Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.3% → 82.2%

Time: 1.5min

Alternatives: 14

Speedup: 1.4×

• could not determine a ground truth

  1. d = -5.349642227923744e-164
  2. h = -1.8837966661259885e+160
  3. l = 3.6255514618222246e-59
  4. M = 2.2218744882517165e+136
  5. D = 1.4080306027872842e+209

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    code = (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	return (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
}

Python

def code(d, h, l, M, D):
	return (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))

Julia

function code(d, h, l, M, D)
	return Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
end

MATLAB

function tmp = code(d, h, l, M, D)
	tmp = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
end

Wolfram

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

Initial Program: 66.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    code = (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	return (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
}

Python

def code(d, h, l, M, D):
	return (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))

Julia

function code(d, h, l, M, D)
	return Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
end

MATLAB

function tmp = code(d, h, l, M, D)
	tmp = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
end

Wolfram

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

Alternative 1: 82.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \frac{\sqrt{-d}}{\sqrt{-h}}\\ \mathbf{if}\;\ell \leq -2 \cdot 10^{+40}:\\ \;\;\;\;t\_1 \cdot \left(t\_0 \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(\frac{D\_m}{d \cdot \frac{2}{M\_m}}\right)}^{2} \cdot -0.5, 1\right)\right)\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-309}:\\ \;\;\;\;t\_1 \cdot \left(t\_0 \cdot \left(1 + \frac{h \cdot \left(-0.5 \cdot {\left(\left(M\_m \cdot 0.5\right) \cdot \frac{D\_m}{d}\right)}^{2}\right)}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(h, \frac{{\left(M\_m \cdot \left(0.5 \cdot \frac{D\_m}{d}\right)\right)}^{2}}{\frac{\ell}{-0.5}}, 1\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l))) (t_1 (/ (sqrt (- d)) (sqrt (- h)))))
   (if (<= l -2e+40)
     (*
      t_1
      (* t_0 (fma (/ h l) (* (pow (/ D_m (* d (/ 2.0 M_m))) 2.0) -0.5) 1.0)))
     (if (<= l -1e-309)
       (*
        t_1
        (*
         t_0
         (+ 1.0 (/ (* h (* -0.5 (pow (* (* M_m 0.5) (/ D_m d)) 2.0))) l))))
       (*
        (fma h (/ (pow (* M_m (* 0.5 (/ D_m d))) 2.0) (/ l -0.5)) 1.0)
        (/ d (* (sqrt h) (sqrt l))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((d / l));
	double t_1 = sqrt(-d) / sqrt(-h);
	double tmp;
	if (l <= -2e+40) {
		tmp = t_1 * (t_0 * fma((h / l), (pow((D_m / (d * (2.0 / M_m))), 2.0) * -0.5), 1.0));
	} else if (l <= -1e-309) {
		tmp = t_1 * (t_0 * (1.0 + ((h * (-0.5 * pow(((M_m * 0.5) * (D_m / d)), 2.0))) / l)));
	} else {
		tmp = fma(h, (pow((M_m * (0.5 * (D_m / d))), 2.0) / (l / -0.5)), 1.0) * (d / (sqrt(h) * sqrt(l)));
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(d / l))
	t_1 = Float64(sqrt(Float64(-d)) / sqrt(Float64(-h)))
	tmp = 0.0
	if (l <= -2e+40)
		tmp = Float64(t_1 * Float64(t_0 * fma(Float64(h / l), Float64((Float64(D_m / Float64(d * Float64(2.0 / M_m))) ^ 2.0) * -0.5), 1.0)));
	elseif (l <= -1e-309)
		tmp = Float64(t_1 * Float64(t_0 * Float64(1.0 + Float64(Float64(h * Float64(-0.5 * (Float64(Float64(M_m * 0.5) * Float64(D_m / d)) ^ 2.0))) / l))));
	else
		tmp = Float64(fma(h, Float64((Float64(M_m * Float64(0.5 * Float64(D_m / d))) ^ 2.0) / Float64(l / -0.5)), 1.0) * Float64(d / Float64(sqrt(h) * sqrt(l))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -2e+40], N[(t$95$1 * N[(t$95$0 * N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(D$95$m / N[(d * N[(2.0 / M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -1e-309], N[(t$95$1 * N[(t$95$0 * N[(1.0 + N[(N[(h * N[(-0.5 * N[Power[N[(N[(M$95$m * 0.5), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(h * N[(N[Power[N[(M$95$m * N[(0.5 * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[(l / -0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.00000000000000006e40

   1. Initial program 56.2%

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

   3. Simplified 54.5%

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

      1. frac-2neg 54.5%

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

      2. sqrt-div 66.5%

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

   6. Applied egg-rr 66.5%

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

   if -2.00000000000000006e40 < l < -1.000000000000002e-309

   1. Initial program 72.2%

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

   3. Simplified 72.2%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*l/ 76.9%

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

   6. Applied egg-rr 76.9%

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

      1. frac-2neg 70.7%

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

      2. sqrt-div 84.9%

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

   8. Applied egg-rr 91.1%

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

   if -1.000000000000002e-309 < l

   1. Initial program 57.1%

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

   4. Applied egg-rr 31.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right)\right)\right)} - 1} \]
   5. Step-by-step derivation

      1. expm1-def 40.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right)\right)\right)\right)} \]

      2. expm1-log1p 73.2%

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

      3. *-commutative 73.2%

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

   6. Simplified 80.2%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2 \cdot 10^{+40}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(\frac{D}{d \cdot \frac{2}{M}}\right)}^{2} \cdot -0.5, 1\right)\right)\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h \cdot \left(-0.5 \cdot {\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}\right)}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(h, \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{-0.5}}, 1\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 77.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{D\_m}{d} \cdot \frac{M\_m}{2}\right)}^{2}\right)\right) \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(h, \frac{{\left(M\_m \cdot \left(0.5 \cdot \frac{D\_m}{d}\right)\right)}^{2}}{\frac{\ell}{-0.5}}, 1\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d -5e-310)
   (*
    (sqrt (/ d h))
    (*
     (+ 1.0 (* (/ h l) (* -0.5 (pow (* (/ D_m d) (/ M_m 2.0)) 2.0))))
     (/ (sqrt (- d)) (sqrt (- l)))))
   (*
    (fma h (/ (pow (* M_m (* 0.5 (/ D_m d))) 2.0) (/ l -0.5)) 1.0)
    (/ d (* (sqrt h) (sqrt l))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -5e-310) {
		tmp = sqrt((d / h)) * ((1.0 + ((h / l) * (-0.5 * pow(((D_m / d) * (M_m / 2.0)), 2.0)))) * (sqrt(-d) / sqrt(-l)));
	} else {
		tmp = fma(h, (pow((M_m * (0.5 * (D_m / d))), 2.0) / (l / -0.5)), 1.0) * (d / (sqrt(h) * sqrt(l)));
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= -5e-310)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(Float64(1.0 + Float64(Float64(h / l) * Float64(-0.5 * (Float64(Float64(D_m / d) * Float64(M_m / 2.0)) ^ 2.0)))) * Float64(sqrt(Float64(-d)) / sqrt(Float64(-l)))));
	else
		tmp = Float64(fma(h, Float64((Float64(M_m * Float64(0.5 * Float64(D_m / d))) ^ 2.0) / Float64(l / -0.5)), 1.0) * Float64(d / Float64(sqrt(h) * sqrt(l))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, -5e-310], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(-0.5 * N[Power[N[(N[(D$95$m / d), $MachinePrecision] * N[(M$95$m / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(h * N[(N[Power[N[(M$95$m * N[(0.5 * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[(l / -0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -4.999999999999985e-310

   1. Initial program 64.4%

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

   3. Simplified 62.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. frac-2neg 62.9%

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

      2. sqrt-div 68.0%

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

   6. Applied egg-rr 68.0%

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

   if -4.999999999999985e-310 < d

   1. Initial program 57.1%

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

   4. Applied egg-rr 31.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right)\right)\right)} - 1} \]
   5. Step-by-step derivation

      1. expm1-def 40.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right)\right)\right)\right)} \]

      2. expm1-log1p 73.2%

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

      3. *-commutative 73.2%

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

   6. Simplified 80.2%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right) \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(h, \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{-0.5}}, 1\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;h \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\left(1 + \frac{h \cdot \left(-0.5 \cdot {\left(\left(M\_m \cdot 0.5\right) \cdot \frac{D\_m}{d}\right)}^{2}\right)}{\ell}\right) \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(h, \frac{{\left(M\_m \cdot \left(0.5 \cdot \frac{D\_m}{d}\right)\right)}^{2}}{\frac{\ell}{-0.5}}, 1\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= h -1e-310)
   (*
    (sqrt (/ d h))
    (*
     (+ 1.0 (/ (* h (* -0.5 (pow (* (* M_m 0.5) (/ D_m d)) 2.0))) l))
     (/ (sqrt (- d)) (sqrt (- l)))))
   (*
    (fma h (/ (pow (* M_m (* 0.5 (/ D_m d))) 2.0) (/ l -0.5)) 1.0)
    (/ d (* (sqrt h) (sqrt l))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (h <= -1e-310) {
		tmp = sqrt((d / h)) * ((1.0 + ((h * (-0.5 * pow(((M_m * 0.5) * (D_m / d)), 2.0))) / l)) * (sqrt(-d) / sqrt(-l)));
	} else {
		tmp = fma(h, (pow((M_m * (0.5 * (D_m / d))), 2.0) / (l / -0.5)), 1.0) * (d / (sqrt(h) * sqrt(l)));
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (h <= -1e-310)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(Float64(1.0 + Float64(Float64(h * Float64(-0.5 * (Float64(Float64(M_m * 0.5) * Float64(D_m / d)) ^ 2.0))) / l)) * Float64(sqrt(Float64(-d)) / sqrt(Float64(-l)))));
	else
		tmp = Float64(fma(h, Float64((Float64(M_m * Float64(0.5 * Float64(D_m / d))) ^ 2.0) / Float64(l / -0.5)), 1.0) * Float64(d / Float64(sqrt(h) * sqrt(l))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[h, -1e-310], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 + N[(N[(h * N[(-0.5 * N[Power[N[(N[(M$95$m * 0.5), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(h * N[(N[Power[N[(M$95$m * N[(0.5 * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[(l / -0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < -9.999999999999969e-311

   1. Initial program 64.4%

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

   3. Simplified 62.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*l/ 64.0%

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

   6. Applied egg-rr 64.0%

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

      1. frac-2neg 62.9%

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

      2. sqrt-div 68.0%

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

   8. Applied egg-rr 72.1%

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

   if -9.999999999999969e-311 < h

   1. Initial program 57.1%

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

   4. Applied egg-rr 31.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right)\right)\right)} - 1} \]
   5. Step-by-step derivation

      1. expm1-def 40.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right)\right)\right)\right)} \]

      2. expm1-log1p 73.2%

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

      3. *-commutative 73.2%

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

   6. Simplified 80.2%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\left(1 + \frac{h \cdot \left(-0.5 \cdot {\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}\right)}{\ell}\right) \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(h, \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{-0.5}}, 1\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{D\_m}{d} \cdot \frac{M\_m}{2}\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(h, \frac{{\left(M\_m \cdot \left(0.5 \cdot \frac{D\_m}{d}\right)\right)}^{2}}{\frac{\ell}{-0.5}}, 1\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l -1e-309)
   (*
    (/ (sqrt (- d)) (sqrt (- h)))
    (*
     (sqrt (/ d l))
     (+ 1.0 (* (/ h l) (* -0.5 (pow (* (/ D_m d) (/ M_m 2.0)) 2.0))))))
   (*
    (fma h (/ (pow (* M_m (* 0.5 (/ D_m d))) 2.0) (/ l -0.5)) 1.0)
    (/ d (* (sqrt h) (sqrt l))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -1e-309) {
		tmp = (sqrt(-d) / sqrt(-h)) * (sqrt((d / l)) * (1.0 + ((h / l) * (-0.5 * pow(((D_m / d) * (M_m / 2.0)), 2.0)))));
	} else {
		tmp = fma(h, (pow((M_m * (0.5 * (D_m / d))), 2.0) / (l / -0.5)), 1.0) * (d / (sqrt(h) * sqrt(l)));
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= -1e-309)
		tmp = Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * Float64(sqrt(Float64(d / l)) * Float64(1.0 + Float64(Float64(h / l) * Float64(-0.5 * (Float64(Float64(D_m / d) * Float64(M_m / 2.0)) ^ 2.0))))));
	else
		tmp = Float64(fma(h, Float64((Float64(M_m * Float64(0.5 * Float64(D_m / d))) ^ 2.0) / Float64(l / -0.5)), 1.0) * Float64(d / Float64(sqrt(h) * sqrt(l))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, -1e-309], N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(-0.5 * N[Power[N[(N[(D$95$m / d), $MachinePrecision] * N[(M$95$m / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(h * N[(N[Power[N[(M$95$m * N[(0.5 * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[(l / -0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.000000000000002e-309

   1. Initial program 64.4%

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

   3. Simplified 62.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. frac-2neg 62.8%

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

      2. sqrt-div 76.0%

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

   6. Applied egg-rr 75.3%

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

   if -1.000000000000002e-309 < l

   1. Initial program 57.1%

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

   4. Applied egg-rr 31.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right)\right)\right)} - 1} \]
   5. Step-by-step derivation

      1. expm1-def 40.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right)\right)\right)\right)} \]

      2. expm1-log1p 73.2%

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

      3. *-commutative 73.2%

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

   6. Simplified 80.2%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(h, \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{-0.5}}, 1\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h \cdot \left(-0.5 \cdot {\left(\left(M\_m \cdot 0.5\right) \cdot \frac{D\_m}{d}\right)}^{2}\right)}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(h, \frac{{\left(M\_m \cdot \left(0.5 \cdot \frac{D\_m}{d}\right)\right)}^{2}}{\frac{\ell}{-0.5}}, 1\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l -1e-309)
   (*
    (/ (sqrt (- d)) (sqrt (- h)))
    (*
     (sqrt (/ d l))
     (+ 1.0 (/ (* h (* -0.5 (pow (* (* M_m 0.5) (/ D_m d)) 2.0))) l))))
   (*
    (fma h (/ (pow (* M_m (* 0.5 (/ D_m d))) 2.0) (/ l -0.5)) 1.0)
    (/ d (* (sqrt h) (sqrt l))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -1e-309) {
		tmp = (sqrt(-d) / sqrt(-h)) * (sqrt((d / l)) * (1.0 + ((h * (-0.5 * pow(((M_m * 0.5) * (D_m / d)), 2.0))) / l)));
	} else {
		tmp = fma(h, (pow((M_m * (0.5 * (D_m / d))), 2.0) / (l / -0.5)), 1.0) * (d / (sqrt(h) * sqrt(l)));
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= -1e-309)
		tmp = Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * Float64(sqrt(Float64(d / l)) * Float64(1.0 + Float64(Float64(h * Float64(-0.5 * (Float64(Float64(M_m * 0.5) * Float64(D_m / d)) ^ 2.0))) / l))));
	else
		tmp = Float64(fma(h, Float64((Float64(M_m * Float64(0.5 * Float64(D_m / d))) ^ 2.0) / Float64(l / -0.5)), 1.0) * Float64(d / Float64(sqrt(h) * sqrt(l))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, -1e-309], N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(h * N[(-0.5 * N[Power[N[(N[(M$95$m * 0.5), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(h * N[(N[Power[N[(M$95$m * N[(0.5 * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[(l / -0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.000000000000002e-309

   1. Initial program 64.4%

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

   3. Simplified 62.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*l/ 64.0%

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

   6. Applied egg-rr 64.0%

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

      1. frac-2neg 62.8%

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

      2. sqrt-div 76.0%

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

   8. Applied egg-rr 76.3%

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

   if -1.000000000000002e-309 < l

   1. Initial program 57.1%

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

   4. Applied egg-rr 31.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right)\right)\right)} - 1} \]
   5. Step-by-step derivation

      1. expm1-def 40.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right)\right)\right)\right)} \]

      2. expm1-log1p 73.2%

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

      3. *-commutative 73.2%

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

   6. Simplified 80.2%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h \cdot \left(-0.5 \cdot {\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}\right)}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(h, \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{-0.5}}, 1\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := M\_m \cdot \frac{D\_m \cdot 0.5}{d}\\ \mathbf{if}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \left(\frac{t\_0}{\ell} \cdot \frac{t\_0}{-2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(h, \frac{{\left(M\_m \cdot \left(0.5 \cdot \frac{D\_m}{d}\right)\right)}^{2}}{\frac{\ell}{-0.5}}, 1\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* M_m (/ (* D_m 0.5) d))))
   (if (<= d -5e-310)
     (*
      (sqrt (/ d h))
      (* (sqrt (/ d l)) (+ 1.0 (* h (* (/ t_0 l) (/ t_0 -2.0))))))
     (*
      (fma h (/ (pow (* M_m (* 0.5 (/ D_m d))) 2.0) (/ l -0.5)) 1.0)
      (/ d (* (sqrt h) (sqrt l)))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = M_m * ((D_m * 0.5) / d);
	double tmp;
	if (d <= -5e-310) {
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 + (h * ((t_0 / l) * (t_0 / -2.0)))));
	} else {
		tmp = fma(h, (pow((M_m * (0.5 * (D_m / d))), 2.0) / (l / -0.5)), 1.0) * (d / (sqrt(h) * sqrt(l)));
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(M_m * Float64(Float64(D_m * 0.5) / d))
	tmp = 0.0
	if (d <= -5e-310)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * Float64(1.0 + Float64(h * Float64(Float64(t_0 / l) * Float64(t_0 / -2.0))))));
	else
		tmp = Float64(fma(h, Float64((Float64(M_m * Float64(0.5 * Float64(D_m / d))) ^ 2.0) / Float64(l / -0.5)), 1.0) * Float64(d / Float64(sqrt(h) * sqrt(l))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(M$95$m * N[(N[(D$95$m * 0.5), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -5e-310], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(h * N[(N[(t$95$0 / l), $MachinePrecision] * N[(t$95$0 / -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(h * N[(N[Power[N[(M$95$m * N[(0.5 * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[(l / -0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if d < -4.999999999999985e-310

   1. Initial program 64.4%

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

   3. Simplified 62.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing

   6. Applied egg-rr 33.5%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{h}{\ell} \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)\right)} - 1\right)}\right)\right) \]
   7. Step-by-step derivation

      1. expm1-def 33.5%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{h}{\ell} \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)\right)\right)}\right)\right) \]

      2. expm1-log1p 62.9%

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

      3. associate-*l/ 64.0%

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

      4. *-commutative 64.0%

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

      5. associate-*l/ 66.2%

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

      6. *-commutative 66.2%

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

      7. associate-/l* 66.2%

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

      8. associate-*l* 66.2%

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

   8. Simplified 66.2%

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

      1. unpow2 66.2%

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

      2. div-inv 66.2%

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

      3. metadata-eval 66.2%

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

      4. metadata-eval 66.2%

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

      5. times-frac 66.9%

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

      6. associate-*r/ 66.9%

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

      7. associate-*r/ 66.9%

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

      8. metadata-eval 66.9%

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

   10. Applied egg-rr 66.9%

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

   if -4.999999999999985e-310 < d

   1. Initial program 57.1%

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

   4. Applied egg-rr 31.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right)\right)\right)} - 1} \]
   5. Step-by-step derivation

      1. expm1-def 40.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right)\right)\right)\right)} \]

      2. expm1-log1p 73.2%

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

      3. *-commutative 73.2%

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

   6. Simplified 80.2%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \left(\frac{M \cdot \frac{D \cdot 0.5}{d}}{\ell} \cdot \frac{M \cdot \frac{D \cdot 0.5}{d}}{-2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(h, \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{-0.5}}, 1\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := M\_m \cdot \frac{D\_m \cdot 0.5}{d}\\ \mathbf{if}\;\ell \leq 3.85 \cdot 10^{-305}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \left(\frac{t\_0}{\ell} \cdot \frac{t\_0}{-2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - \frac{{\left(D\_m \cdot \frac{M\_m}{d \cdot 2}\right)}^{2}}{\frac{\frac{\ell}{h}}{0.5}}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* M_m (/ (* D_m 0.5) d))))
   (if (<= l 3.85e-305)
     (*
      (sqrt (/ d h))
      (* (sqrt (/ d l)) (+ 1.0 (* h (* (/ t_0 l) (/ t_0 -2.0))))))
     (*
      (/ d (* (sqrt h) (sqrt l)))
      (- 1.0 (/ (pow (* D_m (/ M_m (* d 2.0))) 2.0) (/ (/ l h) 0.5)))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = M_m * ((D_m * 0.5) / d);
	double tmp;
	if (l <= 3.85e-305) {
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 + (h * ((t_0 / l) * (t_0 / -2.0)))));
	} else {
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 - (pow((D_m * (M_m / (d * 2.0))), 2.0) / ((l / h) / 0.5)));
	}
	return tmp;
}

Fortran

M_m = abs(M)
D_m = abs(D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = m_m * ((d_m * 0.5d0) / d)
    if (l <= 3.85d-305) then
        tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0d0 + (h * ((t_0 / l) * (t_0 / (-2.0d0))))))
    else
        tmp = (d / (sqrt(h) * sqrt(l))) * (1.0d0 - (((d_m * (m_m / (d * 2.0d0))) ** 2.0d0) / ((l / h) / 0.5d0)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = M_m * ((D_m * 0.5) / d);
	double tmp;
	if (l <= 3.85e-305) {
		tmp = Math.sqrt((d / h)) * (Math.sqrt((d / l)) * (1.0 + (h * ((t_0 / l) * (t_0 / -2.0)))));
	} else {
		tmp = (d / (Math.sqrt(h) * Math.sqrt(l))) * (1.0 - (Math.pow((D_m * (M_m / (d * 2.0))), 2.0) / ((l / h) / 0.5)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = M_m * ((D_m * 0.5) / d)
	tmp = 0
	if l <= 3.85e-305:
		tmp = math.sqrt((d / h)) * (math.sqrt((d / l)) * (1.0 + (h * ((t_0 / l) * (t_0 / -2.0)))))
	else:
		tmp = (d / (math.sqrt(h) * math.sqrt(l))) * (1.0 - (math.pow((D_m * (M_m / (d * 2.0))), 2.0) / ((l / h) / 0.5)))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(M_m * Float64(Float64(D_m * 0.5) / d))
	tmp = 0.0
	if (l <= 3.85e-305)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * Float64(1.0 + Float64(h * Float64(Float64(t_0 / l) * Float64(t_0 / -2.0))))));
	else
		tmp = Float64(Float64(d / Float64(sqrt(h) * sqrt(l))) * Float64(1.0 - Float64((Float64(D_m * Float64(M_m / Float64(d * 2.0))) ^ 2.0) / Float64(Float64(l / h) / 0.5))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = M_m * ((D_m * 0.5) / d);
	tmp = 0.0;
	if (l <= 3.85e-305)
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 + (h * ((t_0 / l) * (t_0 / -2.0)))));
	else
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 - (((D_m * (M_m / (d * 2.0))) ^ 2.0) / ((l / h) / 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(M$95$m * N[(N[(D$95$m * 0.5), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, 3.85e-305], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(h * N[(N[(t$95$0 / l), $MachinePrecision] * N[(t$95$0 / -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[Power[N[(D$95$m * N[(M$95$m / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[(N[(l / h), $MachinePrecision] / 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 3.8499999999999999e-305

   1. Initial program 64.5%

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

   3. Simplified 63.0%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing

   6. Applied egg-rr 32.7%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{h}{\ell} \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)\right)} - 1\right)}\right)\right) \]
   7. Step-by-step derivation

      1. expm1-def 32.7%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{h}{\ell} \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)\right)\right)}\right)\right) \]

      2. expm1-log1p 63.0%

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

      3. associate-*l/ 64.0%

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

      4. *-commutative 64.0%

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

      5. associate-*l/ 66.2%

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

      6. *-commutative 66.2%

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

      7. associate-/l* 66.2%

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

      8. associate-*l* 66.2%

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

   8. Simplified 66.2%

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

      1. unpow2 66.2%

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

      2. div-inv 66.2%

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

      3. metadata-eval 66.2%

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

      4. metadata-eval 66.2%

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

      5. times-frac 66.9%

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

      6. associate-*r/ 66.9%

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

      7. associate-*r/ 66.9%

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

      8. metadata-eval 66.9%

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

   10. Applied egg-rr 66.9%

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

   if 3.8499999999999999e-305 < l

   1. Initial program 56.8%

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

   4. Applied egg-rr 31.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right)\right)\right)} - 1} \]
   5. Step-by-step derivation

      1. expm1-def 41.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right)\right)\right)\right)} \]

      2. expm1-log1p 73.4%

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

      3. associate-*l/ 79.7%

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

      4. *-commutative 79.7%

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

      5. associate-/l* 75.2%

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

      6. associate-/l* 75.2%

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

      7. *-commutative 75.2%

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

      8. /-rgt-identity 75.2%

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

      9. associate-/l* 75.2%

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

      10. metadata-eval 75.2%

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

      11. times-frac 74.3%

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

      12. associate-*r/ 75.9%

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

      13. *-commutative 75.9%

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

   6. Simplified 75.9%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3.85 \cdot 10^{-305}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \left(\frac{M \cdot \frac{D \cdot 0.5}{d}}{\ell} \cdot \frac{M \cdot \frac{D \cdot 0.5}{d}}{-2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - \frac{{\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}}{\frac{\frac{\ell}{h}}{0.5}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 70.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := M\_m \cdot \frac{D\_m \cdot 0.5}{d}\\ \mathbf{if}\;\ell \leq 3.55 \cdot 10^{+207}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \left(\frac{t\_0}{\ell} \cdot \frac{t\_0}{-2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot \sqrt{\frac{1}{h}}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* M_m (/ (* D_m 0.5) d))))
   (if (<= l 3.55e+207)
     (*
      (sqrt (/ d h))
      (* (sqrt (/ d l)) (+ 1.0 (* h (* (/ t_0 l) (/ t_0 -2.0))))))
     (* d (* (pow l -0.5) (sqrt (/ 1.0 h)))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = M_m * ((D_m * 0.5) / d);
	double tmp;
	if (l <= 3.55e+207) {
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 + (h * ((t_0 / l) * (t_0 / -2.0)))));
	} else {
		tmp = d * (pow(l, -0.5) * sqrt((1.0 / h)));
	}
	return tmp;
}

Fortran

M_m = abs(M)
D_m = abs(D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = m_m * ((d_m * 0.5d0) / d)
    if (l <= 3.55d+207) then
        tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0d0 + (h * ((t_0 / l) * (t_0 / (-2.0d0))))))
    else
        tmp = d * ((l ** (-0.5d0)) * sqrt((1.0d0 / h)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = M_m * ((D_m * 0.5) / d);
	double tmp;
	if (l <= 3.55e+207) {
		tmp = Math.sqrt((d / h)) * (Math.sqrt((d / l)) * (1.0 + (h * ((t_0 / l) * (t_0 / -2.0)))));
	} else {
		tmp = d * (Math.pow(l, -0.5) * Math.sqrt((1.0 / h)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = M_m * ((D_m * 0.5) / d)
	tmp = 0
	if l <= 3.55e+207:
		tmp = math.sqrt((d / h)) * (math.sqrt((d / l)) * (1.0 + (h * ((t_0 / l) * (t_0 / -2.0)))))
	else:
		tmp = d * (math.pow(l, -0.5) * math.sqrt((1.0 / h)))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(M_m * Float64(Float64(D_m * 0.5) / d))
	tmp = 0.0
	if (l <= 3.55e+207)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * Float64(1.0 + Float64(h * Float64(Float64(t_0 / l) * Float64(t_0 / -2.0))))));
	else
		tmp = Float64(d * Float64((l ^ -0.5) * sqrt(Float64(1.0 / h))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = M_m * ((D_m * 0.5) / d);
	tmp = 0.0;
	if (l <= 3.55e+207)
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 + (h * ((t_0 / l) * (t_0 / -2.0)))));
	else
		tmp = d * ((l ^ -0.5) * sqrt((1.0 / h)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(M$95$m * N[(N[(D$95$m * 0.5), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, 3.55e+207], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(h * N[(N[(t$95$0 / l), $MachinePrecision] * N[(t$95$0 / -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[l, -0.5], $MachinePrecision] * N[Sqrt[N[(1.0 / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 3.55000000000000021e207

   1. Initial program 63.9%

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

   3. Simplified 63.5%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing

   6. Applied egg-rr 32.0%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{h}{\ell} \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)\right)} - 1\right)}\right)\right) \]
   7. Step-by-step derivation

      1. expm1-def 32.0%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{h}{\ell} \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)\right)\right)}\right)\right) \]

      2. expm1-log1p 63.5%

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

      3. associate-*l/ 66.6%

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

      4. *-commutative 66.6%

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

      5. associate-*l/ 67.5%

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

      6. *-commutative 67.5%

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

      7. associate-/l* 67.5%

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

      8. associate-*l* 67.5%

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

   8. Simplified 67.5%

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

      1. unpow2 67.5%

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

      2. div-inv 67.5%

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

      3. metadata-eval 67.5%

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

      4. metadata-eval 67.5%

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

      5. times-frac 68.7%

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

      6. associate-*r/ 68.7%

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

      7. associate-*r/ 68.7%

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

      8. metadata-eval 68.7%

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

   10. Applied egg-rr 68.7%

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

   if 3.55000000000000021e207 < l

   1. Initial program 28.3%

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

   3. Taylor expanded in d around inf 45.2%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative 45.2%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      2. associate-/r* 49.2%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   5. Simplified 49.2%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]
   6. Step-by-step derivation

      1. pow1/2 49.2%

         \[\leadsto d \cdot \color{blue}{{\left(\frac{\frac{1}{\ell}}{h}\right)}^{0.5}} \]

      2. div-inv 49.2%

         \[\leadsto d \cdot {\color{blue}{\left(\frac{1}{\ell} \cdot \frac{1}{h}\right)}}^{0.5} \]

      3. unpow-prod-down 57.3%

         \[\leadsto d \cdot \color{blue}{\left({\left(\frac{1}{\ell}\right)}^{0.5} \cdot {\left(\frac{1}{h}\right)}^{0.5}\right)} \]

      4. pow1/2 57.3%

         \[\leadsto d \cdot \left(\color{blue}{\sqrt{\frac{1}{\ell}}} \cdot {\left(\frac{1}{h}\right)}^{0.5}\right) \]

      5. inv-pow 57.3%

         \[\leadsto d \cdot \left(\sqrt{\color{blue}{{\ell}^{-1}}} \cdot {\left(\frac{1}{h}\right)}^{0.5}\right) \]

      6. sqrt-pow1 57.5%

         \[\leadsto d \cdot \left(\color{blue}{{\ell}^{\left(\frac{-1}{2}\right)}} \cdot {\left(\frac{1}{h}\right)}^{0.5}\right) \]

      7. metadata-eval 57.5%

         \[\leadsto d \cdot \left({\ell}^{\color{blue}{-0.5}} \cdot {\left(\frac{1}{h}\right)}^{0.5}\right) \]

   7. Applied egg-rr 57.5%

      \[\leadsto d \cdot \color{blue}{\left({\ell}^{-0.5} \cdot {\left(\frac{1}{h}\right)}^{0.5}\right)} \]
   8. Step-by-step derivation

      1. unpow1/2 57.5%

         \[\leadsto d \cdot \left({\ell}^{-0.5} \cdot \color{blue}{\sqrt{\frac{1}{h}}}\right) \]

   9. Simplified 57.5%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3.55 \cdot 10^{+207}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \left(\frac{M \cdot \frac{D \cdot 0.5}{d}}{\ell} \cdot \frac{M \cdot \frac{D \cdot 0.5}{d}}{-2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot \sqrt{\frac{1}{h}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 45.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -9.5 \cdot 10^{-108}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;d \leq 8 \cdot 10^{-261}:\\ \;\;\;\;d \cdot \sqrt[3]{{\left(\frac{1}{\ell \cdot h}\right)}^{1.5}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot \sqrt{\frac{1}{h}}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d -9.5e-108)
   (* (sqrt (/ d l)) (sqrt (/ d h)))
   (if (<= d 8e-261)
     (* d (cbrt (pow (/ 1.0 (* l h)) 1.5)))
     (* d (* (pow l -0.5) (sqrt (/ 1.0 h)))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -9.5e-108) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else if (d <= 8e-261) {
		tmp = d * cbrt(pow((1.0 / (l * h)), 1.5));
	} else {
		tmp = d * (pow(l, -0.5) * sqrt((1.0 / h)));
	}
	return tmp;
}

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -9.5e-108) {
		tmp = Math.sqrt((d / l)) * Math.sqrt((d / h));
	} else if (d <= 8e-261) {
		tmp = d * Math.cbrt(Math.pow((1.0 / (l * h)), 1.5));
	} else {
		tmp = d * (Math.pow(l, -0.5) * Math.sqrt((1.0 / h)));
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= -9.5e-108)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	elseif (d <= 8e-261)
		tmp = Float64(d * cbrt((Float64(1.0 / Float64(l * h)) ^ 1.5)));
	else
		tmp = Float64(d * Float64((l ^ -0.5) * sqrt(Float64(1.0 / h))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, -9.5e-108], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 8e-261], N[(d * N[Power[N[Power[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[l, -0.5], $MachinePrecision] * N[Sqrt[N[(1.0 / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -9.5000000000000005e-108

   1. Initial program 71.0%

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

   3. Simplified 69.8%

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

   5. Taylor expanded in h around 0 47.6%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{1}\right) \]

   if -9.5000000000000005e-108 < d < 7.99999999999999987e-261

   1. Initial program 41.1%

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

   3. Taylor expanded in d around inf 15.8%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative 15.8%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      2. associate-/r* 15.8%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   5. Simplified 15.8%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]
   6. Step-by-step derivation

      1. add-cbrt-cube 26.8%

         \[\leadsto d \cdot \color{blue}{\sqrt[3]{\left(\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}}} \]

      2. pow1/3 26.8%

         \[\leadsto d \cdot \color{blue}{{\left(\left(\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)}^{0.3333333333333333}} \]

      3. add-sqr-sqrt 26.8%

         \[\leadsto d \cdot {\left(\color{blue}{\frac{\frac{1}{\ell}}{h}} \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)}^{0.3333333333333333} \]

      4. pow1 26.8%

         \[\leadsto d \cdot {\left(\color{blue}{{\left(\frac{\frac{1}{\ell}}{h}\right)}^{1}} \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)}^{0.3333333333333333} \]

      5. pow1/2 26.8%

         \[\leadsto d \cdot {\left({\left(\frac{\frac{1}{\ell}}{h}\right)}^{1} \cdot \color{blue}{{\left(\frac{\frac{1}{\ell}}{h}\right)}^{0.5}}\right)}^{0.3333333333333333} \]

      6. pow-prod-up 26.8%

         \[\leadsto d \cdot {\color{blue}{\left({\left(\frac{\frac{1}{\ell}}{h}\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333} \]

      7. associate-/l/ 26.8%

         \[\leadsto d \cdot {\left({\color{blue}{\left(\frac{1}{h \cdot \ell}\right)}}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

      8. metadata-eval 26.8%

         \[\leadsto d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} \]

   7. Applied egg-rr 26.8%

      \[\leadsto d \cdot \color{blue}{{\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}} \]
   8. Step-by-step derivation

      1. unpow1/3 26.8%

         \[\leadsto d \cdot \color{blue}{\sqrt[3]{{\left(\frac{1}{h \cdot \ell}\right)}^{1.5}}} \]

      2. *-commutative 26.8%

         \[\leadsto d \cdot \sqrt[3]{{\left(\frac{1}{\color{blue}{\ell \cdot h}}\right)}^{1.5}} \]

   9. Simplified 26.8%

      \[\leadsto d \cdot \color{blue}{\sqrt[3]{{\left(\frac{1}{\ell \cdot h}\right)}^{1.5}}} \]

   if 7.99999999999999987e-261 < d

   1. Initial program 62.1%

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

   3. Taylor expanded in d around inf 40.9%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative 40.9%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      2. associate-/r* 41.7%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   5. Simplified 41.7%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]
   6. Step-by-step derivation

      1. pow1/2 41.7%

         \[\leadsto d \cdot \color{blue}{{\left(\frac{\frac{1}{\ell}}{h}\right)}^{0.5}} \]

      2. div-inv 41.7%

         \[\leadsto d \cdot {\color{blue}{\left(\frac{1}{\ell} \cdot \frac{1}{h}\right)}}^{0.5} \]

      3. unpow-prod-down 51.9%

         \[\leadsto d \cdot \color{blue}{\left({\left(\frac{1}{\ell}\right)}^{0.5} \cdot {\left(\frac{1}{h}\right)}^{0.5}\right)} \]

      4. pow1/2 51.9%

         \[\leadsto d \cdot \left(\color{blue}{\sqrt{\frac{1}{\ell}}} \cdot {\left(\frac{1}{h}\right)}^{0.5}\right) \]

      5. inv-pow 51.9%

         \[\leadsto d \cdot \left(\sqrt{\color{blue}{{\ell}^{-1}}} \cdot {\left(\frac{1}{h}\right)}^{0.5}\right) \]

      6. sqrt-pow1 51.9%

         \[\leadsto d \cdot \left(\color{blue}{{\ell}^{\left(\frac{-1}{2}\right)}} \cdot {\left(\frac{1}{h}\right)}^{0.5}\right) \]

      7. metadata-eval 51.9%

         \[\leadsto d \cdot \left({\ell}^{\color{blue}{-0.5}} \cdot {\left(\frac{1}{h}\right)}^{0.5}\right) \]

   7. Applied egg-rr 51.9%

      \[\leadsto d \cdot \color{blue}{\left({\ell}^{-0.5} \cdot {\left(\frac{1}{h}\right)}^{0.5}\right)} \]
   8. Step-by-step derivation

      1. unpow1/2 51.9%

         \[\leadsto d \cdot \left({\ell}^{-0.5} \cdot \color{blue}{\sqrt{\frac{1}{h}}}\right) \]

   9. Simplified 51.9%

      \[\leadsto d \cdot \color{blue}{\left({\ell}^{-0.5} \cdot \sqrt{\frac{1}{h}}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 45.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -9.5 \cdot 10^{-108}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;d \leq 8 \cdot 10^{-261}:\\ \;\;\;\;d \cdot \sqrt[3]{{\left(\frac{1}{\ell \cdot h}\right)}^{1.5}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot \sqrt{\frac{1}{h}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 31.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;h \leq 2.8 \cdot 10^{-298}:\\ \;\;\;\;d \cdot \sqrt[3]{{\left(\frac{1}{\ell \cdot h}\right)}^{1.5}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{\ell}^{-0.5}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= h 2.8e-298)
   (* d (cbrt (pow (/ 1.0 (* l h)) 1.5)))
   (* d (/ (pow l -0.5) (sqrt h)))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (h <= 2.8e-298) {
		tmp = d * cbrt(pow((1.0 / (l * h)), 1.5));
	} else {
		tmp = d * (pow(l, -0.5) / sqrt(h));
	}
	return tmp;
}

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (h <= 2.8e-298) {
		tmp = d * Math.cbrt(Math.pow((1.0 / (l * h)), 1.5));
	} else {
		tmp = d * (Math.pow(l, -0.5) / Math.sqrt(h));
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (h <= 2.8e-298)
		tmp = Float64(d * cbrt((Float64(1.0 / Float64(l * h)) ^ 1.5)));
	else
		tmp = Float64(d * Float64((l ^ -0.5) / sqrt(h)));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[h, 2.8e-298], N[(d * N[Power[N[Power[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[l, -0.5], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < 2.79999999999999992e-298

   1. Initial program 64.0%

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

   3. Taylor expanded in d around inf 14.0%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative 14.0%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      2. associate-/r* 14.0%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   5. Simplified 14.0%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]
   6. Step-by-step derivation

      1. add-cbrt-cube 19.3%

         \[\leadsto d \cdot \color{blue}{\sqrt[3]{\left(\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}}} \]

      2. pow1/3 19.2%

         \[\leadsto d \cdot \color{blue}{{\left(\left(\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)}^{0.3333333333333333}} \]

      3. add-sqr-sqrt 19.2%

         \[\leadsto d \cdot {\left(\color{blue}{\frac{\frac{1}{\ell}}{h}} \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)}^{0.3333333333333333} \]

      4. pow1 19.2%

         \[\leadsto d \cdot {\left(\color{blue}{{\left(\frac{\frac{1}{\ell}}{h}\right)}^{1}} \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)}^{0.3333333333333333} \]

      5. pow1/2 19.2%

         \[\leadsto d \cdot {\left({\left(\frac{\frac{1}{\ell}}{h}\right)}^{1} \cdot \color{blue}{{\left(\frac{\frac{1}{\ell}}{h}\right)}^{0.5}}\right)}^{0.3333333333333333} \]

      6. pow-prod-up 19.2%

         \[\leadsto d \cdot {\color{blue}{\left({\left(\frac{\frac{1}{\ell}}{h}\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333} \]

      7. associate-/l/ 19.2%

         \[\leadsto d \cdot {\left({\color{blue}{\left(\frac{1}{h \cdot \ell}\right)}}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

      8. metadata-eval 19.2%

         \[\leadsto d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} \]

   7. Applied egg-rr 19.2%

      \[\leadsto d \cdot \color{blue}{{\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}} \]
   8. Step-by-step derivation

      1. unpow1/3 19.3%

         \[\leadsto d \cdot \color{blue}{\sqrt[3]{{\left(\frac{1}{h \cdot \ell}\right)}^{1.5}}} \]

      2. *-commutative 19.3%

         \[\leadsto d \cdot \sqrt[3]{{\left(\frac{1}{\color{blue}{\ell \cdot h}}\right)}^{1.5}} \]

   9. Simplified 19.3%

      \[\leadsto d \cdot \color{blue}{\sqrt[3]{{\left(\frac{1}{\ell \cdot h}\right)}^{1.5}}} \]

   if 2.79999999999999992e-298 < h

   1. Initial program 57.5%

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

   3. Taylor expanded in d around inf 36.9%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative 36.9%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      2. associate-/r* 37.6%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   5. Simplified 37.6%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]
   6. Step-by-step derivation

      1. expm1-log1p-u 36.5%

         \[\leadsto d \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\right)} \]

      2. expm1-udef 22.4%

         \[\leadsto d \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{\frac{1}{\ell}}{h}}\right)} - 1\right)} \]

      3. sqrt-div 28.8%

         \[\leadsto d \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}}\right)} - 1\right) \]

      4. inv-pow 28.8%

         \[\leadsto d \cdot \left(e^{\mathsf{log1p}\left(\frac{\sqrt{\color{blue}{{\ell}^{-1}}}}{\sqrt{h}}\right)} - 1\right) \]

      5. sqrt-pow1 28.8%

         \[\leadsto d \cdot \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{{\ell}^{\left(\frac{-1}{2}\right)}}}{\sqrt{h}}\right)} - 1\right) \]

      6. metadata-eval 28.8%

         \[\leadsto d \cdot \left(e^{\mathsf{log1p}\left(\frac{{\ell}^{\color{blue}{-0.5}}}{\sqrt{h}}\right)} - 1\right) \]

   7. Applied egg-rr 28.8%

      \[\leadsto d \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\ell}^{-0.5}}{\sqrt{h}}\right)} - 1\right)} \]
   8. Step-by-step derivation

      1. expm1-def 45.1%

         \[\leadsto d \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{-0.5}}{\sqrt{h}}\right)\right)} \]

      2. expm1-log1p 46.9%

         \[\leadsto d \cdot \color{blue}{\frac{{\ell}^{-0.5}}{\sqrt{h}}} \]

   9. Simplified 46.9%

      \[\leadsto d \cdot \color{blue}{\frac{{\ell}^{-0.5}}{\sqrt{h}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq 2.8 \cdot 10^{-298}:\\ \;\;\;\;d \cdot \sqrt[3]{{\left(\frac{1}{\ell \cdot h}\right)}^{1.5}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{\ell}^{-0.5}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 32.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.4 \cdot 10^{-307}:\\ \;\;\;\;d \cdot \sqrt[3]{{\left(\frac{1}{\ell \cdot h}\right)}^{1.5}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot \sqrt{\frac{1}{h}}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l 1.4e-307)
   (* d (cbrt (pow (/ 1.0 (* l h)) 1.5)))
   (* d (* (pow l -0.5) (sqrt (/ 1.0 h))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= 1.4e-307) {
		tmp = d * cbrt(pow((1.0 / (l * h)), 1.5));
	} else {
		tmp = d * (pow(l, -0.5) * sqrt((1.0 / h)));
	}
	return tmp;
}

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= 1.4e-307) {
		tmp = d * Math.cbrt(Math.pow((1.0 / (l * h)), 1.5));
	} else {
		tmp = d * (Math.pow(l, -0.5) * Math.sqrt((1.0 / h)));
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= 1.4e-307)
		tmp = Float64(d * cbrt((Float64(1.0 / Float64(l * h)) ^ 1.5)));
	else
		tmp = Float64(d * Float64((l ^ -0.5) * sqrt(Float64(1.0 / h))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, 1.4e-307], N[(d * N[Power[N[Power[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[l, -0.5], $MachinePrecision] * N[Sqrt[N[(1.0 / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.4e-307

   1. Initial program 64.7%

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

   3. Taylor expanded in d around inf 13.2%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative 13.2%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      2. associate-/r* 13.2%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   5. Simplified 13.2%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]
   6. Step-by-step derivation

      1. add-cbrt-cube 18.5%

         \[\leadsto d \cdot \color{blue}{\sqrt[3]{\left(\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}}} \]

      2. pow1/3 18.5%

         \[\leadsto d \cdot \color{blue}{{\left(\left(\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)}^{0.3333333333333333}} \]

      3. add-sqr-sqrt 18.5%

         \[\leadsto d \cdot {\left(\color{blue}{\frac{\frac{1}{\ell}}{h}} \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)}^{0.3333333333333333} \]

      4. pow1 18.5%

         \[\leadsto d \cdot {\left(\color{blue}{{\left(\frac{\frac{1}{\ell}}{h}\right)}^{1}} \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)}^{0.3333333333333333} \]

      5. pow1/2 18.5%

         \[\leadsto d \cdot {\left({\left(\frac{\frac{1}{\ell}}{h}\right)}^{1} \cdot \color{blue}{{\left(\frac{\frac{1}{\ell}}{h}\right)}^{0.5}}\right)}^{0.3333333333333333} \]

      6. pow-prod-up 18.5%

         \[\leadsto d \cdot {\color{blue}{\left({\left(\frac{\frac{1}{\ell}}{h}\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333} \]

      7. associate-/l/ 18.5%

         \[\leadsto d \cdot {\left({\color{blue}{\left(\frac{1}{h \cdot \ell}\right)}}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

      8. metadata-eval 18.5%

         \[\leadsto d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} \]

   7. Applied egg-rr 18.5%

      \[\leadsto d \cdot \color{blue}{{\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}} \]
   8. Step-by-step derivation

      1. unpow1/3 18.5%

         \[\leadsto d \cdot \color{blue}{\sqrt[3]{{\left(\frac{1}{h \cdot \ell}\right)}^{1.5}}} \]

      2. *-commutative 18.5%

         \[\leadsto d \cdot \sqrt[3]{{\left(\frac{1}{\color{blue}{\ell \cdot h}}\right)}^{1.5}} \]

   9. Simplified 18.5%

      \[\leadsto d \cdot \color{blue}{\sqrt[3]{{\left(\frac{1}{\ell \cdot h}\right)}^{1.5}}} \]

   if 1.4e-307 < l

   1. Initial program 56.7%

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

   3. Taylor expanded in d around inf 37.7%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative 37.7%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      2. associate-/r* 38.4%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   5. Simplified 38.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]
   6. Step-by-step derivation

      1. pow1/2 38.4%

         \[\leadsto d \cdot \color{blue}{{\left(\frac{\frac{1}{\ell}}{h}\right)}^{0.5}} \]

      2. div-inv 38.4%

         \[\leadsto d \cdot {\color{blue}{\left(\frac{1}{\ell} \cdot \frac{1}{h}\right)}}^{0.5} \]

      3. unpow-prod-down 47.7%

         \[\leadsto d \cdot \color{blue}{\left({\left(\frac{1}{\ell}\right)}^{0.5} \cdot {\left(\frac{1}{h}\right)}^{0.5}\right)} \]

      4. pow1/2 47.7%

         \[\leadsto d \cdot \left(\color{blue}{\sqrt{\frac{1}{\ell}}} \cdot {\left(\frac{1}{h}\right)}^{0.5}\right) \]

      5. inv-pow 47.7%

         \[\leadsto d \cdot \left(\sqrt{\color{blue}{{\ell}^{-1}}} \cdot {\left(\frac{1}{h}\right)}^{0.5}\right) \]

      6. sqrt-pow1 47.7%

         \[\leadsto d \cdot \left(\color{blue}{{\ell}^{\left(\frac{-1}{2}\right)}} \cdot {\left(\frac{1}{h}\right)}^{0.5}\right) \]

      7. metadata-eval 47.7%

         \[\leadsto d \cdot \left({\ell}^{\color{blue}{-0.5}} \cdot {\left(\frac{1}{h}\right)}^{0.5}\right) \]

   7. Applied egg-rr 47.7%

      \[\leadsto d \cdot \color{blue}{\left({\ell}^{-0.5} \cdot {\left(\frac{1}{h}\right)}^{0.5}\right)} \]
   8. Step-by-step derivation

      1. unpow1/2 47.7%

         \[\leadsto d \cdot \left({\ell}^{-0.5} \cdot \color{blue}{\sqrt{\frac{1}{h}}}\right) \]

   9. Simplified 47.7%

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

4. Final simplification 33.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.4 \cdot 10^{-307}:\\ \;\;\;\;d \cdot \sqrt[3]{{\left(\frac{1}{\ell \cdot h}\right)}^{1.5}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot \sqrt{\frac{1}{h}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 30.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;h \leq 4.5 \cdot 10^{-296}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{\ell}^{-0.5}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= h 4.5e-296)
   (* d (sqrt (/ 1.0 (* l h))))
   (* d (/ (pow l -0.5) (sqrt h)))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (h <= 4.5e-296) {
		tmp = d * sqrt((1.0 / (l * h)));
	} else {
		tmp = d * (pow(l, -0.5) / sqrt(h));
	}
	return tmp;
}

Fortran

M_m = abs(M)
D_m = abs(D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (h <= 4.5d-296) then
        tmp = d * sqrt((1.0d0 / (l * h)))
    else
        tmp = d * ((l ** (-0.5d0)) / sqrt(h))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (h <= 4.5e-296) {
		tmp = d * Math.sqrt((1.0 / (l * h)));
	} else {
		tmp = d * (Math.pow(l, -0.5) / Math.sqrt(h));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if h <= 4.5e-296:
		tmp = d * math.sqrt((1.0 / (l * h)))
	else:
		tmp = d * (math.pow(l, -0.5) / math.sqrt(h))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (h <= 4.5e-296)
		tmp = Float64(d * sqrt(Float64(1.0 / Float64(l * h))));
	else
		tmp = Float64(d * Float64((l ^ -0.5) / sqrt(h)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (h <= 4.5e-296)
		tmp = d * sqrt((1.0 / (l * h)));
	else
		tmp = d * ((l ^ -0.5) / sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[h, 4.5e-296], N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[l, -0.5], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < 4.5000000000000002e-296

   1. Initial program 64.0%

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

   3. Taylor expanded in d around inf 14.0%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

   if 4.5000000000000002e-296 < h

   1. Initial program 57.5%

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

   3. Taylor expanded in d around inf 36.9%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative 36.9%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      2. associate-/r* 37.6%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   5. Simplified 37.6%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]
   6. Step-by-step derivation

      1. expm1-log1p-u 36.5%

         \[\leadsto d \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\right)} \]

      2. expm1-udef 22.4%

         \[\leadsto d \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{\frac{1}{\ell}}{h}}\right)} - 1\right)} \]

      3. sqrt-div 28.8%

         \[\leadsto d \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}}\right)} - 1\right) \]

      4. inv-pow 28.8%

         \[\leadsto d \cdot \left(e^{\mathsf{log1p}\left(\frac{\sqrt{\color{blue}{{\ell}^{-1}}}}{\sqrt{h}}\right)} - 1\right) \]

      5. sqrt-pow1 28.8%

         \[\leadsto d \cdot \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{{\ell}^{\left(\frac{-1}{2}\right)}}}{\sqrt{h}}\right)} - 1\right) \]

      6. metadata-eval 28.8%

         \[\leadsto d \cdot \left(e^{\mathsf{log1p}\left(\frac{{\ell}^{\color{blue}{-0.5}}}{\sqrt{h}}\right)} - 1\right) \]

   7. Applied egg-rr 28.8%

      \[\leadsto d \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\ell}^{-0.5}}{\sqrt{h}}\right)} - 1\right)} \]
   8. Step-by-step derivation

      1. expm1-def 45.1%

         \[\leadsto d \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{-0.5}}{\sqrt{h}}\right)\right)} \]

      2. expm1-log1p 46.9%

         \[\leadsto d \cdot \color{blue}{\frac{{\ell}^{-0.5}}{\sqrt{h}}} \]

   9. Simplified 46.9%

      \[\leadsto d \cdot \color{blue}{\frac{{\ell}^{-0.5}}{\sqrt{h}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 30.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq 4.5 \cdot 10^{-296}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{\ell}^{-0.5}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 27.0% accurate, 3.1× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ d \cdot \sqrt{\frac{1}{\ell \cdot h}} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m) :precision binary64 (* d (sqrt (/ 1.0 (* l h)))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	return d * sqrt((1.0 / (l * h)));
}

Fortran

M_m = abs(M)
D_m = abs(D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    code = d * sqrt((1.0d0 / (l * h)))
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	return d * Math.sqrt((1.0 / (l * h)));
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	return d * math.sqrt((1.0 / (l * h)))

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	return Float64(d * sqrt(Float64(1.0 / Float64(l * h))))
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp = code(d, h, l, M_m, D_m)
	tmp = d * sqrt((1.0 / (l * h)));
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 60.7%

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

3. Taylor expanded in d around inf 25.3%

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

4. Final simplification 25.3%

   \[\leadsto d \cdot \sqrt{\frac{1}{\ell \cdot h}} \]
5. Add Preprocessing

Alternative 14: 27.2% accurate, 3.1× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m) :precision binary64 (* d (sqrt (/ (/ 1.0 l) h))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	return d * sqrt(((1.0 / l) / h));
}

Fortran

M_m = abs(M)
D_m = abs(D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    code = d * sqrt(((1.0d0 / l) / h))
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	return d * Math.sqrt(((1.0 / l) / h));
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	return d * math.sqrt(((1.0 / l) / h))

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	return Float64(d * sqrt(Float64(Float64(1.0 / l) / h)))
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp = code(d, h, l, M_m, D_m)
	tmp = d * sqrt(((1.0 / l) / h));
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := N[(d * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 60.7%

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

3. Taylor expanded in d around inf 25.3%

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
4. Step-by-step derivation

   1. *-commutative 25.3%

      \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   2. associate-/r* 25.7%

      \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

5. Simplified 25.7%

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]

6. Final simplification 25.7%

   \[\leadsto d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024040 
(FPCore (d h l M D)
  :name "Henrywood and Agarwal, Equation (12)"
  :precision binary64
  (* (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0))) (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))