Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.8% → 95.3%

Time: 29.3s

Alternatives: 8

Speedup: 3.8×

• 39.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (/
  2.0
  (*
   (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
   (- (+ 1.0 (pow (/ k t) 2.0)) 1.0))))

C

double code(double t, double l, double k) {
	return 2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t), 2.0)) - 1.0));
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 / (((((t ** 3.0d0) / (l * l)) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t) ** 2.0d0)) - 1.0d0))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 / ((((Math.pow(t, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * ((1.0 + Math.pow((k / t), 2.0)) - 1.0));
}

Python

def code(t, l, k):
	return 2.0 / ((((math.pow(t, 3.0) / (l * l)) * math.sin(k)) * math.tan(k)) * ((1.0 + math.pow((k / t), 2.0)) - 1.0))

Julia

function code(t, l, k)
	return Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t) ^ 2.0)) - 1.0)))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 / (((((t ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + ((k / t) ^ 2.0)) - 1.0));
end

Wolfram

code[t_, l_, k_] := N[(2.0 / N[(N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 35.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (/
  2.0
  (*
   (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
   (- (+ 1.0 (pow (/ k t) 2.0)) 1.0))))

C

double code(double t, double l, double k) {
	return 2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t), 2.0)) - 1.0));
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 / (((((t ** 3.0d0) / (l * l)) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t) ** 2.0d0)) - 1.0d0))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 / ((((Math.pow(t, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * ((1.0 + Math.pow((k / t), 2.0)) - 1.0));
}

Python

def code(t, l, k):
	return 2.0 / ((((math.pow(t, 3.0) / (l * l)) * math.sin(k)) * math.tan(k)) * ((1.0 + math.pow((k / t), 2.0)) - 1.0))

Julia

function code(t, l, k)
	return Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t) ^ 2.0)) - 1.0)))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 / (((((t ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + ((k / t) ^ 2.0)) - 1.0));
end

Wolfram

code[t_, l_, k_] := N[(2.0 / N[(N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 95.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 5 \cdot 10^{-16}:\\ \;\;\;\;\frac{2}{{\left(k\_m \cdot \left(\frac{k\_m}{\ell} \cdot \sqrt{t\_m}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{t\_m} \cdot \cos k\_m\right) \cdot {\left(\frac{k\_m \cdot \sin k\_m}{\ell}\right)}^{-2}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 5e-16)
    (/ 2.0 (pow (* k_m (* (/ k_m l) (sqrt t_m))) 2.0))
    (* (* (/ 2.0 t_m) (cos k_m)) (pow (/ (* k_m (sin k_m)) l) -2.0)))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 5e-16) {
		tmp = 2.0 / pow((k_m * ((k_m / l) * sqrt(t_m))), 2.0);
	} else {
		tmp = ((2.0 / t_m) * cos(k_m)) * pow(((k_m * sin(k_m)) / l), -2.0);
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 5d-16) then
        tmp = 2.0d0 / ((k_m * ((k_m / l) * sqrt(t_m))) ** 2.0d0)
    else
        tmp = ((2.0d0 / t_m) * cos(k_m)) * (((k_m * sin(k_m)) / l) ** (-2.0d0))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 5e-16) {
		tmp = 2.0 / Math.pow((k_m * ((k_m / l) * Math.sqrt(t_m))), 2.0);
	} else {
		tmp = ((2.0 / t_m) * Math.cos(k_m)) * Math.pow(((k_m * Math.sin(k_m)) / l), -2.0);
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 5e-16:
		tmp = 2.0 / math.pow((k_m * ((k_m / l) * math.sqrt(t_m))), 2.0)
	else:
		tmp = ((2.0 / t_m) * math.cos(k_m)) * math.pow(((k_m * math.sin(k_m)) / l), -2.0)
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 5e-16)
		tmp = Float64(2.0 / (Float64(k_m * Float64(Float64(k_m / l) * sqrt(t_m))) ^ 2.0));
	else
		tmp = Float64(Float64(Float64(2.0 / t_m) * cos(k_m)) * (Float64(Float64(k_m * sin(k_m)) / l) ^ -2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 5e-16)
		tmp = 2.0 / ((k_m * ((k_m / l) * sqrt(t_m))) ^ 2.0);
	else
		tmp = ((2.0 / t_m) * cos(k_m)) * (((k_m * sin(k_m)) / l) ^ -2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 5e-16], N[(2.0 / N[Power[N[(k$95$m * N[(N[(k$95$m / l), $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / t$95$m), $MachinePrecision] * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(k$95$m * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 5.0000000000000004e-16

   1. Initial program 33.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 20.8%

         \[\leadsto \frac{2}{\color{blue}{\sqrt{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \cdot \sqrt{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}}} \]

      2. pow2 20.8%

         \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\right)}^{2}}} \]

   4. Applied egg-rr 34.0%

      \[\leadsto \frac{2}{\color{blue}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2}}} \]

   5. Taylor expanded in k around inf 52.6%

      \[\leadsto \frac{2}{{\color{blue}{\left(\frac{k \cdot \sin k}{\ell} \cdot \sqrt{\frac{t}{\cos k}}\right)}}^{2}} \]
   6. Step-by-step derivation

      1. associate-/l* 54.8%

         \[\leadsto \frac{2}{{\left(\color{blue}{\left(k \cdot \frac{\sin k}{\ell}\right)} \cdot \sqrt{\frac{t}{\cos k}}\right)}^{2}} \]

      2. associate-*l* 54.8%

         \[\leadsto \frac{2}{{\color{blue}{\left(k \cdot \left(\frac{\sin k}{\ell} \cdot \sqrt{\frac{t}{\cos k}}\right)\right)}}^{2}} \]

   7. Simplified 54.8%

      \[\leadsto \frac{2}{{\color{blue}{\left(k \cdot \left(\frac{\sin k}{\ell} \cdot \sqrt{\frac{t}{\cos k}}\right)\right)}}^{2}} \]

   8. Taylor expanded in k around 0 45.6%

      \[\leadsto \frac{2}{{\left(k \cdot \color{blue}{\left(\frac{k}{\ell} \cdot \sqrt{t}\right)}\right)}^{2}} \]

   if 5.0000000000000004e-16 < k

   1. Initial program 34.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 17.7%

         \[\leadsto \frac{2}{\color{blue}{\sqrt{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \cdot \sqrt{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}}} \]

      2. pow2 17.7%

         \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\right)}^{2}}} \]

   4. Applied egg-rr 17.8%

      \[\leadsto \frac{2}{\color{blue}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2}}} \]

   5. Taylor expanded in k around inf 42.4%

      \[\leadsto \frac{2}{{\color{blue}{\left(\frac{k \cdot \sin k}{\ell} \cdot \sqrt{\frac{t}{\cos k}}\right)}}^{2}} \]
   6. Step-by-step derivation

      1. associate-/l* 42.4%

         \[\leadsto \frac{2}{{\left(\color{blue}{\left(k \cdot \frac{\sin k}{\ell}\right)} \cdot \sqrt{\frac{t}{\cos k}}\right)}^{2}} \]

      2. associate-*l* 42.4%

         \[\leadsto \frac{2}{{\color{blue}{\left(k \cdot \left(\frac{\sin k}{\ell} \cdot \sqrt{\frac{t}{\cos k}}\right)\right)}}^{2}} \]

   7. Simplified 42.4%

      \[\leadsto \frac{2}{{\color{blue}{\left(k \cdot \left(\frac{\sin k}{\ell} \cdot \sqrt{\frac{t}{\cos k}}\right)\right)}}^{2}} \]
   8. Step-by-step derivation

      1. *-un-lft-identity 42.4%

         \[\leadsto \color{blue}{1 \cdot \frac{2}{{\left(k \cdot \left(\frac{\sin k}{\ell} \cdot \sqrt{\frac{t}{\cos k}}\right)\right)}^{2}}} \]

      2. associate-*r* 42.4%

         \[\leadsto 1 \cdot \frac{2}{{\color{blue}{\left(\left(k \cdot \frac{\sin k}{\ell}\right) \cdot \sqrt{\frac{t}{\cos k}}\right)}}^{2}} \]

      3. unpow-prod-down 39.7%

         \[\leadsto 1 \cdot \frac{2}{\color{blue}{{\left(k \cdot \frac{\sin k}{\ell}\right)}^{2} \cdot {\left(\sqrt{\frac{t}{\cos k}}\right)}^{2}}} \]

      4. pow2 39.7%

         \[\leadsto 1 \cdot \frac{2}{{\left(k \cdot \frac{\sin k}{\ell}\right)}^{2} \cdot \color{blue}{\left(\sqrt{\frac{t}{\cos k}} \cdot \sqrt{\frac{t}{\cos k}}\right)}} \]

      5. add-sqr-sqrt 95.3%

         \[\leadsto 1 \cdot \frac{2}{{\left(k \cdot \frac{\sin k}{\ell}\right)}^{2} \cdot \color{blue}{\frac{t}{\cos k}}} \]

   9. Applied egg-rr 95.3%

      \[\leadsto \color{blue}{1 \cdot \frac{2}{{\left(k \cdot \frac{\sin k}{\ell}\right)}^{2} \cdot \frac{t}{\cos k}}} \]
   10. Step-by-step derivation

       1. *-lft-identity 95.3%

          \[\leadsto \color{blue}{\frac{2}{{\left(k \cdot \frac{\sin k}{\ell}\right)}^{2} \cdot \frac{t}{\cos k}}} \]

       2. *-commutative 95.3%

          \[\leadsto \frac{2}{\color{blue}{\frac{t}{\cos k} \cdot {\left(k \cdot \frac{\sin k}{\ell}\right)}^{2}}} \]

       3. associate-/r* 95.4%

          \[\leadsto \color{blue}{\frac{\frac{2}{\frac{t}{\cos k}}}{{\left(k \cdot \frac{\sin k}{\ell}\right)}^{2}}} \]

   11. Simplified 95.4%

       \[\leadsto \color{blue}{\frac{\frac{2}{\frac{t}{\cos k}}}{{\left(k \cdot \frac{\sin k}{\ell}\right)}^{2}}} \]
   12. Step-by-step derivation

       1. div-inv 95.2%

          \[\leadsto \color{blue}{\frac{2}{\frac{t}{\cos k}} \cdot \frac{1}{{\left(k \cdot \frac{\sin k}{\ell}\right)}^{2}}} \]

       2. associate-/r/ 95.2%

          \[\leadsto \color{blue}{\left(\frac{2}{t} \cdot \cos k\right)} \cdot \frac{1}{{\left(k \cdot \frac{\sin k}{\ell}\right)}^{2}} \]

       3. pow-flip 96.1%

          \[\leadsto \left(\frac{2}{t} \cdot \cos k\right) \cdot \color{blue}{{\left(k \cdot \frac{\sin k}{\ell}\right)}^{\left(-2\right)}} \]

       4. associate-*r/ 96.1%

          \[\leadsto \left(\frac{2}{t} \cdot \cos k\right) \cdot {\color{blue}{\left(\frac{k \cdot \sin k}{\ell}\right)}}^{\left(-2\right)} \]

       5. metadata-eval 96.1%

          \[\leadsto \left(\frac{2}{t} \cdot \cos k\right) \cdot {\left(\frac{k \cdot \sin k}{\ell}\right)}^{\color{blue}{-2}} \]

   13. Applied egg-rr 96.1%

       \[\leadsto \color{blue}{\left(\frac{2}{t} \cdot \cos k\right) \cdot {\left(\frac{k \cdot \sin k}{\ell}\right)}^{-2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{-16}:\\ \;\;\;\;\frac{2}{{\left(k \cdot \left(\frac{k}{\ell} \cdot \sqrt{t}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{t} \cdot \cos k\right) \cdot {\left(\frac{k \cdot \sin k}{\ell}\right)}^{-2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 1.15 \cdot 10^{-15}:\\ \;\;\;\;\frac{2}{{\left(k\_m \cdot \left(\frac{k\_m}{\ell} \cdot \sqrt{t\_m}\right)\right)}^{2}}\\ \mathbf{elif}\;k\_m \leq 8.8 \cdot 10^{+74}:\\ \;\;\;\;\frac{\frac{2}{\frac{t\_m}{\cos k\_m}}}{\frac{{k\_m}^{4}}{{\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{t\_m}}{{\left(k\_m \cdot \frac{\sin k\_m}{\ell}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 1.15e-15)
    (/ 2.0 (pow (* k_m (* (/ k_m l) (sqrt t_m))) 2.0))
    (if (<= k_m 8.8e+74)
      (/ (/ 2.0 (/ t_m (cos k_m))) (/ (pow k_m 4.0) (pow l 2.0)))
      (/ (/ 2.0 t_m) (pow (* k_m (/ (sin k_m) l)) 2.0))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 1.15e-15) {
		tmp = 2.0 / pow((k_m * ((k_m / l) * sqrt(t_m))), 2.0);
	} else if (k_m <= 8.8e+74) {
		tmp = (2.0 / (t_m / cos(k_m))) / (pow(k_m, 4.0) / pow(l, 2.0));
	} else {
		tmp = (2.0 / t_m) / pow((k_m * (sin(k_m) / l)), 2.0);
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 1.15d-15) then
        tmp = 2.0d0 / ((k_m * ((k_m / l) * sqrt(t_m))) ** 2.0d0)
    else if (k_m <= 8.8d+74) then
        tmp = (2.0d0 / (t_m / cos(k_m))) / ((k_m ** 4.0d0) / (l ** 2.0d0))
    else
        tmp = (2.0d0 / t_m) / ((k_m * (sin(k_m) / l)) ** 2.0d0)
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 1.15e-15) {
		tmp = 2.0 / Math.pow((k_m * ((k_m / l) * Math.sqrt(t_m))), 2.0);
	} else if (k_m <= 8.8e+74) {
		tmp = (2.0 / (t_m / Math.cos(k_m))) / (Math.pow(k_m, 4.0) / Math.pow(l, 2.0));
	} else {
		tmp = (2.0 / t_m) / Math.pow((k_m * (Math.sin(k_m) / l)), 2.0);
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 1.15e-15:
		tmp = 2.0 / math.pow((k_m * ((k_m / l) * math.sqrt(t_m))), 2.0)
	elif k_m <= 8.8e+74:
		tmp = (2.0 / (t_m / math.cos(k_m))) / (math.pow(k_m, 4.0) / math.pow(l, 2.0))
	else:
		tmp = (2.0 / t_m) / math.pow((k_m * (math.sin(k_m) / l)), 2.0)
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 1.15e-15)
		tmp = Float64(2.0 / (Float64(k_m * Float64(Float64(k_m / l) * sqrt(t_m))) ^ 2.0));
	elseif (k_m <= 8.8e+74)
		tmp = Float64(Float64(2.0 / Float64(t_m / cos(k_m))) / Float64((k_m ^ 4.0) / (l ^ 2.0)));
	else
		tmp = Float64(Float64(2.0 / t_m) / (Float64(k_m * Float64(sin(k_m) / l)) ^ 2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 1.15e-15)
		tmp = 2.0 / ((k_m * ((k_m / l) * sqrt(t_m))) ^ 2.0);
	elseif (k_m <= 8.8e+74)
		tmp = (2.0 / (t_m / cos(k_m))) / ((k_m ^ 4.0) / (l ^ 2.0));
	else
		tmp = (2.0 / t_m) / ((k_m * (sin(k_m) / l)) ^ 2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 1.15e-15], N[(2.0 / N[Power[N[(k$95$m * N[(N[(k$95$m / l), $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 8.8e+74], N[(N[(2.0 / N[(t$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[k$95$m, 4.0], $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / t$95$m), $MachinePrecision] / N[Power[N[(k$95$m * N[(N[Sin[k$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 1.14999999999999995e-15

   1. Initial program 33.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 20.8%

         \[\leadsto \frac{2}{\color{blue}{\sqrt{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \cdot \sqrt{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}}} \]

      2. pow2 20.8%

         \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\right)}^{2}}} \]

   4. Applied egg-rr 34.0%

      \[\leadsto \frac{2}{\color{blue}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2}}} \]

   5. Taylor expanded in k around inf 52.6%

      \[\leadsto \frac{2}{{\color{blue}{\left(\frac{k \cdot \sin k}{\ell} \cdot \sqrt{\frac{t}{\cos k}}\right)}}^{2}} \]
   6. Step-by-step derivation

      1. associate-/l* 54.8%

         \[\leadsto \frac{2}{{\left(\color{blue}{\left(k \cdot \frac{\sin k}{\ell}\right)} \cdot \sqrt{\frac{t}{\cos k}}\right)}^{2}} \]

      2. associate-*l* 54.8%

         \[\leadsto \frac{2}{{\color{blue}{\left(k \cdot \left(\frac{\sin k}{\ell} \cdot \sqrt{\frac{t}{\cos k}}\right)\right)}}^{2}} \]

   7. Simplified 54.8%

      \[\leadsto \frac{2}{{\color{blue}{\left(k \cdot \left(\frac{\sin k}{\ell} \cdot \sqrt{\frac{t}{\cos k}}\right)\right)}}^{2}} \]

   8. Taylor expanded in k around 0 45.6%

      \[\leadsto \frac{2}{{\left(k \cdot \color{blue}{\left(\frac{k}{\ell} \cdot \sqrt{t}\right)}\right)}^{2}} \]

   if 1.14999999999999995e-15 < k < 8.8000000000000005e74

   1. Initial program 21.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 14.5%

         \[\leadsto \frac{2}{\color{blue}{\sqrt{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \cdot \sqrt{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}}} \]

      2. pow2 14.5%

         \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\right)}^{2}}} \]

   4. Applied egg-rr 21.5%

      \[\leadsto \frac{2}{\color{blue}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2}}} \]

   5. Taylor expanded in k around inf 42.6%

      \[\leadsto \frac{2}{{\color{blue}{\left(\frac{k \cdot \sin k}{\ell} \cdot \sqrt{\frac{t}{\cos k}}\right)}}^{2}} \]
   6. Step-by-step derivation

      1. associate-/l* 42.5%

         \[\leadsto \frac{2}{{\left(\color{blue}{\left(k \cdot \frac{\sin k}{\ell}\right)} \cdot \sqrt{\frac{t}{\cos k}}\right)}^{2}} \]

      2. associate-*l* 42.5%

         \[\leadsto \frac{2}{{\color{blue}{\left(k \cdot \left(\frac{\sin k}{\ell} \cdot \sqrt{\frac{t}{\cos k}}\right)\right)}}^{2}} \]

   7. Simplified 42.5%

      \[\leadsto \frac{2}{{\color{blue}{\left(k \cdot \left(\frac{\sin k}{\ell} \cdot \sqrt{\frac{t}{\cos k}}\right)\right)}}^{2}} \]
   8. Step-by-step derivation

      1. *-un-lft-identity 42.5%

         \[\leadsto \color{blue}{1 \cdot \frac{2}{{\left(k \cdot \left(\frac{\sin k}{\ell} \cdot \sqrt{\frac{t}{\cos k}}\right)\right)}^{2}}} \]

      2. associate-*r* 42.5%

         \[\leadsto 1 \cdot \frac{2}{{\color{blue}{\left(\left(k \cdot \frac{\sin k}{\ell}\right) \cdot \sqrt{\frac{t}{\cos k}}\right)}}^{2}} \]

      3. unpow-prod-down 35.7%

         \[\leadsto 1 \cdot \frac{2}{\color{blue}{{\left(k \cdot \frac{\sin k}{\ell}\right)}^{2} \cdot {\left(\sqrt{\frac{t}{\cos k}}\right)}^{2}}} \]

      4. pow2 35.7%

         \[\leadsto 1 \cdot \frac{2}{{\left(k \cdot \frac{\sin k}{\ell}\right)}^{2} \cdot \color{blue}{\left(\sqrt{\frac{t}{\cos k}} \cdot \sqrt{\frac{t}{\cos k}}\right)}} \]

      5. add-sqr-sqrt 92.4%

         \[\leadsto 1 \cdot \frac{2}{{\left(k \cdot \frac{\sin k}{\ell}\right)}^{2} \cdot \color{blue}{\frac{t}{\cos k}}} \]

   9. Applied egg-rr 92.4%

      \[\leadsto \color{blue}{1 \cdot \frac{2}{{\left(k \cdot \frac{\sin k}{\ell}\right)}^{2} \cdot \frac{t}{\cos k}}} \]
   10. Step-by-step derivation

       1. *-lft-identity 92.4%

          \[\leadsto \color{blue}{\frac{2}{{\left(k \cdot \frac{\sin k}{\ell}\right)}^{2} \cdot \frac{t}{\cos k}}} \]

       2. *-commutative 92.4%

          \[\leadsto \frac{2}{\color{blue}{\frac{t}{\cos k} \cdot {\left(k \cdot \frac{\sin k}{\ell}\right)}^{2}}} \]

       3. associate-/r* 92.5%

          \[\leadsto \color{blue}{\frac{\frac{2}{\frac{t}{\cos k}}}{{\left(k \cdot \frac{\sin k}{\ell}\right)}^{2}}} \]

   11. Simplified 92.5%

       \[\leadsto \color{blue}{\frac{\frac{2}{\frac{t}{\cos k}}}{{\left(k \cdot \frac{\sin k}{\ell}\right)}^{2}}} \]

   12. Taylor expanded in k around 0 57.2%

       \[\leadsto \frac{\frac{2}{\frac{t}{\cos k}}}{\color{blue}{\frac{{k}^{4}}{{\ell}^{2}}}} \]

   if 8.8000000000000005e74 < k

   1. Initial program 37.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 18.6%

         \[\leadsto \frac{2}{\color{blue}{\sqrt{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \cdot \sqrt{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}}} \]

      2. pow2 18.6%

         \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\right)}^{2}}} \]

   4. Applied egg-rr 16.8%

      \[\leadsto \frac{2}{\color{blue}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2}}} \]

   5. Taylor expanded in k around inf 42.4%

      \[\leadsto \frac{2}{{\color{blue}{\left(\frac{k \cdot \sin k}{\ell} \cdot \sqrt{\frac{t}{\cos k}}\right)}}^{2}} \]
   6. Step-by-step derivation

      1. associate-/l* 42.4%

         \[\leadsto \frac{2}{{\left(\color{blue}{\left(k \cdot \frac{\sin k}{\ell}\right)} \cdot \sqrt{\frac{t}{\cos k}}\right)}^{2}} \]

      2. associate-*l* 42.4%

         \[\leadsto \frac{2}{{\color{blue}{\left(k \cdot \left(\frac{\sin k}{\ell} \cdot \sqrt{\frac{t}{\cos k}}\right)\right)}}^{2}} \]

   7. Simplified 42.4%

      \[\leadsto \frac{2}{{\color{blue}{\left(k \cdot \left(\frac{\sin k}{\ell} \cdot \sqrt{\frac{t}{\cos k}}\right)\right)}}^{2}} \]
   8. Step-by-step derivation

      1. *-un-lft-identity 42.4%

         \[\leadsto \color{blue}{1 \cdot \frac{2}{{\left(k \cdot \left(\frac{\sin k}{\ell} \cdot \sqrt{\frac{t}{\cos k}}\right)\right)}^{2}}} \]

      2. associate-*r* 42.4%

         \[\leadsto 1 \cdot \frac{2}{{\color{blue}{\left(\left(k \cdot \frac{\sin k}{\ell}\right) \cdot \sqrt{\frac{t}{\cos k}}\right)}}^{2}} \]

      3. unpow-prod-down 40.7%

         \[\leadsto 1 \cdot \frac{2}{\color{blue}{{\left(k \cdot \frac{\sin k}{\ell}\right)}^{2} \cdot {\left(\sqrt{\frac{t}{\cos k}}\right)}^{2}}} \]

      4. pow2 40.7%

         \[\leadsto 1 \cdot \frac{2}{{\left(k \cdot \frac{\sin k}{\ell}\right)}^{2} \cdot \color{blue}{\left(\sqrt{\frac{t}{\cos k}} \cdot \sqrt{\frac{t}{\cos k}}\right)}} \]

      5. add-sqr-sqrt 96.1%

         \[\leadsto 1 \cdot \frac{2}{{\left(k \cdot \frac{\sin k}{\ell}\right)}^{2} \cdot \color{blue}{\frac{t}{\cos k}}} \]

   9. Applied egg-rr 96.1%

      \[\leadsto \color{blue}{1 \cdot \frac{2}{{\left(k \cdot \frac{\sin k}{\ell}\right)}^{2} \cdot \frac{t}{\cos k}}} \]
   10. Step-by-step derivation

       1. *-lft-identity 96.1%

          \[\leadsto \color{blue}{\frac{2}{{\left(k \cdot \frac{\sin k}{\ell}\right)}^{2} \cdot \frac{t}{\cos k}}} \]

       2. *-commutative 96.1%

          \[\leadsto \frac{2}{\color{blue}{\frac{t}{\cos k} \cdot {\left(k \cdot \frac{\sin k}{\ell}\right)}^{2}}} \]

       3. associate-/r* 96.2%

          \[\leadsto \color{blue}{\frac{\frac{2}{\frac{t}{\cos k}}}{{\left(k \cdot \frac{\sin k}{\ell}\right)}^{2}}} \]

   11. Simplified 96.2%

       \[\leadsto \color{blue}{\frac{\frac{2}{\frac{t}{\cos k}}}{{\left(k \cdot \frac{\sin k}{\ell}\right)}^{2}}} \]

   12. Taylor expanded in k around 0 62.4%

       \[\leadsto \frac{\frac{2}{\color{blue}{t}}}{{\left(k \cdot \frac{\sin k}{\ell}\right)}^{2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 49.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.15 \cdot 10^{-15}:\\ \;\;\;\;\frac{2}{{\left(k \cdot \left(\frac{k}{\ell} \cdot \sqrt{t}\right)\right)}^{2}}\\ \mathbf{elif}\;k \leq 8.8 \cdot 10^{+74}:\\ \;\;\;\;\frac{\frac{2}{\frac{t}{\cos k}}}{\frac{{k}^{4}}{{\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{t}}{{\left(k \cdot \frac{\sin k}{\ell}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 1.3 \cdot 10^{+37}:\\ \;\;\;\;\frac{2}{{\left(k\_m \cdot \left(\frac{k\_m}{\ell} \cdot \sqrt{t\_m}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{t\_m}}{{\left(k\_m \cdot \frac{\sin k\_m}{\ell}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 1.3e+37)
    (/ 2.0 (pow (* k_m (* (/ k_m l) (sqrt t_m))) 2.0))
    (/ (/ 2.0 t_m) (pow (* k_m (/ (sin k_m) l)) 2.0)))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 1.3e+37) {
		tmp = 2.0 / pow((k_m * ((k_m / l) * sqrt(t_m))), 2.0);
	} else {
		tmp = (2.0 / t_m) / pow((k_m * (sin(k_m) / l)), 2.0);
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 1.3d+37) then
        tmp = 2.0d0 / ((k_m * ((k_m / l) * sqrt(t_m))) ** 2.0d0)
    else
        tmp = (2.0d0 / t_m) / ((k_m * (sin(k_m) / l)) ** 2.0d0)
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 1.3e+37) {
		tmp = 2.0 / Math.pow((k_m * ((k_m / l) * Math.sqrt(t_m))), 2.0);
	} else {
		tmp = (2.0 / t_m) / Math.pow((k_m * (Math.sin(k_m) / l)), 2.0);
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 1.3e+37:
		tmp = 2.0 / math.pow((k_m * ((k_m / l) * math.sqrt(t_m))), 2.0)
	else:
		tmp = (2.0 / t_m) / math.pow((k_m * (math.sin(k_m) / l)), 2.0)
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 1.3e+37)
		tmp = Float64(2.0 / (Float64(k_m * Float64(Float64(k_m / l) * sqrt(t_m))) ^ 2.0));
	else
		tmp = Float64(Float64(2.0 / t_m) / (Float64(k_m * Float64(sin(k_m) / l)) ^ 2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 1.3e+37)
		tmp = 2.0 / ((k_m * ((k_m / l) * sqrt(t_m))) ^ 2.0);
	else
		tmp = (2.0 / t_m) / ((k_m * (sin(k_m) / l)) ^ 2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 1.3e+37], N[(2.0 / N[Power[N[(k$95$m * N[(N[(k$95$m / l), $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / t$95$m), $MachinePrecision] / N[Power[N[(k$95$m * N[(N[Sin[k$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.3e37

   1. Initial program 32.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 19.9%

         \[\leadsto \frac{2}{\color{blue}{\sqrt{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \cdot \sqrt{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}}} \]

      2. pow2 19.9%

         \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\right)}^{2}}} \]

   4. Applied egg-rr 33.0%

      \[\leadsto \frac{2}{\color{blue}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2}}} \]

   5. Taylor expanded in k around inf 51.7%

      \[\leadsto \frac{2}{{\color{blue}{\left(\frac{k \cdot \sin k}{\ell} \cdot \sqrt{\frac{t}{\cos k}}\right)}}^{2}} \]
   6. Step-by-step derivation

      1. associate-/l* 53.8%

         \[\leadsto \frac{2}{{\left(\color{blue}{\left(k \cdot \frac{\sin k}{\ell}\right)} \cdot \sqrt{\frac{t}{\cos k}}\right)}^{2}} \]

      2. associate-*l* 53.8%

         \[\leadsto \frac{2}{{\color{blue}{\left(k \cdot \left(\frac{\sin k}{\ell} \cdot \sqrt{\frac{t}{\cos k}}\right)\right)}}^{2}} \]

   7. Simplified 53.8%

      \[\leadsto \frac{2}{{\color{blue}{\left(k \cdot \left(\frac{\sin k}{\ell} \cdot \sqrt{\frac{t}{\cos k}}\right)\right)}}^{2}} \]

   8. Taylor expanded in k around 0 44.4%

      \[\leadsto \frac{2}{{\left(k \cdot \color{blue}{\left(\frac{k}{\ell} \cdot \sqrt{t}\right)}\right)}^{2}} \]

   if 1.3e37 < k

   1. Initial program 37.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 20.4%

         \[\leadsto \frac{2}{\color{blue}{\sqrt{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \cdot \sqrt{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}}} \]

      2. pow2 20.4%

         \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\right)}^{2}}} \]

   4. Applied egg-rr 18.8%

      \[\leadsto \frac{2}{\color{blue}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2}}} \]

   5. Taylor expanded in k around inf 43.9%

      \[\leadsto \frac{2}{{\color{blue}{\left(\frac{k \cdot \sin k}{\ell} \cdot \sqrt{\frac{t}{\cos k}}\right)}}^{2}} \]
   6. Step-by-step derivation

      1. associate-/l* 43.9%

         \[\leadsto \frac{2}{{\left(\color{blue}{\left(k \cdot \frac{\sin k}{\ell}\right)} \cdot \sqrt{\frac{t}{\cos k}}\right)}^{2}} \]

      2. associate-*l* 43.9%

         \[\leadsto \frac{2}{{\color{blue}{\left(k \cdot \left(\frac{\sin k}{\ell} \cdot \sqrt{\frac{t}{\cos k}}\right)\right)}}^{2}} \]

   7. Simplified 43.9%

      \[\leadsto \frac{2}{{\color{blue}{\left(k \cdot \left(\frac{\sin k}{\ell} \cdot \sqrt{\frac{t}{\cos k}}\right)\right)}}^{2}} \]
   8. Step-by-step derivation

      1. *-un-lft-identity 43.9%

         \[\leadsto \color{blue}{1 \cdot \frac{2}{{\left(k \cdot \left(\frac{\sin k}{\ell} \cdot \sqrt{\frac{t}{\cos k}}\right)\right)}^{2}}} \]

      2. associate-*r* 43.9%

         \[\leadsto 1 \cdot \frac{2}{{\color{blue}{\left(\left(k \cdot \frac{\sin k}{\ell}\right) \cdot \sqrt{\frac{t}{\cos k}}\right)}}^{2}} \]

      3. unpow-prod-down 42.4%

         \[\leadsto 1 \cdot \frac{2}{\color{blue}{{\left(k \cdot \frac{\sin k}{\ell}\right)}^{2} \cdot {\left(\sqrt{\frac{t}{\cos k}}\right)}^{2}}} \]

      4. pow2 42.4%

         \[\leadsto 1 \cdot \frac{2}{{\left(k \cdot \frac{\sin k}{\ell}\right)}^{2} \cdot \color{blue}{\left(\sqrt{\frac{t}{\cos k}} \cdot \sqrt{\frac{t}{\cos k}}\right)}} \]

      5. add-sqr-sqrt 96.4%

         \[\leadsto 1 \cdot \frac{2}{{\left(k \cdot \frac{\sin k}{\ell}\right)}^{2} \cdot \color{blue}{\frac{t}{\cos k}}} \]

   9. Applied egg-rr 96.4%

      \[\leadsto \color{blue}{1 \cdot \frac{2}{{\left(k \cdot \frac{\sin k}{\ell}\right)}^{2} \cdot \frac{t}{\cos k}}} \]
   10. Step-by-step derivation

       1. *-lft-identity 96.4%

          \[\leadsto \color{blue}{\frac{2}{{\left(k \cdot \frac{\sin k}{\ell}\right)}^{2} \cdot \frac{t}{\cos k}}} \]

       2. *-commutative 96.4%

          \[\leadsto \frac{2}{\color{blue}{\frac{t}{\cos k} \cdot {\left(k \cdot \frac{\sin k}{\ell}\right)}^{2}}} \]

       3. associate-/r* 96.5%

          \[\leadsto \color{blue}{\frac{\frac{2}{\frac{t}{\cos k}}}{{\left(k \cdot \frac{\sin k}{\ell}\right)}^{2}}} \]

   11. Simplified 96.5%

       \[\leadsto \color{blue}{\frac{\frac{2}{\frac{t}{\cos k}}}{{\left(k \cdot \frac{\sin k}{\ell}\right)}^{2}}} \]

   12. Taylor expanded in k around 0 60.8%

       \[\leadsto \frac{\frac{2}{\color{blue}{t}}}{{\left(k \cdot \frac{\sin k}{\ell}\right)}^{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.3 \cdot 10^{+37}:\\ \;\;\;\;\frac{2}{{\left(k \cdot \left(\frac{k}{\ell} \cdot \sqrt{t}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{t}}{{\left(k \cdot \frac{\sin k}{\ell}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{2}{{\left(k\_m \cdot \left(k\_m \cdot \frac{\sqrt{t\_m}}{\ell}\right)\right)}^{2}} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (/ 2.0 (pow (* k_m (* k_m (/ (sqrt t_m) l))) 2.0))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	return t_s * (2.0 / pow((k_m * (k_m * (sqrt(t_m) / l))), 2.0));
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = t_s * (2.0d0 / ((k_m * (k_m * (sqrt(t_m) / l))) ** 2.0d0))
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	return t_s * (2.0 / Math.pow((k_m * (k_m * (Math.sqrt(t_m) / l))), 2.0));
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	return t_s * (2.0 / math.pow((k_m * (k_m * (math.sqrt(t_m) / l))), 2.0))

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	return Float64(t_s * Float64(2.0 / (Float64(k_m * Float64(k_m * Float64(sqrt(t_m) / l))) ^ 2.0)))
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k_m)
	tmp = t_s * (2.0 / ((k_m * (k_m * (sqrt(t_m) / l))) ^ 2.0));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * N[(2.0 / N[Power[N[(k$95$m * N[(k$95$m * N[(N[Sqrt[t$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.0%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. add-sqr-sqrt 20.0%

      \[\leadsto \frac{2}{\color{blue}{\sqrt{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \cdot \sqrt{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}}} \]

   2. pow2 20.0%

      \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\right)}^{2}}} \]

4. Applied egg-rr 29.7%

   \[\leadsto \frac{2}{\color{blue}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2}}} \]

5. Taylor expanded in k around inf 49.9%

   \[\leadsto \frac{2}{{\color{blue}{\left(\frac{k \cdot \sin k}{\ell} \cdot \sqrt{\frac{t}{\cos k}}\right)}}^{2}} \]
6. Step-by-step derivation

   1. associate-/l* 51.5%

      \[\leadsto \frac{2}{{\left(\color{blue}{\left(k \cdot \frac{\sin k}{\ell}\right)} \cdot \sqrt{\frac{t}{\cos k}}\right)}^{2}} \]

   2. associate-*l* 51.5%

      \[\leadsto \frac{2}{{\color{blue}{\left(k \cdot \left(\frac{\sin k}{\ell} \cdot \sqrt{\frac{t}{\cos k}}\right)\right)}}^{2}} \]

7. Simplified 51.5%

   \[\leadsto \frac{2}{{\color{blue}{\left(k \cdot \left(\frac{\sin k}{\ell} \cdot \sqrt{\frac{t}{\cos k}}\right)\right)}}^{2}} \]

8. Taylor expanded in k around 0 41.1%

   \[\leadsto \frac{2}{{\left(k \cdot \color{blue}{\left(\frac{k}{\ell} \cdot \sqrt{t}\right)}\right)}^{2}} \]
9. Step-by-step derivation

   1. associate-*l/ 41.1%

      \[\leadsto \frac{2}{{\left(k \cdot \color{blue}{\frac{k \cdot \sqrt{t}}{\ell}}\right)}^{2}} \]

   2. associate-*r/ 40.7%

      \[\leadsto \frac{2}{{\left(k \cdot \color{blue}{\left(k \cdot \frac{\sqrt{t}}{\ell}\right)}\right)}^{2}} \]

10. Simplified 40.7%

    \[\leadsto \frac{2}{{\left(k \cdot \color{blue}{\left(k \cdot \frac{\sqrt{t}}{\ell}\right)}\right)}^{2}} \]

11. Final simplification 40.7%

    \[\leadsto \frac{2}{{\left(k \cdot \left(k \cdot \frac{\sqrt{t}}{\ell}\right)\right)}^{2}} \]
12. Add Preprocessing

Alternative 5: 76.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{2}{{\left(k\_m \cdot \left(\frac{k\_m}{\ell} \cdot \sqrt{t\_m}\right)\right)}^{2}} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (/ 2.0 (pow (* k_m (* (/ k_m l) (sqrt t_m))) 2.0))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	return t_s * (2.0 / pow((k_m * ((k_m / l) * sqrt(t_m))), 2.0));
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = t_s * (2.0d0 / ((k_m * ((k_m / l) * sqrt(t_m))) ** 2.0d0))
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	return t_s * (2.0 / Math.pow((k_m * ((k_m / l) * Math.sqrt(t_m))), 2.0));
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	return t_s * (2.0 / math.pow((k_m * ((k_m / l) * math.sqrt(t_m))), 2.0))

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	return Float64(t_s * Float64(2.0 / (Float64(k_m * Float64(Float64(k_m / l) * sqrt(t_m))) ^ 2.0)))
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k_m)
	tmp = t_s * (2.0 / ((k_m * ((k_m / l) * sqrt(t_m))) ^ 2.0));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * N[(2.0 / N[Power[N[(k$95$m * N[(N[(k$95$m / l), $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.0%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. add-sqr-sqrt 20.0%

      \[\leadsto \frac{2}{\color{blue}{\sqrt{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \cdot \sqrt{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}}} \]

   2. pow2 20.0%

      \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\right)}^{2}}} \]

4. Applied egg-rr 29.7%

   \[\leadsto \frac{2}{\color{blue}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2}}} \]

5. Taylor expanded in k around inf 49.9%

   \[\leadsto \frac{2}{{\color{blue}{\left(\frac{k \cdot \sin k}{\ell} \cdot \sqrt{\frac{t}{\cos k}}\right)}}^{2}} \]
6. Step-by-step derivation

   1. associate-/l* 51.5%

      \[\leadsto \frac{2}{{\left(\color{blue}{\left(k \cdot \frac{\sin k}{\ell}\right)} \cdot \sqrt{\frac{t}{\cos k}}\right)}^{2}} \]

   2. associate-*l* 51.5%

      \[\leadsto \frac{2}{{\color{blue}{\left(k \cdot \left(\frac{\sin k}{\ell} \cdot \sqrt{\frac{t}{\cos k}}\right)\right)}}^{2}} \]

7. Simplified 51.5%

   \[\leadsto \frac{2}{{\color{blue}{\left(k \cdot \left(\frac{\sin k}{\ell} \cdot \sqrt{\frac{t}{\cos k}}\right)\right)}}^{2}} \]

8. Taylor expanded in k around 0 41.1%

   \[\leadsto \frac{2}{{\left(k \cdot \color{blue}{\left(\frac{k}{\ell} \cdot \sqrt{t}\right)}\right)}^{2}} \]

9. Final simplification 41.1%

   \[\leadsto \frac{2}{{\left(k \cdot \left(\frac{k}{\ell} \cdot \sqrt{t}\right)\right)}^{2}} \]
10. Add Preprocessing

Alternative 6: 63.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{2}{\frac{t\_m \cdot {k\_m}^{4}}{{\ell}^{2}}} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (/ 2.0 (/ (* t_m (pow k_m 4.0)) (pow l 2.0)))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	return t_s * (2.0 / ((t_m * pow(k_m, 4.0)) / pow(l, 2.0)));
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = t_s * (2.0d0 / ((t_m * (k_m ** 4.0d0)) / (l ** 2.0d0)))
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	return t_s * (2.0 / ((t_m * Math.pow(k_m, 4.0)) / Math.pow(l, 2.0)));
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	return t_s * (2.0 / ((t_m * math.pow(k_m, 4.0)) / math.pow(l, 2.0)))

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	return Float64(t_s * Float64(2.0 / Float64(Float64(t_m * (k_m ^ 4.0)) / (l ^ 2.0))))
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k_m)
	tmp = t_s * (2.0 / ((t_m * (k_m ^ 4.0)) / (l ^ 2.0)));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * N[(2.0 / N[(N[(t$95$m * N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.0%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0 60.8%

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]

4. Final simplification 60.8%

   \[\leadsto \frac{2}{\frac{t \cdot {k}^{4}}{{\ell}^{2}}} \]
5. Add Preprocessing

Alternative 7: 63.2% accurate, 3.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(\frac{2}{t\_m} \cdot {k\_m}^{-4}\right)\right) \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (* (* l l) (* (/ 2.0 t_m) (pow k_m -4.0)))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	return t_s * ((l * l) * ((2.0 / t_m) * pow(k_m, -4.0)));
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = t_s * ((l * l) * ((2.0d0 / t_m) * (k_m ** (-4.0d0))))
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	return t_s * ((l * l) * ((2.0 / t_m) * Math.pow(k_m, -4.0)));
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	return t_s * ((l * l) * ((2.0 / t_m) * math.pow(k_m, -4.0)))

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	return Float64(t_s * Float64(Float64(l * l) * Float64(Float64(2.0 / t_m) * (k_m ^ -4.0))))
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k_m)
	tmp = t_s * ((l * l) * ((2.0 / t_m) * (k_m ^ -4.0)));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * N[(N[(l * l), $MachinePrecision] * N[(N[(2.0 / t$95$m), $MachinePrecision] * N[Power[k$95$m, -4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.0%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

3. Simplified 41.8%

   \[\leadsto \color{blue}{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)} \]
4. Add Preprocessing

5. Taylor expanded in k around 0 60.8%

   \[\leadsto \frac{2}{\color{blue}{{k}^{4} \cdot t}} \cdot \left(\ell \cdot \ell\right) \]
6. Step-by-step derivation

   1. div-inv 60.8%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{1}{{k}^{4} \cdot t}\right)} \cdot \left(\ell \cdot \ell\right) \]

   2. *-commutative 60.8%

      \[\leadsto \left(2 \cdot \frac{1}{\color{blue}{t \cdot {k}^{4}}}\right) \cdot \left(\ell \cdot \ell\right) \]

7. Applied egg-rr 60.8%

   \[\leadsto \color{blue}{\left(2 \cdot \frac{1}{t \cdot {k}^{4}}\right)} \cdot \left(\ell \cdot \ell\right) \]
8. Step-by-step derivation

   1. associate-*r/ 60.8%

      \[\leadsto \color{blue}{\frac{2 \cdot 1}{t \cdot {k}^{4}}} \cdot \left(\ell \cdot \ell\right) \]

   2. metadata-eval 60.8%

      \[\leadsto \frac{\color{blue}{2}}{t \cdot {k}^{4}} \cdot \left(\ell \cdot \ell\right) \]

   3. associate-/r* 60.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{t}}{{k}^{4}}} \cdot \left(\ell \cdot \ell\right) \]

9. Simplified 60.8%

   \[\leadsto \color{blue}{\frac{\frac{2}{t}}{{k}^{4}}} \cdot \left(\ell \cdot \ell\right) \]
10. Step-by-step derivation

    1. div-inv 60.8%

       \[\leadsto \color{blue}{\left(\frac{2}{t} \cdot \frac{1}{{k}^{4}}\right)} \cdot \left(\ell \cdot \ell\right) \]

    2. pow-flip 60.8%

       \[\leadsto \left(\frac{2}{t} \cdot \color{blue}{{k}^{\left(-4\right)}}\right) \cdot \left(\ell \cdot \ell\right) \]

    3. metadata-eval 60.8%

       \[\leadsto \left(\frac{2}{t} \cdot {k}^{\color{blue}{-4}}\right) \cdot \left(\ell \cdot \ell\right) \]

11. Applied egg-rr 60.8%

    \[\leadsto \color{blue}{\left(\frac{2}{t} \cdot {k}^{-4}\right)} \cdot \left(\ell \cdot \ell\right) \]

12. Final simplification 60.8%

    \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(\frac{2}{t} \cdot {k}^{-4}\right) \]
13. Add Preprocessing

Alternative 8: 63.3% accurate, 3.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\frac{2}{t\_m \cdot {k\_m}^{4}} \cdot \left(\ell \cdot \ell\right)\right) \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (* (/ 2.0 (* t_m (pow k_m 4.0))) (* l l))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	return t_s * ((2.0 / (t_m * pow(k_m, 4.0))) * (l * l));
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = t_s * ((2.0d0 / (t_m * (k_m ** 4.0d0))) * (l * l))
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	return t_s * ((2.0 / (t_m * Math.pow(k_m, 4.0))) * (l * l));
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	return t_s * ((2.0 / (t_m * math.pow(k_m, 4.0))) * (l * l))

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	return Float64(t_s * Float64(Float64(2.0 / Float64(t_m * (k_m ^ 4.0))) * Float64(l * l)))
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k_m)
	tmp = t_s * ((2.0 / (t_m * (k_m ^ 4.0))) * (l * l));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * N[(N[(2.0 / N[(t$95$m * N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.0%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

3. Simplified 41.8%

   \[\leadsto \color{blue}{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)} \]
4. Add Preprocessing

5. Taylor expanded in k around 0 60.8%

   \[\leadsto \frac{2}{\color{blue}{{k}^{4} \cdot t}} \cdot \left(\ell \cdot \ell\right) \]

6. Final simplification 60.8%

   \[\leadsto \frac{2}{t \cdot {k}^{4}} \cdot \left(\ell \cdot \ell\right) \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024071 
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10-)"
  :precision binary64
  (/ 2.0 (* (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k)) (- (+ 1.0 (pow (/ k t) 2.0)) 1.0))))