Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.6% → 82.0%

Time: 12.7s

Alternatives: 12

Speedup: 6.6×

• 26.6% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l, k)
use fmin_fmax_functions
    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: 54.6% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l, k)
use fmin_fmax_functions
    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: 82.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\frac{\frac{\mathsf{fma}\left(2, \frac{{\left(\sin k \cdot t\right)}^{2}}{\cos k}, \frac{{\left(\sin k \cdot k\right)}^{2}}{\cos k}\right)}{\ell}}{\ell} \cdot t} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (/
  2.0
  (*
   (/
    (/
     (fma
      2.0
      (/ (pow (* (sin k) t) 2.0) (cos k))
      (/ (pow (* (sin k) k) 2.0) (cos k)))
     l)
    l)
   t)))

C

double code(double t, double l, double k) {
	return 2.0 / (((fma(2.0, (pow((sin(k) * t), 2.0) / cos(k)), (pow((sin(k) * k), 2.0) / cos(k))) / l) / l) * t);
}

Julia

function code(t, l, k)
	return Float64(2.0 / Float64(Float64(Float64(fma(2.0, Float64((Float64(sin(k) * t) ^ 2.0) / cos(k)), Float64((Float64(sin(k) * k) ^ 2.0) / cos(k))) / l) / l) * t))
end

Wolfram

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

Derivation

1. Initial program 54.6%

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

2. Taylor expanded in t around 0

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

4. Applied rewrites 68.4%

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

5. Taylor expanded in l around 0

   \[\leadsto \frac{2}{\frac{2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{\cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{\cos k}}{{\ell}^{2}} \cdot t} \]

7. Applied rewrites 74.1%

   \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\frac{{\left(\sin k \cdot t\right)}^{2}}{\cos k}, 2, \frac{{\left(\sin k \cdot k\right)}^{2}}{\cos k}\right)}{\ell \cdot \ell} \cdot t} \]

9. Applied rewrites 82.0%

   \[\leadsto \frac{2}{\frac{\frac{\mathsf{fma}\left(2, \frac{{\left(\sin k \cdot t\right)}^{2}}{\cos k}, \frac{{\left(\sin k \cdot k\right)}^{2}}{\cos k}\right)}{\ell}}{\ell} \cdot t} \]
10. Add Preprocessing

Alternative 2: 76.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\\ \mathbf{if}\;\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot t\_1 \leq 4 \cdot 10^{+27}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \sin k\right) \cdot \tan k\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{\mathsf{fma}\left(2, \frac{{\left(\mathsf{fma}\left(\left(k \cdot k\right) \cdot t, -0.16666666666666666, t\right) \cdot k\right)}^{2}}{\cos k}, \frac{{\left(\sin k \cdot k\right)}^{2}}{\cos k}\right)}{\ell}}{\ell} \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0)))
   (if (<= (* (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k)) t_1) 4e+27)
     (/ 2.0 (* (* (* (/ (/ (* (* t t) t) l) l) (sin k)) (tan k)) t_1))
     (/
      2.0
      (*
       (/
        (/
         (fma
          2.0
          (/
           (pow (* (fma (* (* k k) t) -0.16666666666666666 t) k) 2.0)
           (cos k))
          (/ (pow (* (sin k) k) 2.0) (cos k)))
         l)
        l)
       t)))))

C

double code(double t, double l, double k) {
	double t_1 = (1.0 + pow((k / t), 2.0)) + 1.0;
	double tmp;
	if (((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * t_1) <= 4e+27) {
		tmp = 2.0 / (((((((t * t) * t) / l) / l) * sin(k)) * tan(k)) * t_1);
	} else {
		tmp = 2.0 / (((fma(2.0, (pow((fma(((k * k) * t), -0.16666666666666666, t) * k), 2.0) / cos(k)), (pow((sin(k) * k), 2.0) / cos(k))) / l) / l) * t);
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(Float64(1.0 + (Float64(k / t) ^ 2.0)) + 1.0)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * t_1) <= 4e+27)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(Float64(t * t) * t) / l) / l) * sin(k)) * tan(k)) * t_1));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(fma(2.0, Float64((Float64(fma(Float64(Float64(k * k) * t), -0.16666666666666666, t) * k) ^ 2.0) / cos(k)), Float64((Float64(sin(k) * k) ^ 2.0) / cos(k))) / l) / l) * t));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[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] * t$95$1), $MachinePrecision], 4e+27], N[(2.0 / N[(N[(N[(N[(N[(N[(N[(t * t), $MachinePrecision] * t), $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(2.0 * N[(N[Power[N[(N[(N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision] * -0.16666666666666666 + t), $MachinePrecision] * k), $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[(N[Sin[k], $MachinePrecision] * k), $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))) < 4.0000000000000001e27

   1. Initial program 84.1%

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

   3. Applied rewrites 87.0%

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

   if 4.0000000000000001e27 < (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))

   1. Initial program 30.7%

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

   2. Taylor expanded in t around 0

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

   4. Applied rewrites 54.0%

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

   5. Taylor expanded in l around 0

      \[\leadsto \frac{2}{\frac{2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{\cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{\cos k}}{{\ell}^{2}} \cdot t} \]

   7. Applied rewrites 61.5%

      \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\frac{{\left(\sin k \cdot t\right)}^{2}}{\cos k}, 2, \frac{{\left(\sin k \cdot k\right)}^{2}}{\cos k}\right)}{\ell \cdot \ell} \cdot t} \]

   9. Applied rewrites 73.4%

      \[\leadsto \frac{2}{\frac{\frac{\mathsf{fma}\left(2, \frac{{\left(\sin k \cdot t\right)}^{2}}{\cos k}, \frac{{\left(\sin k \cdot k\right)}^{2}}{\cos k}\right)}{\ell}}{\ell} \cdot t} \]

   10. Taylor expanded in k around 0

       \[\leadsto \frac{2}{\frac{\frac{\mathsf{fma}\left(2, \frac{{\left(k \cdot \left(t + \frac{-1}{6} \cdot \left({k}^{2} \cdot t\right)\right)\right)}^{2}}{\cos k}, \frac{{\left(\sin k \cdot k\right)}^{2}}{\cos k}\right)}{\ell}}{\ell} \cdot t} \]

   12. Applied rewrites 67.4%

       \[\leadsto \frac{2}{\frac{\frac{\mathsf{fma}\left(2, \frac{{\left(\mathsf{fma}\left(\left(k \cdot k\right) \cdot t, -0.16666666666666666, t\right) \cdot k\right)}^{2}}{\cos k}, \frac{{\left(\sin k \cdot k\right)}^{2}}{\cos k}\right)}{\ell}}{\ell} \cdot t} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 71.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos k \cdot \ell\\ \mathbf{if}\;t \leq 3.6 \cdot 10^{-39}:\\ \;\;\;\;\frac{2}{\frac{\frac{{\left(\sin k \cdot k\right)}^{2}}{t\_1}}{\ell} \cdot t}\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{+139}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{{\left(\sin k \cdot t\right)}^{2} \cdot 2}{t\_1}}{\ell} \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (cos k) l)))
   (if (<= t 3.6e-39)
     (/ 2.0 (* (/ (/ (pow (* (sin k) k) 2.0) t_1) l) t))
     (if (<= t 2.1e+139)
       (/
        2.0
        (*
         (* (* (/ (/ (* (* t t) t) l) l) (sin k)) (tan k))
         (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0)))
       (/ 2.0 (* (/ (/ (* (pow (* (sin k) t) 2.0) 2.0) t_1) l) t))))))

C

double code(double t, double l, double k) {
	double t_1 = cos(k) * l;
	double tmp;
	if (t <= 3.6e-39) {
		tmp = 2.0 / (((pow((sin(k) * k), 2.0) / t_1) / l) * t);
	} else if (t <= 2.1e+139) {
		tmp = 2.0 / (((((((t * t) * t) / l) / l) * sin(k)) * tan(k)) * ((1.0 + pow((k / t), 2.0)) + 1.0));
	} else {
		tmp = 2.0 / ((((pow((sin(k) * t), 2.0) * 2.0) / t_1) / l) * t);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = cos(k) * l
    if (t <= 3.6d-39) then
        tmp = 2.0d0 / (((((sin(k) * k) ** 2.0d0) / t_1) / l) * t)
    else if (t <= 2.1d+139) then
        tmp = 2.0d0 / (((((((t * t) * t) / l) / l) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t) ** 2.0d0)) + 1.0d0))
    else
        tmp = 2.0d0 / ((((((sin(k) * t) ** 2.0d0) * 2.0d0) / t_1) / l) * t)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.cos(k) * l;
	double tmp;
	if (t <= 3.6e-39) {
		tmp = 2.0 / (((Math.pow((Math.sin(k) * k), 2.0) / t_1) / l) * t);
	} else if (t <= 2.1e+139) {
		tmp = 2.0 / (((((((t * t) * t) / l) / l) * Math.sin(k)) * Math.tan(k)) * ((1.0 + Math.pow((k / t), 2.0)) + 1.0));
	} else {
		tmp = 2.0 / ((((Math.pow((Math.sin(k) * t), 2.0) * 2.0) / t_1) / l) * t);
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.cos(k) * l
	tmp = 0
	if t <= 3.6e-39:
		tmp = 2.0 / (((math.pow((math.sin(k) * k), 2.0) / t_1) / l) * t)
	elif t <= 2.1e+139:
		tmp = 2.0 / (((((((t * t) * t) / l) / l) * math.sin(k)) * math.tan(k)) * ((1.0 + math.pow((k / t), 2.0)) + 1.0))
	else:
		tmp = 2.0 / ((((math.pow((math.sin(k) * t), 2.0) * 2.0) / t_1) / l) * t)
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(cos(k) * l)
	tmp = 0.0
	if (t <= 3.6e-39)
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(sin(k) * k) ^ 2.0) / t_1) / l) * t));
	elseif (t <= 2.1e+139)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(Float64(t * t) * t) / l) / l) * sin(k)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t) ^ 2.0)) + 1.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64((Float64(sin(k) * t) ^ 2.0) * 2.0) / t_1) / l) * t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = cos(k) * l;
	tmp = 0.0;
	if (t <= 3.6e-39)
		tmp = 2.0 / (((((sin(k) * k) ^ 2.0) / t_1) / l) * t);
	elseif (t <= 2.1e+139)
		tmp = 2.0 / (((((((t * t) * t) / l) / l) * sin(k)) * tan(k)) * ((1.0 + ((k / t) ^ 2.0)) + 1.0));
	else
		tmp = 2.0 / ((((((sin(k) * t) ^ 2.0) * 2.0) / t_1) / l) * t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]}, If[LessEqual[t, 3.6e-39], N[(2.0 / N[(N[(N[(N[Power[N[(N[Sin[k], $MachinePrecision] * k), $MachinePrecision], 2.0], $MachinePrecision] / t$95$1), $MachinePrecision] / l), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.1e+139], N[(2.0 / N[(N[(N[(N[(N[(N[(N[(t * t), $MachinePrecision] * t), $MachinePrecision] / l), $MachinePrecision] / l), $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], N[(2.0 / N[(N[(N[(N[(N[Power[N[(N[Sin[k], $MachinePrecision] * t), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision] / t$95$1), $MachinePrecision] / l), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < 3.6000000000000001e-39

   1. Initial program 49.9%

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

   2. Taylor expanded in t around 0

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

   4. Applied rewrites 67.8%

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

   5. Taylor expanded in l around 0

      \[\leadsto \frac{2}{\frac{2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{\cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{\cos k}}{{\ell}^{2}} \cdot t} \]

   7. Applied rewrites 73.5%

      \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\frac{{\left(\sin k \cdot t\right)}^{2}}{\cos k}, 2, \frac{{\left(\sin k \cdot k\right)}^{2}}{\cos k}\right)}{\ell \cdot \ell} \cdot t} \]

   9. Applied rewrites 81.2%

      \[\leadsto \frac{2}{\frac{\frac{\mathsf{fma}\left(2, \frac{{\left(\sin k \cdot t\right)}^{2}}{\cos k}, \frac{{\left(\sin k \cdot k\right)}^{2}}{\cos k}\right)}{\ell}}{\ell} \cdot t} \]

   10. Taylor expanded in t around 0

       \[\leadsto \frac{2}{\frac{\frac{{k}^{2} \cdot {\sin k}^{2}}{\ell \cdot \cos k}}{\ell} \cdot t} \]

   12. Applied rewrites 68.0%

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

   if 3.6000000000000001e-39 < t < 2.0999999999999999e139

   1. Initial program 70.1%

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

   3. Applied rewrites 73.9%

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

   if 2.0999999999999999e139 < t

   1. Initial program 61.7%

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

   2. Taylor expanded in t around 0

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

   4. Applied rewrites 65.5%

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

   5. Taylor expanded in l around 0

      \[\leadsto \frac{2}{\frac{2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{\cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{\cos k}}{{\ell}^{2}} \cdot t} \]

   7. Applied rewrites 75.1%

      \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\frac{{\left(\sin k \cdot t\right)}^{2}}{\cos k}, 2, \frac{{\left(\sin k \cdot k\right)}^{2}}{\cos k}\right)}{\ell \cdot \ell} \cdot t} \]

   9. Applied rewrites 85.0%

      \[\leadsto \frac{2}{\frac{\frac{\mathsf{fma}\left(2, \frac{{\left(\sin k \cdot t\right)}^{2}}{\cos k}, \frac{{\left(\sin k \cdot k\right)}^{2}}{\cos k}\right)}{\ell}}{\ell} \cdot t} \]

   10. Taylor expanded in t around inf

       \[\leadsto \frac{2}{\frac{2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{\ell \cdot \cos k}}{\ell} \cdot t} \]

   12. Applied rewrites 84.7%

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

Alternative 4: 70.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos k \cdot \ell\\ \mathbf{if}\;t \leq 3.6 \cdot 10^{-39}:\\ \;\;\;\;\frac{2}{\frac{\frac{{\left(\sin k \cdot k\right)}^{2}}{t\_1}}{\ell} \cdot t}\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{+139}:\\ \;\;\;\;\frac{2}{\left(\frac{\left(\left(t \cdot t\right) \cdot t\right) \cdot \sin k}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{{\left(\sin k \cdot t\right)}^{2} \cdot 2}{t\_1}}{\ell} \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (cos k) l)))
   (if (<= t 3.6e-39)
     (/ 2.0 (* (/ (/ (pow (* (sin k) k) 2.0) t_1) l) t))
     (if (<= t 2.1e+139)
       (/
        2.0
        (*
         (* (/ (* (* (* t t) t) (sin k)) (* l l)) (tan k))
         (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0)))
       (/ 2.0 (* (/ (/ (* (pow (* (sin k) t) 2.0) 2.0) t_1) l) t))))))

C

double code(double t, double l, double k) {
	double t_1 = cos(k) * l;
	double tmp;
	if (t <= 3.6e-39) {
		tmp = 2.0 / (((pow((sin(k) * k), 2.0) / t_1) / l) * t);
	} else if (t <= 2.1e+139) {
		tmp = 2.0 / ((((((t * t) * t) * sin(k)) / (l * l)) * tan(k)) * ((1.0 + pow((k / t), 2.0)) + 1.0));
	} else {
		tmp = 2.0 / ((((pow((sin(k) * t), 2.0) * 2.0) / t_1) / l) * t);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = cos(k) * l
    if (t <= 3.6d-39) then
        tmp = 2.0d0 / (((((sin(k) * k) ** 2.0d0) / t_1) / l) * t)
    else if (t <= 2.1d+139) then
        tmp = 2.0d0 / ((((((t * t) * t) * sin(k)) / (l * l)) * tan(k)) * ((1.0d0 + ((k / t) ** 2.0d0)) + 1.0d0))
    else
        tmp = 2.0d0 / ((((((sin(k) * t) ** 2.0d0) * 2.0d0) / t_1) / l) * t)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.cos(k) * l;
	double tmp;
	if (t <= 3.6e-39) {
		tmp = 2.0 / (((Math.pow((Math.sin(k) * k), 2.0) / t_1) / l) * t);
	} else if (t <= 2.1e+139) {
		tmp = 2.0 / ((((((t * t) * t) * Math.sin(k)) / (l * l)) * Math.tan(k)) * ((1.0 + Math.pow((k / t), 2.0)) + 1.0));
	} else {
		tmp = 2.0 / ((((Math.pow((Math.sin(k) * t), 2.0) * 2.0) / t_1) / l) * t);
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.cos(k) * l
	tmp = 0
	if t <= 3.6e-39:
		tmp = 2.0 / (((math.pow((math.sin(k) * k), 2.0) / t_1) / l) * t)
	elif t <= 2.1e+139:
		tmp = 2.0 / ((((((t * t) * t) * math.sin(k)) / (l * l)) * math.tan(k)) * ((1.0 + math.pow((k / t), 2.0)) + 1.0))
	else:
		tmp = 2.0 / ((((math.pow((math.sin(k) * t), 2.0) * 2.0) / t_1) / l) * t)
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(cos(k) * l)
	tmp = 0.0
	if (t <= 3.6e-39)
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(sin(k) * k) ^ 2.0) / t_1) / l) * t));
	elseif (t <= 2.1e+139)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(t * t) * t) * sin(k)) / Float64(l * l)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t) ^ 2.0)) + 1.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64((Float64(sin(k) * t) ^ 2.0) * 2.0) / t_1) / l) * t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = cos(k) * l;
	tmp = 0.0;
	if (t <= 3.6e-39)
		tmp = 2.0 / (((((sin(k) * k) ^ 2.0) / t_1) / l) * t);
	elseif (t <= 2.1e+139)
		tmp = 2.0 / ((((((t * t) * t) * sin(k)) / (l * l)) * tan(k)) * ((1.0 + ((k / t) ^ 2.0)) + 1.0));
	else
		tmp = 2.0 / ((((((sin(k) * t) ^ 2.0) * 2.0) / t_1) / l) * t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]}, If[LessEqual[t, 3.6e-39], N[(2.0 / N[(N[(N[(N[Power[N[(N[Sin[k], $MachinePrecision] * k), $MachinePrecision], 2.0], $MachinePrecision] / t$95$1), $MachinePrecision] / l), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.1e+139], N[(2.0 / N[(N[(N[(N[(N[(N[(t * t), $MachinePrecision] * t), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[(l * l), $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], N[(2.0 / N[(N[(N[(N[(N[Power[N[(N[Sin[k], $MachinePrecision] * t), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision] / t$95$1), $MachinePrecision] / l), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < 3.6000000000000001e-39

   1. Initial program 49.9%

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

   2. Taylor expanded in t around 0

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

   4. Applied rewrites 67.8%

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

   5. Taylor expanded in l around 0

      \[\leadsto \frac{2}{\frac{2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{\cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{\cos k}}{{\ell}^{2}} \cdot t} \]

   7. Applied rewrites 73.5%

      \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\frac{{\left(\sin k \cdot t\right)}^{2}}{\cos k}, 2, \frac{{\left(\sin k \cdot k\right)}^{2}}{\cos k}\right)}{\ell \cdot \ell} \cdot t} \]

   9. Applied rewrites 81.2%

      \[\leadsto \frac{2}{\frac{\frac{\mathsf{fma}\left(2, \frac{{\left(\sin k \cdot t\right)}^{2}}{\cos k}, \frac{{\left(\sin k \cdot k\right)}^{2}}{\cos k}\right)}{\ell}}{\ell} \cdot t} \]

   10. Taylor expanded in t around 0

       \[\leadsto \frac{2}{\frac{\frac{{k}^{2} \cdot {\sin k}^{2}}{\ell \cdot \cos k}}{\ell} \cdot t} \]

   12. Applied rewrites 68.0%

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

   if 3.6000000000000001e-39 < t < 2.0999999999999999e139

   1. Initial program 70.1%

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

   3. Applied rewrites 72.3%

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

   if 2.0999999999999999e139 < t

   1. Initial program 61.7%

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

   2. Taylor expanded in t around 0

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

   4. Applied rewrites 65.5%

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

   5. Taylor expanded in l around 0

      \[\leadsto \frac{2}{\frac{2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{\cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{\cos k}}{{\ell}^{2}} \cdot t} \]

   7. Applied rewrites 75.1%

      \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\frac{{\left(\sin k \cdot t\right)}^{2}}{\cos k}, 2, \frac{{\left(\sin k \cdot k\right)}^{2}}{\cos k}\right)}{\ell \cdot \ell} \cdot t} \]

   9. Applied rewrites 85.0%

      \[\leadsto \frac{2}{\frac{\frac{\mathsf{fma}\left(2, \frac{{\left(\sin k \cdot t\right)}^{2}}{\cos k}, \frac{{\left(\sin k \cdot k\right)}^{2}}{\cos k}\right)}{\ell}}{\ell} \cdot t} \]

   10. Taylor expanded in t around inf

       \[\leadsto \frac{2}{\frac{2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{\ell \cdot \cos k}}{\ell} \cdot t} \]

   12. Applied rewrites 84.7%

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

Alternative 5: 70.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos k \cdot \ell\\ \mathbf{if}\;t \leq 1.6 \cdot 10^{-28}:\\ \;\;\;\;\frac{2}{\frac{\frac{{\left(\sin k \cdot k\right)}^{2}}{t\_1}}{\ell} \cdot t}\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{+139}:\\ \;\;\;\;\frac{2}{\left(\frac{\frac{\left(\left(t \cdot t\right) \cdot t\right) \cdot k}{\ell}}{\ell} \cdot \tan k\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{{\left(\sin k \cdot t\right)}^{2} \cdot 2}{t\_1}}{\ell} \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (cos k) l)))
   (if (<= t 1.6e-28)
     (/ 2.0 (* (/ (/ (pow (* (sin k) k) 2.0) t_1) l) t))
     (if (<= t 2.1e+139)
       (/ 2.0 (* (* (/ (/ (* (* (* t t) t) k) l) l) (tan k)) 2.0))
       (/ 2.0 (* (/ (/ (* (pow (* (sin k) t) 2.0) 2.0) t_1) l) t))))))

C

double code(double t, double l, double k) {
	double t_1 = cos(k) * l;
	double tmp;
	if (t <= 1.6e-28) {
		tmp = 2.0 / (((pow((sin(k) * k), 2.0) / t_1) / l) * t);
	} else if (t <= 2.1e+139) {
		tmp = 2.0 / (((((((t * t) * t) * k) / l) / l) * tan(k)) * 2.0);
	} else {
		tmp = 2.0 / ((((pow((sin(k) * t), 2.0) * 2.0) / t_1) / l) * t);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = cos(k) * l
    if (t <= 1.6d-28) then
        tmp = 2.0d0 / (((((sin(k) * k) ** 2.0d0) / t_1) / l) * t)
    else if (t <= 2.1d+139) then
        tmp = 2.0d0 / (((((((t * t) * t) * k) / l) / l) * tan(k)) * 2.0d0)
    else
        tmp = 2.0d0 / ((((((sin(k) * t) ** 2.0d0) * 2.0d0) / t_1) / l) * t)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.cos(k) * l;
	double tmp;
	if (t <= 1.6e-28) {
		tmp = 2.0 / (((Math.pow((Math.sin(k) * k), 2.0) / t_1) / l) * t);
	} else if (t <= 2.1e+139) {
		tmp = 2.0 / (((((((t * t) * t) * k) / l) / l) * Math.tan(k)) * 2.0);
	} else {
		tmp = 2.0 / ((((Math.pow((Math.sin(k) * t), 2.0) * 2.0) / t_1) / l) * t);
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.cos(k) * l
	tmp = 0
	if t <= 1.6e-28:
		tmp = 2.0 / (((math.pow((math.sin(k) * k), 2.0) / t_1) / l) * t)
	elif t <= 2.1e+139:
		tmp = 2.0 / (((((((t * t) * t) * k) / l) / l) * math.tan(k)) * 2.0)
	else:
		tmp = 2.0 / ((((math.pow((math.sin(k) * t), 2.0) * 2.0) / t_1) / l) * t)
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(cos(k) * l)
	tmp = 0.0
	if (t <= 1.6e-28)
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(sin(k) * k) ^ 2.0) / t_1) / l) * t));
	elseif (t <= 2.1e+139)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(Float64(t * t) * t) * k) / l) / l) * tan(k)) * 2.0));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64((Float64(sin(k) * t) ^ 2.0) * 2.0) / t_1) / l) * t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = cos(k) * l;
	tmp = 0.0;
	if (t <= 1.6e-28)
		tmp = 2.0 / (((((sin(k) * k) ^ 2.0) / t_1) / l) * t);
	elseif (t <= 2.1e+139)
		tmp = 2.0 / (((((((t * t) * t) * k) / l) / l) * tan(k)) * 2.0);
	else
		tmp = 2.0 / ((((((sin(k) * t) ^ 2.0) * 2.0) / t_1) / l) * t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]}, If[LessEqual[t, 1.6e-28], N[(2.0 / N[(N[(N[(N[Power[N[(N[Sin[k], $MachinePrecision] * k), $MachinePrecision], 2.0], $MachinePrecision] / t$95$1), $MachinePrecision] / l), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.1e+139], N[(2.0 / N[(N[(N[(N[(N[(N[(N[(t * t), $MachinePrecision] * t), $MachinePrecision] * k), $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[Power[N[(N[Sin[k], $MachinePrecision] * t), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision] / t$95$1), $MachinePrecision] / l), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < 1.59999999999999991e-28

   1. Initial program 50.2%

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

   2. Taylor expanded in t around 0

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

   4. Applied rewrites 67.9%

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

   5. Taylor expanded in l around 0

      \[\leadsto \frac{2}{\frac{2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{\cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{\cos k}}{{\ell}^{2}} \cdot t} \]

   7. Applied rewrites 73.5%

      \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\frac{{\left(\sin k \cdot t\right)}^{2}}{\cos k}, 2, \frac{{\left(\sin k \cdot k\right)}^{2}}{\cos k}\right)}{\ell \cdot \ell} \cdot t} \]

   9. Applied rewrites 81.2%

      \[\leadsto \frac{2}{\frac{\frac{\mathsf{fma}\left(2, \frac{{\left(\sin k \cdot t\right)}^{2}}{\cos k}, \frac{{\left(\sin k \cdot k\right)}^{2}}{\cos k}\right)}{\ell}}{\ell} \cdot t} \]

   10. Taylor expanded in t around 0

       \[\leadsto \frac{2}{\frac{\frac{{k}^{2} \cdot {\sin k}^{2}}{\ell \cdot \cos k}}{\ell} \cdot t} \]

   12. Applied rewrites 68.0%

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

   if 1.59999999999999991e-28 < t < 2.0999999999999999e139

   1. Initial program 70.0%

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

   2. Taylor expanded in t around 0

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

   4. Applied rewrites 55.7%

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

   6. Applied rewrites 57.2%

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

   7. Taylor expanded in k around 0

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

   9. Applied rewrites 51.7%

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

   10. Taylor expanded in t around inf

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

   12. Applied rewrites 67.0%

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

   if 2.0999999999999999e139 < t

   1. Initial program 61.7%

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

   2. Taylor expanded in t around 0

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

   4. Applied rewrites 65.5%

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

   5. Taylor expanded in l around 0

      \[\leadsto \frac{2}{\frac{2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{\cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{\cos k}}{{\ell}^{2}} \cdot t} \]

   7. Applied rewrites 75.1%

      \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\frac{{\left(\sin k \cdot t\right)}^{2}}{\cos k}, 2, \frac{{\left(\sin k \cdot k\right)}^{2}}{\cos k}\right)}{\ell \cdot \ell} \cdot t} \]

   9. Applied rewrites 85.0%

      \[\leadsto \frac{2}{\frac{\frac{\mathsf{fma}\left(2, \frac{{\left(\sin k \cdot t\right)}^{2}}{\cos k}, \frac{{\left(\sin k \cdot k\right)}^{2}}{\cos k}\right)}{\ell}}{\ell} \cdot t} \]

   10. Taylor expanded in t around inf

       \[\leadsto \frac{2}{\frac{2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{\ell \cdot \cos k}}{\ell} \cdot t} \]

   12. Applied rewrites 84.7%

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

Alternative 6: 69.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(t \cdot t\right) \cdot t\right) \cdot k\\ \mathbf{if}\;t \leq 1.6 \cdot 10^{-28}:\\ \;\;\;\;\frac{2}{\frac{\frac{{\left(\sin k \cdot k\right)}^{2}}{\cos k \cdot \ell}}{\ell} \cdot t}\\ \mathbf{elif}\;t \leq 9 \cdot 10^{+54}:\\ \;\;\;\;\frac{2}{\left(\frac{\frac{t\_1}{\ell}}{\ell} \cdot \tan k\right) \cdot 2}\\ \mathbf{elif}\;t \leq 1.16 \cdot 10^{+240}:\\ \;\;\;\;\frac{\left(\ell \cdot \ell\right) \cdot \cos k}{{\left(\sin k \cdot t\right)}^{2} \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k \cdot t\_1} \cdot \ell\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (* (* t t) t) k)))
   (if (<= t 1.6e-28)
     (/ 2.0 (* (/ (/ (pow (* (sin k) k) 2.0) (* (cos k) l)) l) t))
     (if (<= t 9e+54)
       (/ 2.0 (* (* (/ (/ t_1 l) l) (tan k)) 2.0))
       (if (<= t 1.16e+240)
         (/ (* (* l l) (cos k)) (* (pow (* (sin k) t) 2.0) t))
         (* (/ l (* k t_1)) l))))))

C

double code(double t, double l, double k) {
	double t_1 = ((t * t) * t) * k;
	double tmp;
	if (t <= 1.6e-28) {
		tmp = 2.0 / (((pow((sin(k) * k), 2.0) / (cos(k) * l)) / l) * t);
	} else if (t <= 9e+54) {
		tmp = 2.0 / ((((t_1 / l) / l) * tan(k)) * 2.0);
	} else if (t <= 1.16e+240) {
		tmp = ((l * l) * cos(k)) / (pow((sin(k) * t), 2.0) * t);
	} else {
		tmp = (l / (k * t_1)) * l;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((t * t) * t) * k
    if (t <= 1.6d-28) then
        tmp = 2.0d0 / (((((sin(k) * k) ** 2.0d0) / (cos(k) * l)) / l) * t)
    else if (t <= 9d+54) then
        tmp = 2.0d0 / ((((t_1 / l) / l) * tan(k)) * 2.0d0)
    else if (t <= 1.16d+240) then
        tmp = ((l * l) * cos(k)) / (((sin(k) * t) ** 2.0d0) * t)
    else
        tmp = (l / (k * t_1)) * l
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = ((t * t) * t) * k;
	double tmp;
	if (t <= 1.6e-28) {
		tmp = 2.0 / (((Math.pow((Math.sin(k) * k), 2.0) / (Math.cos(k) * l)) / l) * t);
	} else if (t <= 9e+54) {
		tmp = 2.0 / ((((t_1 / l) / l) * Math.tan(k)) * 2.0);
	} else if (t <= 1.16e+240) {
		tmp = ((l * l) * Math.cos(k)) / (Math.pow((Math.sin(k) * t), 2.0) * t);
	} else {
		tmp = (l / (k * t_1)) * l;
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = ((t * t) * t) * k
	tmp = 0
	if t <= 1.6e-28:
		tmp = 2.0 / (((math.pow((math.sin(k) * k), 2.0) / (math.cos(k) * l)) / l) * t)
	elif t <= 9e+54:
		tmp = 2.0 / ((((t_1 / l) / l) * math.tan(k)) * 2.0)
	elif t <= 1.16e+240:
		tmp = ((l * l) * math.cos(k)) / (math.pow((math.sin(k) * t), 2.0) * t)
	else:
		tmp = (l / (k * t_1)) * l
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(Float64(Float64(t * t) * t) * k)
	tmp = 0.0
	if (t <= 1.6e-28)
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(sin(k) * k) ^ 2.0) / Float64(cos(k) * l)) / l) * t));
	elseif (t <= 9e+54)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t_1 / l) / l) * tan(k)) * 2.0));
	elseif (t <= 1.16e+240)
		tmp = Float64(Float64(Float64(l * l) * cos(k)) / Float64((Float64(sin(k) * t) ^ 2.0) * t));
	else
		tmp = Float64(Float64(l / Float64(k * t_1)) * l);
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = ((t * t) * t) * k;
	tmp = 0.0;
	if (t <= 1.6e-28)
		tmp = 2.0 / (((((sin(k) * k) ^ 2.0) / (cos(k) * l)) / l) * t);
	elseif (t <= 9e+54)
		tmp = 2.0 / ((((t_1 / l) / l) * tan(k)) * 2.0);
	elseif (t <= 1.16e+240)
		tmp = ((l * l) * cos(k)) / (((sin(k) * t) ^ 2.0) * t);
	else
		tmp = (l / (k * t_1)) * l;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[(N[(t * t), $MachinePrecision] * t), $MachinePrecision] * k), $MachinePrecision]}, If[LessEqual[t, 1.6e-28], N[(2.0 / N[(N[(N[(N[Power[N[(N[Sin[k], $MachinePrecision] * k), $MachinePrecision], 2.0], $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 9e+54], N[(2.0 / N[(N[(N[(N[(t$95$1 / l), $MachinePrecision] / l), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.16e+240], N[(N[(N[(l * l), $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[(N[Sin[k], $MachinePrecision] * t), $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(k * t$95$1), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if t < 1.59999999999999991e-28

   1. Initial program 50.2%

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

   2. Taylor expanded in t around 0

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

   4. Applied rewrites 67.9%

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

   5. Taylor expanded in l around 0

      \[\leadsto \frac{2}{\frac{2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{\cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{\cos k}}{{\ell}^{2}} \cdot t} \]

   7. Applied rewrites 73.5%

      \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\frac{{\left(\sin k \cdot t\right)}^{2}}{\cos k}, 2, \frac{{\left(\sin k \cdot k\right)}^{2}}{\cos k}\right)}{\ell \cdot \ell} \cdot t} \]

   9. Applied rewrites 81.2%

      \[\leadsto \frac{2}{\frac{\frac{\mathsf{fma}\left(2, \frac{{\left(\sin k \cdot t\right)}^{2}}{\cos k}, \frac{{\left(\sin k \cdot k\right)}^{2}}{\cos k}\right)}{\ell}}{\ell} \cdot t} \]

   10. Taylor expanded in t around 0

       \[\leadsto \frac{2}{\frac{\frac{{k}^{2} \cdot {\sin k}^{2}}{\ell \cdot \cos k}}{\ell} \cdot t} \]

   12. Applied rewrites 68.0%

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

   if 1.59999999999999991e-28 < t < 8.99999999999999968e54

   1. Initial program 75.8%

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

   2. Taylor expanded in t around 0

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

   4. Applied rewrites 63.1%

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

   6. Applied rewrites 64.8%

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

   7. Taylor expanded in k around 0

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

   9. Applied rewrites 55.0%

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

   10. Taylor expanded in t around inf

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

   12. Applied rewrites 67.3%

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

   if 8.99999999999999968e54 < t < 1.1600000000000001e240

   1. Initial program 60.5%

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

   2. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{t}^{3} \cdot {\sin k}^{2}}} \]

   4. Applied rewrites 52.5%

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

   6. Applied rewrites 73.2%

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

   if 1.1600000000000001e240 < t

   1. Initial program 68.0%

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

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

   4. Applied rewrites 56.0%

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

   6. Applied rewrites 65.4%

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

   8. Applied rewrites 65.4%

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

   10. Applied rewrites 77.6%

       \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 7: 65.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(t \cdot t\right) \cdot t\right) \cdot k\\ t_2 := \left(\ell \cdot \ell\right) \cdot \cos k\\ \mathbf{if}\;t \leq 1.6 \cdot 10^{-28}:\\ \;\;\;\;\frac{2}{\frac{{\left(\sin k \cdot k\right)}^{2}}{t\_2} \cdot t}\\ \mathbf{elif}\;t \leq 9 \cdot 10^{+54}:\\ \;\;\;\;\frac{2}{\left(\frac{\frac{t\_1}{\ell}}{\ell} \cdot \tan k\right) \cdot 2}\\ \mathbf{elif}\;t \leq 1.16 \cdot 10^{+240}:\\ \;\;\;\;\frac{t\_2}{{\left(\sin k \cdot t\right)}^{2} \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k \cdot t\_1} \cdot \ell\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (* (* t t) t) k)) (t_2 (* (* l l) (cos k))))
   (if (<= t 1.6e-28)
     (/ 2.0 (* (/ (pow (* (sin k) k) 2.0) t_2) t))
     (if (<= t 9e+54)
       (/ 2.0 (* (* (/ (/ t_1 l) l) (tan k)) 2.0))
       (if (<= t 1.16e+240)
         (/ t_2 (* (pow (* (sin k) t) 2.0) t))
         (* (/ l (* k t_1)) l))))))

C

double code(double t, double l, double k) {
	double t_1 = ((t * t) * t) * k;
	double t_2 = (l * l) * cos(k);
	double tmp;
	if (t <= 1.6e-28) {
		tmp = 2.0 / ((pow((sin(k) * k), 2.0) / t_2) * t);
	} else if (t <= 9e+54) {
		tmp = 2.0 / ((((t_1 / l) / l) * tan(k)) * 2.0);
	} else if (t <= 1.16e+240) {
		tmp = t_2 / (pow((sin(k) * t), 2.0) * t);
	} else {
		tmp = (l / (k * t_1)) * l;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = ((t * t) * t) * k
    t_2 = (l * l) * cos(k)
    if (t <= 1.6d-28) then
        tmp = 2.0d0 / ((((sin(k) * k) ** 2.0d0) / t_2) * t)
    else if (t <= 9d+54) then
        tmp = 2.0d0 / ((((t_1 / l) / l) * tan(k)) * 2.0d0)
    else if (t <= 1.16d+240) then
        tmp = t_2 / (((sin(k) * t) ** 2.0d0) * t)
    else
        tmp = (l / (k * t_1)) * l
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = ((t * t) * t) * k;
	double t_2 = (l * l) * Math.cos(k);
	double tmp;
	if (t <= 1.6e-28) {
		tmp = 2.0 / ((Math.pow((Math.sin(k) * k), 2.0) / t_2) * t);
	} else if (t <= 9e+54) {
		tmp = 2.0 / ((((t_1 / l) / l) * Math.tan(k)) * 2.0);
	} else if (t <= 1.16e+240) {
		tmp = t_2 / (Math.pow((Math.sin(k) * t), 2.0) * t);
	} else {
		tmp = (l / (k * t_1)) * l;
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = ((t * t) * t) * k
	t_2 = (l * l) * math.cos(k)
	tmp = 0
	if t <= 1.6e-28:
		tmp = 2.0 / ((math.pow((math.sin(k) * k), 2.0) / t_2) * t)
	elif t <= 9e+54:
		tmp = 2.0 / ((((t_1 / l) / l) * math.tan(k)) * 2.0)
	elif t <= 1.16e+240:
		tmp = t_2 / (math.pow((math.sin(k) * t), 2.0) * t)
	else:
		tmp = (l / (k * t_1)) * l
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(Float64(Float64(t * t) * t) * k)
	t_2 = Float64(Float64(l * l) * cos(k))
	tmp = 0.0
	if (t <= 1.6e-28)
		tmp = Float64(2.0 / Float64(Float64((Float64(sin(k) * k) ^ 2.0) / t_2) * t));
	elseif (t <= 9e+54)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t_1 / l) / l) * tan(k)) * 2.0));
	elseif (t <= 1.16e+240)
		tmp = Float64(t_2 / Float64((Float64(sin(k) * t) ^ 2.0) * t));
	else
		tmp = Float64(Float64(l / Float64(k * t_1)) * l);
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = ((t * t) * t) * k;
	t_2 = (l * l) * cos(k);
	tmp = 0.0;
	if (t <= 1.6e-28)
		tmp = 2.0 / ((((sin(k) * k) ^ 2.0) / t_2) * t);
	elseif (t <= 9e+54)
		tmp = 2.0 / ((((t_1 / l) / l) * tan(k)) * 2.0);
	elseif (t <= 1.16e+240)
		tmp = t_2 / (((sin(k) * t) ^ 2.0) * t);
	else
		tmp = (l / (k * t_1)) * l;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[(N[(t * t), $MachinePrecision] * t), $MachinePrecision] * k), $MachinePrecision]}, Block[{t$95$2 = N[(N[(l * l), $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, 1.6e-28], N[(2.0 / N[(N[(N[Power[N[(N[Sin[k], $MachinePrecision] * k), $MachinePrecision], 2.0], $MachinePrecision] / t$95$2), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 9e+54], N[(2.0 / N[(N[(N[(N[(t$95$1 / l), $MachinePrecision] / l), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.16e+240], N[(t$95$2 / N[(N[Power[N[(N[Sin[k], $MachinePrecision] * t), $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(k * t$95$1), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < 1.59999999999999991e-28

   1. Initial program 50.2%

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

   2. Taylor expanded in t around 0

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

   4. Applied rewrites 67.9%

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

   5. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} \cdot t} \]

   7. Applied rewrites 62.6%

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

   if 1.59999999999999991e-28 < t < 8.99999999999999968e54

   1. Initial program 75.8%

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

   2. Taylor expanded in t around 0

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

   4. Applied rewrites 63.1%

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

   6. Applied rewrites 64.8%

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

   7. Taylor expanded in k around 0

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

   9. Applied rewrites 55.0%

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

   10. Taylor expanded in t around inf

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

   12. Applied rewrites 67.3%

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

   if 8.99999999999999968e54 < t < 1.1600000000000001e240

   1. Initial program 60.5%

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

   2. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{t}^{3} \cdot {\sin k}^{2}}} \]

   4. Applied rewrites 52.5%

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

   6. Applied rewrites 73.2%

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

   if 1.1600000000000001e240 < t

   1. Initial program 68.0%

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

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

   4. Applied rewrites 56.0%

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

   6. Applied rewrites 65.4%

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

   8. Applied rewrites 65.4%

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

   10. Applied rewrites 77.6%

       \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 8: 63.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 8 \cdot 10^{+134}:\\ \;\;\;\;\frac{2}{\left(\frac{\frac{\left(\left(t \cdot t\right) \cdot t\right) \cdot \sin k}{\ell}}{\ell} \cdot \tan k\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\left(t \cdot t\right) \cdot \left(-0.3333333333333333 - -0.5\right), 2, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\ell}}{\ell} \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 8e+134)
   (/ 2.0 (* (* (/ (/ (* (* (* t t) t) (sin k)) l) l) (tan k)) 2.0))
   (/
    2.0
    (*
     (/
      (/
       (*
        (fma
         (fma (* (* t t) (- -0.3333333333333333 -0.5)) 2.0 1.0)
         (* k k)
         (* (* t t) 2.0))
        (* k k))
       l)
      l)
     t))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 8e+134) {
		tmp = 2.0 / (((((((t * t) * t) * sin(k)) / l) / l) * tan(k)) * 2.0);
	} else {
		tmp = 2.0 / ((((fma(fma(((t * t) * (-0.3333333333333333 - -0.5)), 2.0, 1.0), (k * k), ((t * t) * 2.0)) * (k * k)) / l) / l) * t);
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 8e+134)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(Float64(t * t) * t) * sin(k)) / l) / l) * tan(k)) * 2.0));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(fma(fma(Float64(Float64(t * t) * Float64(-0.3333333333333333 - -0.5)), 2.0, 1.0), Float64(k * k), Float64(Float64(t * t) * 2.0)) * Float64(k * k)) / l) / l) * t));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 8e+134], N[(2.0 / N[(N[(N[(N[(N[(N[(N[(t * t), $MachinePrecision] * t), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(N[(N[(N[(t * t), $MachinePrecision] * N[(-0.3333333333333333 - -0.5), $MachinePrecision]), $MachinePrecision] * 2.0 + 1.0), $MachinePrecision] * N[(k * k), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 7.99999999999999937e134

   1. Initial program 55.9%

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

   2. Taylor expanded in t around 0

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

   4. Applied rewrites 26.9%

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

   6. Applied rewrites 28.1%

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

   7. Taylor expanded in t around inf

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

   9. Applied rewrites 62.0%

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

   if 7.99999999999999937e134 < k

   1. Initial program 46.5%

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

   2. Taylor expanded in t around 0

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

   4. Applied rewrites 57.8%

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

   5. Taylor expanded in l around 0

      \[\leadsto \frac{2}{\frac{2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{\cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{\cos k}}{{\ell}^{2}} \cdot t} \]

   7. Applied rewrites 62.6%

      \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\frac{{\left(\sin k \cdot t\right)}^{2}}{\cos k}, 2, \frac{{\left(\sin k \cdot k\right)}^{2}}{\cos k}\right)}{\ell \cdot \ell} \cdot t} \]

   9. Applied rewrites 66.9%

      \[\leadsto \frac{2}{\frac{\frac{\mathsf{fma}\left(2, \frac{{\left(\sin k \cdot t\right)}^{2}}{\cos k}, \frac{{\left(\sin k \cdot k\right)}^{2}}{\cos k}\right)}{\ell}}{\ell} \cdot t} \]

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 63.0%

       \[\leadsto \frac{2}{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\left(t \cdot t\right) \cdot \left(-0.3333333333333333 - -0.5\right), 2, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\ell}}{\ell} \cdot t} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 62.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 3 \cdot 10^{-118}:\\ \;\;\;\;\frac{2}{\left(\frac{\frac{\left(\left(t \cdot t\right) \cdot t\right) \cdot k}{\ell}}{\ell} \cdot \tan k\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\left(t \cdot t\right) \cdot \left(-0.3333333333333333 - -0.5\right), 2, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\ell}}{\ell} \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 3e-118)
   (/ 2.0 (* (* (/ (/ (* (* (* t t) t) k) l) l) (tan k)) 2.0))
   (/
    2.0
    (*
     (/
      (/
       (*
        (fma
         (fma (* (* t t) (- -0.3333333333333333 -0.5)) 2.0 1.0)
         (* k k)
         (* (* t t) 2.0))
        (* k k))
       l)
      l)
     t))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 3e-118) {
		tmp = 2.0 / (((((((t * t) * t) * k) / l) / l) * tan(k)) * 2.0);
	} else {
		tmp = 2.0 / ((((fma(fma(((t * t) * (-0.3333333333333333 - -0.5)), 2.0, 1.0), (k * k), ((t * t) * 2.0)) * (k * k)) / l) / l) * t);
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 3e-118)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(Float64(t * t) * t) * k) / l) / l) * tan(k)) * 2.0));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(fma(fma(Float64(Float64(t * t) * Float64(-0.3333333333333333 - -0.5)), 2.0, 1.0), Float64(k * k), Float64(Float64(t * t) * 2.0)) * Float64(k * k)) / l) / l) * t));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 3e-118], N[(2.0 / N[(N[(N[(N[(N[(N[(N[(t * t), $MachinePrecision] * t), $MachinePrecision] * k), $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(N[(N[(N[(t * t), $MachinePrecision] * N[(-0.3333333333333333 - -0.5), $MachinePrecision]), $MachinePrecision] * 2.0 + 1.0), $MachinePrecision] * N[(k * k), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.00000000000000018e-118

   1. Initial program 56.0%

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

   2. Taylor expanded in t around 0

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

   4. Applied rewrites 26.7%

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

   6. Applied rewrites 28.0%

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

   7. Taylor expanded in k around 0

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

   9. Applied rewrites 26.5%

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

   10. Taylor expanded in t around inf

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

   12. Applied rewrites 61.7%

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

   if 3.00000000000000018e-118 < k

   1. Initial program 52.0%

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

   2. Taylor expanded in t around 0

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

   4. Applied rewrites 69.2%

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

   5. Taylor expanded in l around 0

      \[\leadsto \frac{2}{\frac{2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{\cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{\cos k}}{{\ell}^{2}} \cdot t} \]

   7. Applied rewrites 73.4%

      \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\frac{{\left(\sin k \cdot t\right)}^{2}}{\cos k}, 2, \frac{{\left(\sin k \cdot k\right)}^{2}}{\cos k}\right)}{\ell \cdot \ell} \cdot t} \]

   9. Applied rewrites 79.9%

      \[\leadsto \frac{2}{\frac{\frac{\mathsf{fma}\left(2, \frac{{\left(\sin k \cdot t\right)}^{2}}{\cos k}, \frac{{\left(\sin k \cdot k\right)}^{2}}{\cos k}\right)}{\ell}}{\ell} \cdot t} \]

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 65.2%

       \[\leadsto \frac{2}{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\left(t \cdot t\right) \cdot \left(-0.3333333333333333 - -0.5\right), 2, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\ell}}{\ell} \cdot t} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 62.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.5 \cdot 10^{-143}:\\ \;\;\;\;\frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\left(t \cdot t\right) \cdot \left(-0.3333333333333333 - -0.5\right), 2, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\ell}}{\ell} \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 2.5e-143)
   (* (/ l (* k (* (* (* t t) t) k))) l)
   (/
    2.0
    (*
     (/
      (/
       (*
        (fma
         (fma (* (* t t) (- -0.3333333333333333 -0.5)) 2.0 1.0)
         (* k k)
         (* (* t t) 2.0))
        (* k k))
       l)
      l)
     t))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.5e-143) {
		tmp = (l / (k * (((t * t) * t) * k))) * l;
	} else {
		tmp = 2.0 / ((((fma(fma(((t * t) * (-0.3333333333333333 - -0.5)), 2.0, 1.0), (k * k), ((t * t) * 2.0)) * (k * k)) / l) / l) * t);
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 2.5e-143)
		tmp = Float64(Float64(l / Float64(k * Float64(Float64(Float64(t * t) * t) * k))) * l);
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(fma(fma(Float64(Float64(t * t) * Float64(-0.3333333333333333 - -0.5)), 2.0, 1.0), Float64(k * k), Float64(Float64(t * t) * 2.0)) * Float64(k * k)) / l) / l) * t));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 2.5e-143], N[(N[(l / N[(k * N[(N[(N[(t * t), $MachinePrecision] * t), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(N[(N[(N[(t * t), $MachinePrecision] * N[(-0.3333333333333333 - -0.5), $MachinePrecision]), $MachinePrecision] * 2.0 + 1.0), $MachinePrecision] * N[(k * k), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.5000000000000001e-143

   1. Initial program 55.6%

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

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

   4. Applied rewrites 49.6%

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

   6. Applied rewrites 54.6%

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

   8. Applied rewrites 54.6%

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

   10. Applied rewrites 61.3%

       \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

   if 2.5000000000000001e-143 < k

   1. Initial program 52.9%

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

   2. Taylor expanded in t around 0

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

   4. Applied rewrites 69.3%

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

   5. Taylor expanded in l around 0

      \[\leadsto \frac{2}{\frac{2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{\cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{\cos k}}{{\ell}^{2}} \cdot t} \]

   7. Applied rewrites 73.9%

      \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\frac{{\left(\sin k \cdot t\right)}^{2}}{\cos k}, 2, \frac{{\left(\sin k \cdot k\right)}^{2}}{\cos k}\right)}{\ell \cdot \ell} \cdot t} \]

   9. Applied rewrites 80.4%

      \[\leadsto \frac{2}{\frac{\frac{\mathsf{fma}\left(2, \frac{{\left(\sin k \cdot t\right)}^{2}}{\cos k}, \frac{{\left(\sin k \cdot k\right)}^{2}}{\cos k}\right)}{\ell}}{\ell} \cdot t} \]

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 66.2%

       \[\leadsto \frac{2}{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\left(t \cdot t\right) \cdot \left(-0.3333333333333333 - -0.5\right), 2, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\ell}}{\ell} \cdot t} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 60.7% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.2 \cdot 10^{+126}:\\ \;\;\;\;\frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\ell \cdot \ell} \cdot \left(k \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 2.2e+126)
   (* (/ l (* k (* (* (* t t) t) k))) l)
   (/ 2.0 (* (/ (* (* k k) t) (* l l)) (* k k)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.2e+126) {
		tmp = (l / (k * (((t * t) * t) * k))) * l;
	} else {
		tmp = 2.0 / ((((k * k) * t) / (l * l)) * (k * k));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 2.2d+126) then
        tmp = (l / (k * (((t * t) * t) * k))) * l
    else
        tmp = 2.0d0 / ((((k * k) * t) / (l * l)) * (k * k))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.2e+126) {
		tmp = (l / (k * (((t * t) * t) * k))) * l;
	} else {
		tmp = 2.0 / ((((k * k) * t) / (l * l)) * (k * k));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 2.2e+126:
		tmp = (l / (k * (((t * t) * t) * k))) * l
	else:
		tmp = 2.0 / ((((k * k) * t) / (l * l)) * (k * k))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 2.2e+126)
		tmp = Float64(Float64(l / Float64(k * Float64(Float64(Float64(t * t) * t) * k))) * l);
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * k) * t) / Float64(l * l)) * Float64(k * k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 2.2e+126)
		tmp = (l / (k * (((t * t) * t) * k))) * l;
	else
		tmp = 2.0 / ((((k * k) * t) / (l * l)) * (k * k));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 2.2e+126], N[(N[(l / N[(k * N[(N[(N[(t * t), $MachinePrecision] * t), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.19999999999999999e126

   1. Initial program 56.0%

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

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

   4. Applied rewrites 51.6%

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

   6. Applied rewrites 55.9%

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

   8. Applied rewrites 55.9%

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

   10. Applied rewrites 60.9%

       \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

   if 2.19999999999999999e126 < k

   1. Initial program 46.3%

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

   2. Taylor expanded in k around 0

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

   4. Applied rewrites 45.6%

      \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(k \cdot k, \left(\left(t \cdot t\right) \cdot t\right) \cdot \frac{0.3333333333333333 + \frac{1}{t \cdot t}}{\ell \cdot \ell}, \frac{2 \cdot \left(\left(t \cdot t\right) \cdot t\right)}{\ell \cdot \ell}\right) \cdot \left(k \cdot k\right)}} \]

   5. Taylor expanded in t around 0

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

   7. Applied rewrites 60.0%

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

Alternative 12: 59.1% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (* (/ l (* k (* (* (* t t) t) k))) l))

C

double code(double t, double l, double k) {
	return (l / (k * (((t * t) * t) * k))) * l;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = (l / (k * (((t * t) * t) * k))) * l
end function

Java

public static double code(double t, double l, double k) {
	return (l / (k * (((t * t) * t) * k))) * l;
}

Python

def code(t, l, k):
	return (l / (k * (((t * t) * t) * k))) * l

Julia

function code(t, l, k)
	return Float64(Float64(l / Float64(k * Float64(Float64(Float64(t * t) * t) * k))) * l)
end

MATLAB

function tmp = code(t, l, k)
	tmp = (l / (k * (((t * t) * t) * k))) * l;
end

Wolfram

code[t_, l_, k_] := N[(N[(l / N[(k * N[(N[(N[(t * t), $MachinePrecision] * t), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]

Derivation

1. Initial program 54.6%

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

2. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

4. Applied rewrites 50.7%

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

6. Applied rewrites 54.8%

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

8. Applied rewrites 54.8%

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

10. Applied rewrites 59.1%

    \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025134 
(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))))