Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.0% → 88.6%

Time: 14.1s

Alternatives: 15

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: 81.0% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{M \cdot D}{2 \cdot d}\\ \mathbf{if}\;1 - {t\_0}^{2} \cdot \frac{h}{\ell} \leq 5 \cdot 10^{+245}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot \left(\frac{M \cdot D}{d} \cdot \frac{M \cdot D}{d \cdot 4}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(t\_0, \frac{\frac{\left(M \cdot D\right) \cdot h}{2 \cdot d}}{-\ell}, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (/ (* M D) (* 2.0 d))))
   (if (<= (- 1.0 (* (pow t_0 2.0) (/ h l))) 5e+245)
     (* w0 (sqrt (- 1.0 (* (/ h l) (* (/ (* M D) d) (/ (* M D) (* d 4.0)))))))
     (* 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 ((1.0 - (pow(t_0, 2.0) * (h / l))) <= 5e+245) {
		tmp = w0 * sqrt((1.0 - ((h / l) * (((M * D) / d) * ((M * D) / (d * 4.0))))));
	} 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(1.0 - Float64((t_0 ^ 2.0) * Float64(h / l))) <= 5e+245)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * Float64(Float64(Float64(M * D) / d) * Float64(Float64(M * D) / Float64(d * 4.0)))))));
	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[(1.0 - N[(N[Power[t$95$0, 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+245], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision] * N[(N[(M * D), $MachinePrecision] / N[(d * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))) < 5.00000000000000034e245

   1. Initial program 99.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. swap-sqr N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. metadata-eval 82.9

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

   4. Applied egg-rr 82.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 99.4

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

   6. Applied egg-rr 99.4%

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

   if 5.00000000000000034e245 < (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)))

   1. Initial program 47.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

   4. Applied egg-rr 71.0%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 5 \cdot 10^{+245}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot \left(\frac{M \cdot D}{d} \cdot \frac{M \cdot D}{d \cdot 4}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{M \cdot D}{2 \cdot d}, \frac{\frac{\left(M \cdot D\right) \cdot h}{2 \cdot d}}{-\ell}, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 87.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{M \cdot D}{2 \cdot d}\\ \mathbf{if}\;1 - {t\_0}^{2} \cdot \frac{h}{\ell} \leq 5 \cdot 10^{+245}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot \left(\frac{M \cdot D}{d} \cdot \frac{M \cdot D}{d \cdot 4}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(t\_0, \frac{-0.5 \cdot \left(\left(M \cdot D\right) \cdot h\right)}{d \cdot \ell}, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (/ (* M D) (* 2.0 d))))
   (if (<= (- 1.0 (* (pow t_0 2.0) (/ h l))) 5e+245)
     (* w0 (sqrt (- 1.0 (* (/ h l) (* (/ (* M D) d) (/ (* M D) (* d 4.0)))))))
     (* w0 (sqrt (fma t_0 (/ (* -0.5 (* (* M D) h)) (* 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 ((1.0 - (pow(t_0, 2.0) * (h / l))) <= 5e+245) {
		tmp = w0 * sqrt((1.0 - ((h / l) * (((M * D) / d) * ((M * D) / (d * 4.0))))));
	} else {
		tmp = w0 * sqrt(fma(t_0, ((-0.5 * ((M * D) * h)) / (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(1.0 - Float64((t_0 ^ 2.0) * Float64(h / l))) <= 5e+245)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * Float64(Float64(Float64(M * D) / d) * Float64(Float64(M * D) / Float64(d * 4.0)))))));
	else
		tmp = Float64(w0 * sqrt(fma(t_0, Float64(Float64(-0.5 * Float64(Float64(M * D) * h)) / 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[N[(1.0 - N[(N[Power[t$95$0, 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+245], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision] * N[(N[(M * D), $MachinePrecision] / N[(d * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(t$95$0 * N[(N[(-0.5 * N[(N[(M * D), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))) < 5.00000000000000034e245

   1. Initial program 99.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. swap-sqr N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. metadata-eval 82.9

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

   4. Applied egg-rr 82.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 99.4

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

   6. Applied egg-rr 99.4%

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

   if 5.00000000000000034e245 < (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)))

   1. Initial program 47.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

   4. Applied egg-rr 71.0%

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

   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. lower-/.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. lower-*.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. lower-*.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. lower-*.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. lower-*.f64 68.4

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

   7. Simplified 68.4%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 5 \cdot 10^{+245}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot \left(\frac{M \cdot D}{d} \cdot \frac{M \cdot D}{d \cdot 4}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{M \cdot D}{2 \cdot d}, \frac{-0.5 \cdot \left(\left(M \cdot D\right) \cdot h\right)}{d \cdot \ell}, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{M \cdot D}{2 \cdot d}\\ \mathbf{if}\;1 - {t\_0}^{2} \cdot \frac{h}{\ell} \leq 5 \cdot 10^{+245}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{M}{d} \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot \frac{h \cdot -0.25}{\ell}\right), D, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(t\_0, \frac{-0.5 \cdot \left(\left(M \cdot D\right) \cdot h\right)}{d \cdot \ell}, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (/ (* M D) (* 2.0 d))))
   (if (<= (- 1.0 (* (pow t_0 2.0) (/ h l))) 5e+245)
     (* w0 (sqrt (fma (* (/ M d) (* (* D (/ M d)) (/ (* h -0.25) l))) D 1.0)))
     (* w0 (sqrt (fma t_0 (/ (* -0.5 (* (* M D) h)) (* 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 ((1.0 - (pow(t_0, 2.0) * (h / l))) <= 5e+245) {
		tmp = w0 * sqrt(fma(((M / d) * ((D * (M / d)) * ((h * -0.25) / l))), D, 1.0));
	} else {
		tmp = w0 * sqrt(fma(t_0, ((-0.5 * ((M * D) * h)) / (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(1.0 - Float64((t_0 ^ 2.0) * Float64(h / l))) <= 5e+245)
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(M / d) * Float64(Float64(D * Float64(M / d)) * Float64(Float64(h * -0.25) / l))), D, 1.0)));
	else
		tmp = Float64(w0 * sqrt(fma(t_0, Float64(Float64(-0.5 * Float64(Float64(M * D) * h)) / 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[N[(1.0 - N[(N[Power[t$95$0, 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+245], N[(w0 * N[Sqrt[N[(N[(N[(M / d), $MachinePrecision] * N[(N[(D * N[(M / d), $MachinePrecision]), $MachinePrecision] * N[(N[(h * -0.25), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * D + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(t$95$0 * N[(N[(-0.5 * N[(N[(M * D), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))) < 5.00000000000000034e245

   1. Initial program 99.4%

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

   3. Taylor expanded in w0 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. cancel-sign-sub-inv N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

   5. Simplified 55.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

   7. Applied egg-rr 74.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. times-frac N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

   9. Applied egg-rr 89.3%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-/.f64 N/A

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

       5. *-commutative N/A

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

       6. lift-/.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. times-frac N/A

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

       10. lift-/.f64 N/A

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

       11. associate-*r* N/A

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

       12. lower-*.f64 N/A

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

       13. lower-*.f64 N/A

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

       14. lower-/.f64 98.9

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

   11. Applied egg-rr 98.9%

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

   if 5.00000000000000034e245 < (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)))

   1. Initial program 47.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

   4. Applied egg-rr 71.0%

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

   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. lower-/.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. lower-*.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. lower-*.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. lower-*.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. lower-*.f64 68.4

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

   7. Simplified 68.4%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 5 \cdot 10^{+245}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{M}{d} \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot \frac{h \cdot -0.25}{\ell}\right), D, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{M \cdot D}{2 \cdot d}, \frac{-0.5 \cdot \left(\left(M \cdot D\right) \cdot h\right)}{d \cdot \ell}, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.3% accurate, 0.7× speedup

Math

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

FPCore

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.4%

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

   3. Taylor expanded in M around 0

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

   5. Simplified 99.8%

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

      1. *-rgt-identity 99.8

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

   7. Applied egg-rr 99.8%

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

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

   1. Initial program 56.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

   4. Applied egg-rr 69.9%

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

   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. lower-/.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. lower-*.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. lower-*.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. lower-*.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. lower-*.f64 67.4

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

   7. Simplified 67.4%

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

4. Final simplification 87.5%

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

Alternative 5: 84.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.4%

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

   3. Taylor expanded in M around 0

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

   5. Simplified 99.8%

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

      1. *-rgt-identity 99.8

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

   7. Applied egg-rr 99.8%

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

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

   1. Initial program 56.4%

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

   3. Taylor expanded in w0 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. cancel-sign-sub-inv N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

   5. Simplified 40.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

   7. Applied egg-rr 51.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. times-frac N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

   9. Applied egg-rr 61.1%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-/.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. associate-*l/ N/A

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

       7. lower-/.f64 N/A

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

   11. Applied egg-rr 65.4%

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

4. Final simplification 86.8%

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

Alternative 6: 82.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.4%

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

   3. Taylor expanded in M around 0

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

   5. Simplified 99.8%

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

      1. *-rgt-identity 99.8

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

   7. Applied egg-rr 99.8%

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

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

   1. Initial program 56.4%

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

   3. Taylor expanded in w0 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. cancel-sign-sub-inv N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

   5. Simplified 40.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

   7. Applied egg-rr 51.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. times-frac N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

   9. Applied egg-rr 61.1%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-/.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. *-commutative N/A

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

       8. lift-*.f64 N/A

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

       9. lift-/.f64 N/A

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

       10. associate-*l/ N/A

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

       11. lift-/.f64 N/A

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

       12. frac-times N/A

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

       13. *-commutative N/A

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

       14. lift-*.f64 N/A

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

       15. associate-*l* N/A

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

       16. lift-*.f64 N/A

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

       17. lift-*.f64 N/A

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

       18. lower-/.f64 N/A

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

   11. Applied egg-rr 60.3%

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

4. Final simplification 84.8%

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

Alternative 7: 82.1% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.4%

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

   3. Taylor expanded in M around 0

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

   5. Simplified 99.8%

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

      1. *-rgt-identity 99.8

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

   7. Applied egg-rr 99.8%

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

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

   1. Initial program 56.4%

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

   3. Taylor expanded in w0 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. cancel-sign-sub-inv N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

   5. Simplified 40.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

   7. Applied egg-rr 51.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 61.9

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 61.9

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

   9. Applied egg-rr 61.9%

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

Alternative 8: 80.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.4%

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

   3. Taylor expanded in M around 0

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

   5. Simplified 99.8%

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

      1. *-rgt-identity 99.8

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

   7. Applied egg-rr 99.8%

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

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

   1. Initial program 56.4%

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

   3. Taylor expanded in w0 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. cancel-sign-sub-inv N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

   5. Simplified 40.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

   7. Applied egg-rr 51.2%

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

4. Final simplification 81.4%

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

Alternative 9: 79.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+283}:\\ \;\;\;\;\left(D \cdot \frac{M \cdot \left(w0 \cdot \left(M \cdot h\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \cdot \left(D \cdot -0.125\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -5e+283)
   (* (* D (/ (* M (* w0 (* M h))) (* d (* d l)))) (* D -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+283) {
		tmp = (D * ((M * (w0 * (M * h))) / (d * (d * l)))) * (D * -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+283)) then
        tmp = (d * ((m * (w0 * (m * h))) / (d_1 * (d_1 * l)))) * (d * (-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+283) {
		tmp = (D * ((M * (w0 * (M * h))) / (d * (d * l)))) * (D * -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+283:
		tmp = (D * ((M * (w0 * (M * h))) / (d * (d * l)))) * (D * -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+283)
		tmp = Float64(Float64(D * Float64(Float64(M * Float64(w0 * Float64(M * h))) / Float64(d * Float64(d * l)))) * Float64(D * -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+283)
		tmp = (D * ((M * (w0 * (M * h))) / (d * (d * l)))) * (D * -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+283], N[(N[(D * N[(N[(M * N[(w0 * N[(M * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(D * -0.125), $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)) < -5.0000000000000004e283

   1. Initial program 57.9%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   5. Simplified 45.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. associate-*r* N/A

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

   7. Applied egg-rr 51.5%

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

   8. Taylor expanded in D around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 45.3

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

   10. Simplified 45.3%

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

   12. Applied egg-rr 54.6%

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

   if -5.0000000000000004e283 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 92.0%

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

   3. Taylor expanded in M around 0

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

   5. Simplified 90.3%

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

      1. *-rgt-identity 90.3

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

   7. Applied egg-rr 90.3%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+283}:\\ \;\;\;\;\left(D \cdot \frac{M \cdot \left(w0 \cdot \left(M \cdot h\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \cdot \left(D \cdot -0.125\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 72.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq 2 \cdot 10^{+171}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(D \cdot D\right) \cdot -0.125, \frac{h \cdot \frac{M \cdot \left(M \cdot w0\right)}{d \cdot \ell}}{d}, w0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (/ (* M D) (* 2.0 d)) 2e+171)
   w0
   (fma (* (* D D) -0.125) (/ (* h (/ (* M (* M w0)) (* 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)) <= 2e+171) {
		tmp = w0;
	} else {
		tmp = fma(((D * D) * -0.125), ((h * ((M * (M * w0)) / (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)) <= 2e+171)
		tmp = w0;
	else
		tmp = fma(Float64(Float64(D * D) * -0.125), Float64(Float64(h * Float64(Float64(M * Float64(M * w0)) / 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], 2e+171], w0, N[(N[(N[(D * D), $MachinePrecision] * -0.125), $MachinePrecision] * N[(N[(h * N[(N[(M * N[(M * w0), $MachinePrecision]), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] + w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) < 1.99999999999999991e171

   1. Initial program 84.1%

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

   3. Taylor expanded in M around 0

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

   5. Simplified 74.7%

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

      1. *-rgt-identity 74.7

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

   7. Applied egg-rr 74.7%

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

   if 1.99999999999999991e171 < (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d))

   1. Initial program 73.5%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   5. Simplified 61.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. associate-*r* N/A

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

   7. Applied egg-rr 69.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 69.8

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

   9. Applied egg-rr 69.8%

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

Alternative 11: 72.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq 2 \cdot 10^{+171}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(D \cdot D, \frac{M \cdot \left(M \cdot w0\right)}{d \cdot \ell} \cdot \frac{h \cdot -0.125}{d}, w0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (/ (* M D) (* 2.0 d)) 2e+171)
   w0
   (fma (* D D) (* (/ (* M (* M w0)) (* d l)) (/ (* h -0.125) 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)) <= 2e+171) {
		tmp = w0;
	} else {
		tmp = fma((D * D), (((M * (M * w0)) / (d * l)) * ((h * -0.125) / 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)) <= 2e+171)
		tmp = w0;
	else
		tmp = fma(Float64(D * D), Float64(Float64(Float64(M * Float64(M * w0)) / Float64(d * l)) * Float64(Float64(h * -0.125) / 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], 2e+171], w0, N[(N[(D * D), $MachinePrecision] * N[(N[(N[(M * N[(M * w0), $MachinePrecision]), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision] * N[(N[(h * -0.125), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] + w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) < 1.99999999999999991e171

   1. Initial program 84.1%

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

   3. Taylor expanded in M around 0

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

   5. Simplified 74.7%

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

      1. *-rgt-identity 74.7

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

   7. Applied egg-rr 74.7%

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

   if 1.99999999999999991e171 < (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d))

   1. Initial program 73.5%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   5. Simplified 61.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 69.8

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

   7. Applied egg-rr 69.8%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq 2 \cdot 10^{+171}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(D \cdot D, \frac{M \cdot \left(M \cdot w0\right)}{d \cdot \ell} \cdot \frac{h \cdot -0.125}{d}, w0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 72.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq 10^{+173}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(D \cdot D, \left(h \cdot -0.125\right) \cdot \frac{M \cdot \left(M \cdot w0\right)}{d \cdot \left(d \cdot \ell\right)}, w0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (/ (* M D) (* 2.0 d)) 1e+173)
   w0
   (fma (* D D) (* (* h -0.125) (/ (* M (* M w0)) (* d (* d l)))) 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+173) {
		tmp = w0;
	} else {
		tmp = fma((D * D), ((h * -0.125) * ((M * (M * w0)) / (d * (d * l)))), 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+173)
		tmp = w0;
	else
		tmp = fma(Float64(D * D), Float64(Float64(h * -0.125) * Float64(Float64(M * Float64(M * w0)) / Float64(d * Float64(d * l)))), 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+173], w0, N[(N[(D * D), $MachinePrecision] * N[(N[(h * -0.125), $MachinePrecision] * N[(N[(M * N[(M * w0), $MachinePrecision]), $MachinePrecision] / N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) < 1e173

   1. Initial program 83.8%

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

   3. Taylor expanded in M around 0

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

   5. Simplified 74.4%

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

      1. *-rgt-identity 74.4

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

   7. Applied egg-rr 74.4%

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

   if 1e173 < (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d))

   1. Initial program 76.3%

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

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

   5. Simplified 63.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 63.7

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 72.0

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

   7. Applied egg-rr 72.0%

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

Alternative 13: 72.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq 5 \cdot 10^{+190}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(D \cdot D\right) \cdot -0.125\right) \cdot \frac{w0 \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\ell \cdot \left(d \cdot d\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (/ (* M D) (* 2.0 d)) 5e+190)
   w0
   (* (* (* D D) -0.125) (/ (* w0 (* h (* M M))) (* l (* d d))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (((M * D) / (2.0 * d)) <= 5e+190) {
		tmp = w0;
	} else {
		tmp = ((D * D) * -0.125) * ((w0 * (h * (M * M))) / (l * (d * d)));
	}
	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)) <= 5d+190) then
        tmp = w0
    else
        tmp = ((d * d) * (-0.125d0)) * ((w0 * (h * (m * m))) / (l * (d_1 * d_1)))
    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 (((M * D) / (2.0 * d)) <= 5e+190) {
		tmp = w0;
	} else {
		tmp = ((D * D) * -0.125) * ((w0 * (h * (M * M))) / (l * (d * d)));
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if ((M * D) / (2.0 * d)) <= 5e+190:
		tmp = w0
	else:
		tmp = ((D * D) * -0.125) * ((w0 * (h * (M * M))) / (l * (d * d)))
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64(Float64(M * D) / Float64(2.0 * d)) <= 5e+190)
		tmp = w0;
	else
		tmp = Float64(Float64(Float64(D * D) * -0.125) * Float64(Float64(w0 * Float64(h * Float64(M * M))) / Float64(l * Float64(d * d))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (((M * D) / (2.0 * d)) <= 5e+190)
		tmp = w0;
	else
		tmp = ((D * D) * -0.125) * ((w0 * (h * (M * M))) / (l * (d * d)));
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 5e+190], w0, N[(N[(N[(D * D), $MachinePrecision] * -0.125), $MachinePrecision] * N[(N[(w0 * N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) < 5.00000000000000036e190

   1. Initial program 83.5%

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

   3. Taylor expanded in M around 0

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

   5. Simplified 73.8%

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

      1. *-rgt-identity 73.8

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

   7. Applied egg-rr 73.8%

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

   if 5.00000000000000036e190 < (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d))

   1. Initial program 78.6%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   5. Simplified 69.2%

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

   6. 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. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 69.2

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

   8. Simplified 69.2%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq 5 \cdot 10^{+190}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(D \cdot D\right) \cdot -0.125\right) \cdot \frac{w0 \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\ell \cdot \left(d \cdot d\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 74.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;D \leq 4.9 \cdot 10^{-54}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{M}{d} \cdot \left(D \cdot \frac{M \cdot \left(h \cdot -0.25\right)}{d \cdot \ell}\right), D, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= D 4.9e-54)
   w0
   (* w0 (sqrt (fma (* (/ M d) (* D (/ (* M (* h -0.25)) (* d l)))) D 1.0)))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (D <= 4.9e-54) {
		tmp = w0;
	} else {
		tmp = w0 * sqrt(fma(((M / d) * (D * ((M * (h * -0.25)) / (d * l)))), D, 1.0));
	}
	return tmp;
}

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (D <= 4.9e-54)
		tmp = w0;
	else
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(M / d) * Float64(D * Float64(Float64(M * Float64(h * -0.25)) / Float64(d * l)))), D, 1.0)));
	end
	return tmp
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[D, 4.9e-54], w0, N[(w0 * N[Sqrt[N[(N[(N[(M / d), $MachinePrecision] * N[(D * N[(N[(M * N[(h * -0.25), $MachinePrecision]), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * D + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 4.90000000000000021e-54

   1. Initial program 85.5%

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

   3. Taylor expanded in M around 0

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

   5. Simplified 74.2%

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

      1. *-rgt-identity 74.2

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

   7. Applied egg-rr 74.2%

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

   if 4.90000000000000021e-54 < D

   1. Initial program 76.7%

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

   3. Taylor expanded in w0 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. cancel-sign-sub-inv N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

   5. Simplified 53.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

   7. Applied egg-rr 70.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. times-frac N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

   9. Applied egg-rr 77.8%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 4.9 \cdot 10^{-54}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{M}{d} \cdot \left(D \cdot \frac{M \cdot \left(h \cdot -0.25\right)}{d \cdot \ell}\right), D, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 67.8% 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 83.1%

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

3. Taylor expanded in M around 0

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

5. Simplified 67.8%

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

   1. *-rgt-identity 67.8

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

7. Applied egg-rr 67.8%

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

Reproduce

herbie shell --seed 2024208 
(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))))))