. accelerator-lowering-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 egg-rr 70.0%

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

   5. Taylor expanded in M around 0

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-lowering-*.f64 68.1

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

   7. Simplified 68.1%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq 10^{-43}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{M \cdot D}{2 \cdot d}, \frac{\left(\left(M \cdot D\right) \cdot h\right) \cdot -0.5}{d \cdot \ell}, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 76.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq 10^{-43}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{\left(M \cdot D\right) \cdot h}{d \cdot \left(\ell \cdot 4\right)} \cdot \frac{M}{d}, -D, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (/ (* M D) (* 2.0 d)) 1e-43)
   w0
   (*
    w0
    (sqrt (fma (* (/ (* (* M D) h) (* d (* l 4.0))) (/ M d)) (- D) 1.0)))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (((M * D) / (2.0 * d)) <= 1e-43) {
		tmp = w0;
	} else {
		tmp = w0 * sqrt(fma(((((M * D) * h) / (d * (l * 4.0))) * (M / d)), -D, 1.0));
	}
	return tmp;
}

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64(Float64(M * D) / Float64(2.0 * d)) <= 1e-43)
		tmp = w0;
	else
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(Float64(Float64(M * D) * h) / Float64(d * Float64(l * 4.0))) * Float64(M / d)), Float64(-D), 1.0)));
	end
	return tmp
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 1e-43], w0, N[(w0 * N[Sqrt[N[(N[(N[(N[(N[(M * D), $MachinePrecision] * h), $MachinePrecision] / N[(d * N[(l * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(M / d), $MachinePrecision]), $MachinePrecision] * (-D) + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) < 1.00000000000000008e-43

   1. Initial program 84.4%

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

   3. Taylor expanded in M around 0

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

   5. Simplified 75.2%

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

   if 1.00000000000000008e-43 < (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d))

   1. Initial program 68.3%

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

      1. unpow2 N/A

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

      2. frac-times N/A

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

      3. associate-/l* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. swap-sqr N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. *-lowering-*.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. metadata-eval 59.4

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

   4. Applied egg-rr 59.4%

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

   6. Applied egg-rr 55.4%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. associate-*l* N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. *-commutative N/A

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

      14. *-lowering-*.f64 N/A

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

      15. neg-lowering-neg.f64 57.8

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

   8. Applied egg-rr 57.8%

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

      1. associate-*l/ N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 N/A

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

      12. /-lowering-/.f64 63.0

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

   10. Applied egg-rr 63.0%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq 10^{-43}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{\left(M \cdot D\right) \cdot h}{d \cdot \left(\ell \cdot 4\right)} \cdot \frac{M}{d}, -D, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 75.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \cdot D \leq 10^{-210}:\\ \;\;\;\;w0\\ \mathbf{elif}\;M \cdot D \leq 2 \cdot 10^{+142}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(-M \cdot D, \left(M \cdot D\right) \cdot \frac{h}{d \cdot \left(d \cdot \left(\ell \cdot 4\right)\right)}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot 0.25\right), \left(-D\right) \cdot \frac{h}{d \cdot \ell}, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* M D) 1e-210)
   w0
   (if (<= (* M D) 2e+142)
     (*
      w0
      (sqrt (fma (- (* M D)) (* (* M D) (/ h (* d (* d (* l 4.0))))) 1.0)))
     (*
      w0
      (sqrt (fma (* (/ D d) (* (* M M) 0.25)) (* (- D) (/ h (* d l))) 1.0))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((M * D) <= 1e-210) {
		tmp = w0;
	} else if ((M * D) <= 2e+142) {
		tmp = w0 * sqrt(fma(-(M * D), ((M * D) * (h / (d * (d * (l * 4.0))))), 1.0));
	} else {
		tmp = w0 * sqrt(fma(((D / d) * ((M * M) * 0.25)), (-D * (h / (d * l))), 1.0));
	}
	return tmp;
}

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64(M * D) <= 1e-210)
		tmp = w0;
	elseif (Float64(M * D) <= 2e+142)
		tmp = Float64(w0 * sqrt(fma(Float64(-Float64(M * D)), Float64(Float64(M * D) * Float64(h / Float64(d * Float64(d * Float64(l * 4.0))))), 1.0)));
	else
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(D / d) * Float64(Float64(M * M) * 0.25)), Float64(Float64(-D) * Float64(h / Float64(d * l))), 1.0)));
	end
	return tmp
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(M * D), $MachinePrecision], 1e-210], w0, If[LessEqual[N[(M * D), $MachinePrecision], 2e+142], N[(w0 * N[Sqrt[N[((-N[(M * D), $MachinePrecision]) * N[(N[(M * D), $MachinePrecision] * N[(h / N[(d * N[(d * N[(l * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(N[(N[(D / d), $MachinePrecision] * N[(N[(M * M), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision] * N[((-D) * N[(h / N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 M D) < 1e-210

   1. Initial program 83.7%

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

   3. Taylor expanded in M around 0

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

   5. Simplified 74.3%

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

   if 1e-210 < (*.f64 M D) < 2.0000000000000001e142

   1. Initial program 83.6%

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

      1. unpow2 N/A

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

      2. frac-times N/A

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

      3. associate-/l* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. swap-sqr N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. *-lowering-*.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. metadata-eval 78.1

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

   4. Applied egg-rr 78.1%

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

   6. Applied egg-rr 78.9%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*l* N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 85.1

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

   8. Applied egg-rr 85.1%

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

   if 2.0000000000000001e142 < (*.f64 M D)

   1. Initial program 67.3%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. distribute-neg-frac2 N/A

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

      5. associate-/l* N/A

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

      6. unpow2 N/A

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

      7. times-frac N/A

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

      8. associate-*r* N/A

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

      9. associate-*l* N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 51.7%

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

      1. frac-times N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. neg-lowering-neg.f64 N/A

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

      6. *-lowering-*.f64 56.0

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

   6. Applied egg-rr 56.0%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot D \leq 10^{-210}:\\ \;\;\;\;w0\\ \mathbf{elif}\;M \cdot D \leq 2 \cdot 10^{+142}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(-M \cdot D, \left(M \cdot D\right) \cdot \frac{h}{d \cdot \left(d \cdot \left(\ell \cdot 4\right)\right)}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot 0.25\right), \left(-D\right) \cdot \frac{h}{d \cdot \ell}, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 76.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \cdot D \leq 10^{-210}:\\ \;\;\;\;w0\\ \mathbf{elif}\;M \cdot D \leq 2 \cdot 10^{+294}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(-M \cdot D, \left(M \cdot D\right) \cdot \frac{h}{d \cdot \left(d \cdot \left(\ell \cdot 4\right)\right)}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{D \cdot D}{d} \cdot \left(h \cdot \left(-0.125 \cdot \left(w0 \cdot \left(M \cdot M\right)\right)\right)\right)}{\ell}}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* M D) 1e-210)
   w0
   (if (<= (* M D) 2e+294)
     (*
      w0
      (sqrt (fma (- (* M D)) (* (* M D) (/ h (* d (* d (* l 4.0))))) 1.0)))
     (/ (/ (* (/ (* D D) d) (* h (* -0.125 (* w0 (* M M))))) l) d))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((M * D) <= 1e-210) {
		tmp = w0;
	} else if ((M * D) <= 2e+294) {
		tmp = w0 * sqrt(fma(-(M * D), ((M * D) * (h / (d * (d * (l * 4.0))))), 1.0));
	} else {
		tmp = ((((D * D) / d) * (h * (-0.125 * (w0 * (M * M))))) / l) / d;
	}
	return tmp;
}

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64(M * D) <= 1e-210)
		tmp = w0;
	elseif (Float64(M * D) <= 2e+294)
		tmp = Float64(w0 * sqrt(fma(Float64(-Float64(M * D)), Float64(Float64(M * D) * Float64(h / Float64(d * Float64(d * Float64(l * 4.0))))), 1.0)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(D * D) / d) * Float64(h * Float64(-0.125 * Float64(w0 * Float64(M * M))))) / l) / d);
	end
	return tmp
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(M * D), $MachinePrecision], 1e-210], w0, If[LessEqual[N[(M * D), $MachinePrecision], 2e+294], N[(w0 * N[Sqrt[N[((-N[(M * D), $MachinePrecision]) * N[(N[(M * D), $MachinePrecision] * N[(h / N[(d * N[(d * N[(l * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(D * D), $MachinePrecision] / d), $MachinePrecision] * N[(h * N[(-0.125 * N[(w0 * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] / d), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 M D) < 1e-210

   1. Initial program 83.7%

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

   3. Taylor expanded in M around 0

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

   5. Simplified 74.3%

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

   if 1e-210 < (*.f64 M D) < 2.00000000000000013e294

   1. Initial program 81.7%

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

      1. unpow2 N/A

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

      2. frac-times N/A

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

      3. associate-/l* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. swap-sqr N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. *-lowering-*.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. metadata-eval 74.2

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

   4. Applied egg-rr 74.2%

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

   6. Applied egg-rr 74.9%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*l* N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 81.1

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

   8. Applied egg-rr 81.1%

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

   if 2.00000000000000013e294 < (*.f64 M D)

   1. Initial program 61.7%

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

   3. Taylor expanded in M around 0

      \[\leadsto \color{blue}{w0 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}} \]
   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. accelerator-lowering-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. Simplified 61.0%

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

   6. Taylor expanded in D around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. associate-*r/ N/A

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

      9. /-lowering-/.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. *-lowering-*.f64 N/A

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

      13. *-lowering-*.f64 N/A

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 N/A

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

      16. *-lowering-*.f64 N/A

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

      17. unpow2 N/A

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

      18. *-lowering-*.f64 61.0

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

   8. Simplified 61.0%

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

      1. associate-*r/ N/A

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

      2. associate-*l* N/A

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

      3. times-frac N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. associate-*r/ N/A

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

      11. *-commutative N/A

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

      12. associate-/r* N/A

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

      13. /-lowering-/.f64 N/A

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

   10. Applied egg-rr 69.9%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot D \leq 10^{-210}:\\ \;\;\;\;w0\\ \mathbf{elif}\;M \cdot D \leq 2 \cdot 10^{+294}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(-M \cdot D, \left(M \cdot D\right) \cdot \frac{h}{d \cdot \left(d \cdot \left(\ell \cdot 4\right)\right)}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{D \cdot D}{d} \cdot \left(h \cdot \left(-0.125 \cdot \left(w0 \cdot \left(M \cdot M\right)\right)\right)\right)}{\ell}}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 72.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq 10^{-43}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{D \cdot \left(\left(M \cdot \left(M \cdot w0\right)\right) \cdot \left(h \cdot -0.125\right)\right)}{d \cdot \left(d \cdot \ell\right)}, D, w0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (/ (* M D) (* 2.0 d)) 1e-43)
   w0
   (fma (/ (* D (* (* M (* M w0)) (* h -0.125))) (* d (* d l))) D w0)))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (((M * D) / (2.0 * d)) <= 1e-43) {
		tmp = w0;
	} else {
		tmp = fma(((D * ((M * (M * w0)) * (h * -0.125))) / (d * (d * l))), D, w0);
	}
	return tmp;
}

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64(Float64(M * D) / Float64(2.0 * d)) <= 1e-43)
		tmp = w0;
	else
		tmp = fma(Float64(Float64(D * Float64(Float64(M * Float64(M * w0)) * Float64(h * -0.125))) / Float64(d * Float64(d * l))), D, w0);
	end
	return tmp
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 1e-43], w0, N[(N[(N[(D * N[(N[(M * N[(M * w0), $MachinePrecision]), $MachinePrecision] * N[(h * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * D + w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) < 1.00000000000000008e-43

   1. Initial program 84.4%

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

   3. Taylor expanded in M around 0

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

   5. Simplified 75.2%

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

   if 1.00000000000000008e-43 < (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d))

   1. Initial program 68.3%

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

   3. Taylor expanded in M around 0

      \[\leadsto \color{blue}{w0 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}} \]
   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. accelerator-lowering-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. Simplified 38.2%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. associate-*r/ N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. *-lowering-*.f64 N/A

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

      16. *-lowering-*.f64 40.8

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

   7. Applied egg-rr 40.8%

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

Alternative 16: 68.7% accurate, 157.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.8%

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

3. Taylor expanded in M around 0

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

5. Simplified 63.3%

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

Reproduce

herbie shell --seed 2024198 
(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))))))