Toniolo and Linder, Equation (10+)

Percentage Accurate: 53.9% → 85.7%

Time: 20.5s

Alternatives: 7

Speedup: N/A×

• 25.8% 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: 53.9% 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: 85.7% accurate, N/A× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t\_m}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)} \leq 10^{+201}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(2, {\left(\sin k \cdot t\_m\right)}^{2}, {\left(\sin k \cdot k\right)}^{2}\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\frac{\cos k}{t\_m}}{{\sin k}^{2}}\right)\right)\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<=
       (/
        2.0
        (*
         (* (* (/ (pow t_m 3.0) (* l l)) (sin k)) (tan k))
         (+ (+ 1.0 (pow (/ k t_m) 2.0)) 1.0)))
       1e+201)
    (/
     2.0
     (*
      (/
       (fma 2.0 (pow (* (sin k) t_m) 2.0) (pow (* (sin k) k) 2.0))
       (* (cos k) (* l l)))
      t_m))
    (* 2.0 (* (/ l k) (* (/ l k) (/ (/ (cos k) t_m) (pow (sin k) 2.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((2.0 / ((((pow(t_m, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t_m), 2.0)) + 1.0))) <= 1e+201) {
		tmp = 2.0 / ((fma(2.0, pow((sin(k) * t_m), 2.0), pow((sin(k) * k), 2.0)) / (cos(k) * (l * l))) * t_m);
	} else {
		tmp = 2.0 * ((l / k) * ((l / k) * ((cos(k) / t_m) / pow(sin(k), 2.0))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(Float64(Float64((t_m ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t_m) ^ 2.0)) + 1.0))) <= 1e+201)
		tmp = Float64(2.0 / Float64(Float64(fma(2.0, (Float64(sin(k) * t_m) ^ 2.0), (Float64(sin(k) * k) ^ 2.0)) / Float64(cos(k) * Float64(l * l))) * t_m));
	else
		tmp = Float64(2.0 * Float64(Float64(l / k) * Float64(Float64(l / k) * Float64(Float64(cos(k) / t_m) / (sin(k) ^ 2.0)))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[N[(2.0 / N[(N[(N[(N[(N[Power[t$95$m, 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$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+201], N[(2.0 / N[(N[(N[(2.0 * N[Power[N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(N[Sin[k], $MachinePrecision] * k), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] * N[(N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.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)))) < 1.00000000000000004e201

   1. Initial program 77.9%

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

   3. 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. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 87.2%

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

   if 1.00000000000000004e201 < (/.f64 #s(literal 2 binary64) (*.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 26.1%

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

   3. 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. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 54.7%

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

   7. Applied rewrites 55.4%

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

   8. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-pow.f64 78.0

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

   10. Applied rewrites 78.0%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. lift-cos.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-pow.f64 N/A

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

       7. lift-sin.f64 N/A

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

       8. associate-*l* N/A

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

       9. lower-*.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. associate-/r* N/A

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

       12. lower-/.f64 N/A

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

       13. lower-/.f64 N/A

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

       14. lift-cos.f64 N/A

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

       15. lift-sin.f64 N/A

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

       16. lift-pow.f64 87.8

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

   12. Applied rewrites 87.8%

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

Alternative 2: 85.5% accurate, N/A× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t\_m}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)} \leq 10^{+201}:\\ \;\;\;\;\frac{2}{\frac{t\_m}{\ell \cdot \ell} \cdot \frac{\mathsf{fma}\left({\left(\sin k \cdot t\_m\right)}^{2}, 2, {\left(\sin k \cdot k\right)}^{2}\right)}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\frac{\cos k}{t\_m}}{{\sin k}^{2}}\right)\right)\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<=
       (/
        2.0
        (*
         (* (* (/ (pow t_m 3.0) (* l l)) (sin k)) (tan k))
         (+ (+ 1.0 (pow (/ k t_m) 2.0)) 1.0)))
       1e+201)
    (/
     2.0
     (*
      (/ t_m (* l l))
      (/ (fma (pow (* (sin k) t_m) 2.0) 2.0 (pow (* (sin k) k) 2.0)) (cos k))))
    (* 2.0 (* (/ l k) (* (/ l k) (/ (/ (cos k) t_m) (pow (sin k) 2.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((2.0 / ((((pow(t_m, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t_m), 2.0)) + 1.0))) <= 1e+201) {
		tmp = 2.0 / ((t_m / (l * l)) * (fma(pow((sin(k) * t_m), 2.0), 2.0, pow((sin(k) * k), 2.0)) / cos(k)));
	} else {
		tmp = 2.0 * ((l / k) * ((l / k) * ((cos(k) / t_m) / pow(sin(k), 2.0))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(Float64(Float64((t_m ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t_m) ^ 2.0)) + 1.0))) <= 1e+201)
		tmp = Float64(2.0 / Float64(Float64(t_m / Float64(l * l)) * Float64(fma((Float64(sin(k) * t_m) ^ 2.0), 2.0, (Float64(sin(k) * k) ^ 2.0)) / cos(k))));
	else
		tmp = Float64(2.0 * Float64(Float64(l / k) * Float64(Float64(l / k) * Float64(Float64(cos(k) / t_m) / (sin(k) ^ 2.0)))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[N[(2.0 / N[(N[(N[(N[(N[Power[t$95$m, 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$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+201], N[(2.0 / N[(N[(t$95$m / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] * 2.0 + N[Power[N[(N[Sin[k], $MachinePrecision] * k), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] * N[(N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.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)))) < 1.00000000000000004e201

   1. Initial program 77.9%

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

   3. 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. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 87.2%

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

   7. Applied rewrites 87.1%

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

   if 1.00000000000000004e201 < (/.f64 #s(literal 2 binary64) (*.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 26.1%

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

   3. 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. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 54.7%

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

   7. Applied rewrites 55.4%

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

   8. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-pow.f64 78.0

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

   10. Applied rewrites 78.0%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. lift-cos.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-pow.f64 N/A

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

       7. lift-sin.f64 N/A

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

       8. associate-*l* N/A

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

       9. lower-*.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. associate-/r* N/A

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

       12. lower-/.f64 N/A

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

       13. lower-/.f64 N/A

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

       14. lift-cos.f64 N/A

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

       15. lift-sin.f64 N/A

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

       16. lift-pow.f64 87.8

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

   12. Applied rewrites 87.8%

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

Alternative 3: 71.2% accurate, N/A× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.76 \cdot 10^{-38}:\\ \;\;\;\;\frac{{\left(\frac{\ell}{k}\right)}^{2}}{{t\_m}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\frac{\cos k}{t\_m}}{{\sin k}^{2}}\right)\right)\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.76e-38)
    (/ (pow (/ l k) 2.0) (pow t_m 3.0))
    (* 2.0 (* (/ l k) (* (/ l k) (/ (/ (cos k) t_m) (pow (sin k) 2.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.76e-38) {
		tmp = pow((l / k), 2.0) / pow(t_m, 3.0);
	} else {
		tmp = 2.0 * ((l / k) * ((l / k) * ((cos(k) / t_m) / pow(sin(k), 2.0))));
	}
	return t_s * tmp;
}

Fortran

t\_m =     private
t\_s =     private
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_s, t_m, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.76d-38) then
        tmp = ((l / k) ** 2.0d0) / (t_m ** 3.0d0)
    else
        tmp = 2.0d0 * ((l / k) * ((l / k) * ((cos(k) / t_m) / (sin(k) ** 2.0d0))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.76e-38) {
		tmp = Math.pow((l / k), 2.0) / Math.pow(t_m, 3.0);
	} else {
		tmp = 2.0 * ((l / k) * ((l / k) * ((Math.cos(k) / t_m) / Math.pow(Math.sin(k), 2.0))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 1.76e-38:
		tmp = math.pow((l / k), 2.0) / math.pow(t_m, 3.0)
	else:
		tmp = 2.0 * ((l / k) * ((l / k) * ((math.cos(k) / t_m) / math.pow(math.sin(k), 2.0))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 1.76e-38)
		tmp = Float64((Float64(l / k) ^ 2.0) / (t_m ^ 3.0));
	else
		tmp = Float64(2.0 * Float64(Float64(l / k) * Float64(Float64(l / k) * Float64(Float64(cos(k) / t_m) / (sin(k) ^ 2.0)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 1.76e-38)
		tmp = ((l / k) ^ 2.0) / (t_m ^ 3.0);
	else
		tmp = 2.0 * ((l / k) * ((l / k) * ((cos(k) / t_m) / (sin(k) ^ 2.0))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 1.76e-38], N[(N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision] / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] * N[(N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.76000000000000001e-38

   1. Initial program 57.8%

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

   3. 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. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 72.7%

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

   7. Applied rewrites 73.1%

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

   8. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-pow.f64 65.1

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

   10. Applied rewrites 65.1%

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

   11. Taylor expanded in k around 0

       \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   12. Step-by-step derivation

   13. Applied rewrites 63.1%

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

   if 1.76000000000000001e-38 < k

   1. Initial program 49.7%

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

   3. 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. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 73.5%

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

   7. Applied rewrites 73.5%

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

   8. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-pow.f64 82.2

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

   10. Applied rewrites 82.2%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. lift-cos.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-pow.f64 N/A

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

       7. lift-sin.f64 N/A

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

       8. associate-*l* N/A

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

       9. lower-*.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. associate-/r* N/A

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

       12. lower-/.f64 N/A

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

       13. lower-/.f64 N/A

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

       14. lift-cos.f64 N/A

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

       15. lift-sin.f64 N/A

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

       16. lift-pow.f64 89.1

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

   12. Applied rewrites 89.1%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.76 \cdot 10^{-38}:\\ \;\;\;\;\frac{{\left(\frac{\ell}{k}\right)}^{2}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\frac{\cos k}{t}}{{\sin k}^{2}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 69.5% accurate, N/A× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.76 \cdot 10^{-38}:\\ \;\;\;\;\frac{{\left(\frac{\ell}{k}\right)}^{2}}{{t\_m}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t\_m \cdot {\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.76e-38)
    (/ (pow (/ l k) 2.0) (pow t_m 3.0))
    (* 2.0 (* (* (/ l k) (/ l k)) (/ (cos k) (* t_m (pow (sin k) 2.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.76e-38) {
		tmp = pow((l / k), 2.0) / pow(t_m, 3.0);
	} else {
		tmp = 2.0 * (((l / k) * (l / k)) * (cos(k) / (t_m * pow(sin(k), 2.0))));
	}
	return t_s * tmp;
}

Fortran

t\_m =     private
t\_s =     private
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_s, t_m, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.76d-38) then
        tmp = ((l / k) ** 2.0d0) / (t_m ** 3.0d0)
    else
        tmp = 2.0d0 * (((l / k) * (l / k)) * (cos(k) / (t_m * (sin(k) ** 2.0d0))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.76e-38) {
		tmp = Math.pow((l / k), 2.0) / Math.pow(t_m, 3.0);
	} else {
		tmp = 2.0 * (((l / k) * (l / k)) * (Math.cos(k) / (t_m * Math.pow(Math.sin(k), 2.0))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 1.76e-38:
		tmp = math.pow((l / k), 2.0) / math.pow(t_m, 3.0)
	else:
		tmp = 2.0 * (((l / k) * (l / k)) * (math.cos(k) / (t_m * math.pow(math.sin(k), 2.0))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 1.76e-38)
		tmp = Float64((Float64(l / k) ^ 2.0) / (t_m ^ 3.0));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(l / k) * Float64(l / k)) * Float64(cos(k) / Float64(t_m * (sin(k) ^ 2.0)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 1.76e-38)
		tmp = ((l / k) ^ 2.0) / (t_m ^ 3.0);
	else
		tmp = 2.0 * (((l / k) * (l / k)) * (cos(k) / (t_m * (sin(k) ^ 2.0))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 1.76e-38], N[(N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision] / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.76000000000000001e-38

   1. Initial program 57.8%

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

   3. 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. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 72.7%

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

   7. Applied rewrites 73.1%

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

   8. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-pow.f64 65.1

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

   10. Applied rewrites 65.1%

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

   11. Taylor expanded in k around 0

       \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   12. Step-by-step derivation

   13. Applied rewrites 63.1%

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

   if 1.76000000000000001e-38 < k

   1. Initial program 49.7%

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

   3. 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. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 73.5%

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

   7. Applied rewrites 73.5%

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

   8. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-pow.f64 82.2

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

   10. Applied rewrites 82.2%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.76 \cdot 10^{-38}:\\ \;\;\;\;\frac{{\left(\frac{\ell}{k}\right)}^{2}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 66.4% accurate, N/A× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.42 \cdot 10^{-33}:\\ \;\;\;\;\frac{{\left(\frac{\ell}{k}\right)}^{2}}{{t\_m}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\cos k}{{\left(t\_m \cdot \sin k\right)}^{2}}\right)\right)\right) \cdot t\_m\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.42e-33)
    (/ (pow (/ l k) 2.0) (pow t_m 3.0))
    (*
     (* 2.0 (* (/ l k) (* (/ l k) (/ (cos k) (pow (* t_m (sin k)) 2.0)))))
     t_m))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.42e-33) {
		tmp = pow((l / k), 2.0) / pow(t_m, 3.0);
	} else {
		tmp = (2.0 * ((l / k) * ((l / k) * (cos(k) / pow((t_m * sin(k)), 2.0))))) * t_m;
	}
	return t_s * tmp;
}

Fortran

t\_m =     private
t\_s =     private
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_s, t_m, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.42d-33) then
        tmp = ((l / k) ** 2.0d0) / (t_m ** 3.0d0)
    else
        tmp = (2.0d0 * ((l / k) * ((l / k) * (cos(k) / ((t_m * sin(k)) ** 2.0d0))))) * t_m
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.42e-33) {
		tmp = Math.pow((l / k), 2.0) / Math.pow(t_m, 3.0);
	} else {
		tmp = (2.0 * ((l / k) * ((l / k) * (Math.cos(k) / Math.pow((t_m * Math.sin(k)), 2.0))))) * t_m;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 1.42e-33:
		tmp = math.pow((l / k), 2.0) / math.pow(t_m, 3.0)
	else:
		tmp = (2.0 * ((l / k) * ((l / k) * (math.cos(k) / math.pow((t_m * math.sin(k)), 2.0))))) * t_m
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 1.42e-33)
		tmp = Float64((Float64(l / k) ^ 2.0) / (t_m ^ 3.0));
	else
		tmp = Float64(Float64(2.0 * Float64(Float64(l / k) * Float64(Float64(l / k) * Float64(cos(k) / (Float64(t_m * sin(k)) ^ 2.0))))) * t_m);
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 1.42e-33)
		tmp = ((l / k) ^ 2.0) / (t_m ^ 3.0);
	else
		tmp = (2.0 * ((l / k) * ((l / k) * (cos(k) / ((t_m * sin(k)) ^ 2.0))))) * t_m;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 1.42e-33], N[(N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision] / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[Power[N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.42000000000000007e-33

   1. Initial program 58.0%

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

   3. Taylor expanded in 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. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 72.8%

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

   7. Applied rewrites 73.3%

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

   8. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-pow.f64 65.3

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

   10. Applied rewrites 65.3%

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

   11. Taylor expanded in k around 0

       \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   12. Step-by-step derivation

   13. Applied rewrites 63.3%

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

   if 1.42000000000000007e-33 < k

   1. Initial program 49.1%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 53.3%

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

   6. Taylor expanded in t around inf

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

      1. *-commutative N/A

         \[\leadsto \left(-4 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{4} \cdot {\sin k}^{2}} + 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left({t}^{2} \cdot {\sin k}^{2}\right)}\right) \cdot t \]

      2. lower-*.f64 N/A

         \[\leadsto \left(-4 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{4} \cdot {\sin k}^{2}} + 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left({t}^{2} \cdot {\sin k}^{2}\right)}\right) \cdot t \]

   8. Applied rewrites 46.7%

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

   9. Taylor expanded in t around 0

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

       1. lower-*.f64 N/A

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

       2. times-frac N/A

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

       3. lower-*.f64 N/A

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

       4. pow2 N/A

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

       5. pow2 N/A

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

       6. times-frac N/A

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

       7. lower-*.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. lower-/.f64 N/A

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

       10. lower-/.f64 N/A

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

       11. lift-cos.f64 N/A

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

       12. pow-prod-down N/A

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

       13. lower-pow.f64 N/A

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

       14. lower-*.f64 N/A

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

       15. lift-sin.f64 73.8

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

   11. Applied rewrites 73.8%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. lift-cos.f64 N/A

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

       5. lift-pow.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-sin.f64 N/A

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

       8. associate-*l* N/A

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

       9. lower-*.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. lift-cos.f64 N/A

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

       12. lift-sin.f64 N/A

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

       13. lift-*.f64 N/A

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

       14. lift-pow.f64 N/A

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

       15. lift-/.f64 75.2

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

   13. Applied rewrites 75.2%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.42 \cdot 10^{-33}:\\ \;\;\;\;\frac{{\left(\frac{\ell}{k}\right)}^{2}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\cos k}{{\left(t \cdot \sin k\right)}^{2}}\right)\right)\right) \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 65.8% accurate, N/A× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.42 \cdot 10^{-33}:\\ \;\;\;\;\frac{{\left(\frac{\ell}{k}\right)}^{2}}{{t\_m}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{{\left(t\_m \cdot \sin k\right)}^{2}}\right)\right) \cdot t\_m\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.42e-33)
    (/ (pow (/ l k) 2.0) (pow t_m 3.0))
    (*
     (* 2.0 (* (* (/ l k) (/ l k)) (/ (cos k) (pow (* t_m (sin k)) 2.0))))
     t_m))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.42e-33) {
		tmp = pow((l / k), 2.0) / pow(t_m, 3.0);
	} else {
		tmp = (2.0 * (((l / k) * (l / k)) * (cos(k) / pow((t_m * sin(k)), 2.0)))) * t_m;
	}
	return t_s * tmp;
}

Fortran

t\_m =     private
t\_s =     private
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_s, t_m, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.42d-33) then
        tmp = ((l / k) ** 2.0d0) / (t_m ** 3.0d0)
    else
        tmp = (2.0d0 * (((l / k) * (l / k)) * (cos(k) / ((t_m * sin(k)) ** 2.0d0)))) * t_m
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.42e-33) {
		tmp = Math.pow((l / k), 2.0) / Math.pow(t_m, 3.0);
	} else {
		tmp = (2.0 * (((l / k) * (l / k)) * (Math.cos(k) / Math.pow((t_m * Math.sin(k)), 2.0)))) * t_m;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 1.42e-33:
		tmp = math.pow((l / k), 2.0) / math.pow(t_m, 3.0)
	else:
		tmp = (2.0 * (((l / k) * (l / k)) * (math.cos(k) / math.pow((t_m * math.sin(k)), 2.0)))) * t_m
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 1.42e-33)
		tmp = Float64((Float64(l / k) ^ 2.0) / (t_m ^ 3.0));
	else
		tmp = Float64(Float64(2.0 * Float64(Float64(Float64(l / k) * Float64(l / k)) * Float64(cos(k) / (Float64(t_m * sin(k)) ^ 2.0)))) * t_m);
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 1.42e-33)
		tmp = ((l / k) ^ 2.0) / (t_m ^ 3.0);
	else
		tmp = (2.0 * (((l / k) * (l / k)) * (cos(k) / ((t_m * sin(k)) ^ 2.0)))) * t_m;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 1.42e-33], N[(N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision] / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[Power[N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.42000000000000007e-33

   1. Initial program 58.0%

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

   3. Taylor expanded in 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. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 72.8%

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

   7. Applied rewrites 73.3%

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

   8. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-pow.f64 65.3

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

   10. Applied rewrites 65.3%

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

   11. Taylor expanded in k around 0

       \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   12. Step-by-step derivation

   13. Applied rewrites 63.3%

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

   if 1.42000000000000007e-33 < k

   1. Initial program 49.1%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 53.3%

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

   6. Taylor expanded in t around inf

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

      1. *-commutative N/A

         \[\leadsto \left(-4 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{4} \cdot {\sin k}^{2}} + 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left({t}^{2} \cdot {\sin k}^{2}\right)}\right) \cdot t \]

      2. lower-*.f64 N/A

         \[\leadsto \left(-4 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{4} \cdot {\sin k}^{2}} + 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left({t}^{2} \cdot {\sin k}^{2}\right)}\right) \cdot t \]

   8. Applied rewrites 46.7%

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

   9. Taylor expanded in t around 0

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

       1. lower-*.f64 N/A

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

       2. times-frac N/A

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

       3. lower-*.f64 N/A

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

       4. pow2 N/A

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

       5. pow2 N/A

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

       6. times-frac N/A

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

       7. lower-*.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. lower-/.f64 N/A

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

       10. lower-/.f64 N/A

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

       11. lift-cos.f64 N/A

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

       12. pow-prod-down N/A

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

       13. lower-pow.f64 N/A

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

       14. lower-*.f64 N/A

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

       15. lift-sin.f64 73.8

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

   11. Applied rewrites 73.8%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.42 \cdot 10^{-33}:\\ \;\;\;\;\frac{{\left(\frac{\ell}{k}\right)}^{2}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{{\left(t \cdot \sin k\right)}^{2}}\right)\right) \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 60.3% accurate, N/A× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2}}{{t\_m}^{3}} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (/ (pow (/ l k) 2.0) (pow t_m 3.0))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (pow((l / k), 2.0) / pow(t_m, 3.0));
}

Fortran

t\_m =     private
t\_s =     private
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_s, t_m, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (((l / k) ** 2.0d0) / (t_m ** 3.0d0))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (Math.pow((l / k), 2.0) / Math.pow(t_m, 3.0));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (math.pow((l / k), 2.0) / math.pow(t_m, 3.0))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64((Float64(l / k) ^ 2.0) / (t_m ^ 3.0)))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (((l / k) ^ 2.0) / (t_m ^ 3.0));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision] / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.2%

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

3. 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. Step-by-step derivation

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 73.0%

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

7. Applied rewrites 73.3%

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

8. Taylor expanded in t around 0

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

   1. lower-*.f64 N/A

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

   2. times-frac N/A

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

   3. lower-*.f64 N/A

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

   4. pow2 N/A

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

   5. pow2 N/A

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

   6. times-frac N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lift-cos.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lift-sin.f64 N/A

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

   14. lift-pow.f64 70.6

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

10. Applied rewrites 70.6%

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

11. Taylor expanded in k around 0

    \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
12. Step-by-step derivation

13. Applied rewrites 58.4%

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

14. Final simplification 58.4%

    \[\leadsto \frac{{\left(\frac{\ell}{k}\right)}^{2}}{{t}^{3}} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2025057 
(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))))